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so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . | what do differentiating a function with respect to other , and partial differential give graphically ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | but we do n't have that tool yet . but using just the tools from algebra , we could at least start to think about , what is the average rate of change over the interval from x1 to x2 ? well , what 's the average rate of change ? well , that 's just how much did my y change -- so that 's my change in y -- for this chang... | so , if x1 gets infinitely close to x2 , does that mean that you can find the instantaneous rate of change over that interval ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | and what we call a line that intersects a curve in exactly two places ? well , we figured out , we call that a secant line . so this right over here is a secant line . so the big idea here is we 're extending the idea of slope . | is secant line somewhat related to trignometric identites ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | it 's shorthand for change in whatever . so change in y over change in x . so let 's think about what this is going to be for this example right over here . | is taking the derivative the same thing as saying what the instantaneous change of a curve is at a given point ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | so this is the point x sub 1 , y sub 1 . so just as a review , the slope of this line , and a line by definition , has a constant slope between any two points that you pick . the slope of this line , which is often denoted by the letter m , is your rate of change of y with respect to x . | now we know that we can not have an equation that forms a vertical line on a graph ( 90 degrees ) , does this mean that an equation is kind of a never ending line on the graph ( never ending as in the slope just increases ) ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | and just as a reminder , we can figure out the slope between two points . two points define a line . and between those two points , we can find the rate of change of y , with respect to x . | how does one define 'integral calculus , intuitively ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | and the y value over here is y sub 1 . so this is the point x sub 1 , y sub 1 . so just as a review , the slope of this line , and a line by definition , has a constant slope between any two points that you pick . | is the sub-number just to label the constants / variables in a specific order ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | so we are moving from x0 to x1 . so our change in x . so this is our change in x , right over here . | if two cars are travelling with velocities va and vb then velocity of car a with respect to b is written as va-vb ( vector subtraction ) .but in the case of derivatives why rate of change of y with respect to x is ^y/^t/^x/^t not ^y/^t-^x/^t ( ^ stands for change ) ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | and what we call a line that intersects a curve in exactly two places ? well , we figured out , we call that a secant line . so this right over here is a secant line . so the big idea here is we 're extending the idea of slope . | is there any use in drawing secant line ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | and what we call a line that intersects a curve in exactly two places ? well , we figured out , we call that a secant line . so this right over here is a secant line . so the big idea here is we 're extending the idea of slope . | in other words , is there any physical significance in drawing a secant line ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | but let 's just use our algebraic tools . we can at least figure out the average rate of change of a curve , or a function , over an interval . that is the same exact thing as the slope of a secant line . | in the first curve ( where the curve is linear ) , is it true ( if x axis is taken as time and y-axis is taken as the distance moved as a function of time ) that the average velocity and the instantaneous velocity are same ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | so that 's some line . and just as a reminder , we can figure out the slope between two points . two points define a line . and between those two points , we can find the rate of change of y , with respect to x . | will the secant be equal to the average slope between the two points no matter how steep the graph becomes in between the two points ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | so that 's some line . and just as a reminder , we can figure out the slope between two points . two points define a line . and between those two points , we can find the rate of change of y , with respect to x . | i mean , if the function zig zagged between the two points , would the secant still give the average slope ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | so the big idea here is we 're extending the idea of slope . we said , ok , we already knew how to find the slope of a line . a curve , we do n't have the tools yet , but calculus is about to give it to us . | how do i find the derivative of sin ( 30 ) ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | so this right here you probably remember from algebra class . the definition of slope is the rate of change of y with respect to x . or it 's the rate of change of our vertical axis , i should say , with respect to our horizontal axis . | if the slope of a curve by definition means the rate of change in y with respect to x , does it not imply that we can only determine the slope , that is , the rate of change in y with respect to x , between two given points only ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | and what we call a line that intersects a curve in exactly two places ? well , we figured out , we call that a secant line . so this right over here is a secant line . so the big idea here is we 're extending the idea of slope . | guys what is a secant line ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | and just as a reminder , we can figure out the slope between two points . two points define a line . and between those two points , we can find the rate of change of y , with respect to x . | a secant line is a line that joins two points of a curve while a tangent line touches one point of a curve right ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | so this is the point x sub 1 , y sub 1 . so just as a review , the slope of this line , and a line by definition , has a constant slope between any two points that you pick . the slope of this line , which is often denoted by the letter m , is your rate of change of y with respect to x . | so are derivatives for finding the slope of curved lines ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | now just as a little bit of foreshadowing , where is this all going ? how will we eventually get the tools , so that we can figure out the instantaneous rate of change , not just the average ? well just imagine what happens if this point right over here got closer and closer to this point . | i 've heard it 's also used to find instantaneous velocity ( in physics ) , but is n't instantaneous just one spot ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | so this is the point x sub 1 , y sub 1 . so just as a review , the slope of this line , and a line by definition , has a constant slope between any two points that you pick . the slope of this line , which is often denoted by the letter m , is your rate of change of y with respect to x . | what is the slope of the line y = -3x + 5 ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | so , there 's a whole number ratio in simplest form , where this square plus this square equals this square . '' `` yeah , that 's the pythagorean theorem , i made it . '' `` yeah , though for this triangle you do n't even need the full theorem . | did pythagoras really invent the pythagorean theorem ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | oh , and he killed a guy . i mean maybe it was a long time ago and he was afraid of beans . as in beans , they just like freaked him out or something , i do n't know . | is pythagoras really afraid of beans ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | so i like to imagine how they thought of this proof , no algebra required . okay , so pythagoras is all like , `` there 's totally a ratio , you can make this with whole numbers . '' and this guy 's like , '' is not ! '' | did vi heart make that up ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | just as this is divsible right down the center so too is it divisible the other way ! and the number squares on this side , which are the number of squares in just one leg is an even number . and for a number of squares to be even what does the number have to be , pythagoras , oh my brother ? '' | can someone provide an algebraic proof showing that a number must be irrational ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | oh , and he killed a guy . i mean maybe it was a long time ago and he was afraid of beans . as in beans , they just like freaked him out or something , i do n't know . | is pythagoras really afraid of beans ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | and for a number of squares to be even what does the number have to be , pythagoras , oh my brother ? '' `` if leg squared is even then the leg is even . but it ca n't be even , because it 's already odd . '' | why does it have to be even ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | so this proof stayed secret . until someone spilled the beans . according to some , it was all a guy named hippasus and pythagoras threw him of a boat to drown him as punishment for ruining what had been perfect . | 0 , was pythagoras really scared of beans ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | as in beans , they just like freaked him out or something , i do n't know . but mostly i want to talk about the murdery part . see , pythagoras and his cult of pythagoreans had this cool-kids club where they 'd talk about proportions all day . | why does vi heart talk so fast ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | `` is too ! '' `` fine have it your way . so , there 's a whole number ratio in simplest form , where this square plus this square equals this square . '' | by the way , is vi 's name just vi or is it short for something ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | oh , and he killed a guy . i mean maybe it was a long time ago and he was afraid of beans . as in beans , they just like freaked him out or something , i do n't know . | why was pythagoras afraid of beans ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | oh , and he killed a guy . i mean maybe it was a long time ago and he was afraid of beans . as in beans , they just like freaked him out or something , i do n't know . | was pythagoras really afraid of beans ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | or maybe it was someone else who discovered it or hippasus or someone else who was killed by the pythagoreans long after pythagoras was dead or maybe they just got exiled , who knows ? and how did pythagoras die ? well , according to one guy some guys got mad because they did n't get into the cool kids ' club . | how did pythagoras invent or rather proved the pythagoras theorem before the invention of zero ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | oh , and he killed a guy . i mean maybe it was a long time ago and he was afraid of beans . as in beans , they just like freaked him out or something , i do n't know . | was pythagoras really afraid of beans ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | it ca n't be both . '' `` unless it does n't exist ! what you forget pythagoras is that if this is a square then the two sides are the same . | but if `` does n't exist , '' would n't that mean it is =0 ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | oh , and he killed a guy . i mean maybe it was a long time ago and he was afraid of beans . as in beans , they just like freaked him out or something , i do n't know . | why was pythagorus afraid of beans ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | but mostly i want to talk about the murdery part . see , pythagoras and his cult of pythagoreans had this cool-kids club where they 'd talk about proportions all day . they 'd be like , `` hey , i drew a two by three rectangle using a straight-edge and compass . | why is there a cult dedicated to math ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | one leg 's square plus the other leg 's square equals hypotenuse 's square . three literally squared plus four made into a square . those two squares have the same area as a five by five square . | so i was wondering whether pi squared is rational ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | one leg 's square plus the other leg 's square equals hypotenuse 's square . three literally squared plus four made into a square . those two squares have the same area as a five by five square . | so is pi squared an irrational or a rational ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | just as this is divsible right down the center so too is it divisible the other way ! and the number squares on this side , which are the number of squares in just one leg is an even number . and for a number of squares to be even what does the number have to be , pythagoras , oh my brother ? '' | what is an irrational number ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | oh , and he killed a guy . i mean maybe it was a long time ago and he was afraid of beans . as in beans , they just like freaked him out or something , i do n't know . | was pythagoras really afraid of beans ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | oh , and he killed a guy . i mean maybe it was a long time ago and he was afraid of beans . as in beans , they just like freaked him out or something , i do n't know . | what does `` nan '' mean ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | so this proof stayed secret . until someone spilled the beans . according to some , it was all a guy named hippasus and pythagoras threw him of a boat to drown him as punishment for ruining what had been perfect . | does the theory have to do with beans ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | i 'd like to imagine what it would be like to think of numbers the way they did . maybe you think of numbers as being on a line . numbers one way , zero , and negative numbers the other , and there are numbers between them : fractions , rationals , filling in the gaps . but pythagoras did n't think about numbers like t... | was hippasus the philosopher who was drowned for showing that irrational numbers exist ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | before that , arithmetic was nightmarishly hard , so if you can multiply multi-digit numbers together you can go back in time and impress the beans out of pythagoras . and before that there was no concept of zero , except in india where zero was discovered around here . and if you keep going back you get to the year on... | where vi says there 's no year zero , technically , would n't there be a year zero , they just would n't call it zero , because it had n't been invented yet ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | people claim pythagoras did n't like beans because he thought they were bad for digestion , or gave you bad dreams or reminded him of male genitalia or because he did n't want a clubhouse full of flatulating mathematicians or he just did n't like them metaphorically . he and his followers were or were n't vegetarian di... | vi says that they were only allowed to eat certain colors of birds , but before , she said they were vegetarian , so does n't that defeat the whole point of vegetarian ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | pythagoras wanted to find the perfect ratio he knew it must exist , but meanwhile someone from his very own pythagorean brotherhood proved there was n't a ratio , the square root of two is irrational , that in decimal notation ( once decimal notation was invented ) the digits go on forever . usually this proof is given... | can anyone recomend a book of geometric constructions that demonstrate math concepts without algebra ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | so , there 's a whole number ratio in simplest form , where this square plus this square equals this square . '' `` yeah , that 's the pythagorean theorem , i made it . '' `` yeah , though for this triangle you do n't even need the full theorem . | who invented the pythagorean theorem ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | so this proof stayed secret . until someone spilled the beans . according to some , it was all a guy named hippasus and pythagoras threw him of a boat to drown him as punishment for ruining what had been perfect . | why is pythagoras afriad of beans ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | so what 's the square root of two and how do we make it into a whole number ratio ? square root two is very close to 1.4 which would be a whole number ratio of 10:14 but 10 squared plus 10 squared is definitely not 14 squared , and a ratio of 1,000 to 1,414 is even closer , and a ratio of 100,000,000 to 141,421,356 is ... | so if 1 squared + 1 squared is supposedly 2 squared , does n't that just make it so 1 squared is actually not 1 , but 1.41421356237 ( square root of 2 ) ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | or maybe it was someone else who discovered it or hippasus or someone else who was killed by the pythagoreans long after pythagoras was dead or maybe they just got exiled , who knows ? and how did pythagoras die ? well , according to one guy some guys got mad because they did n't get into the cool kids ' club . | why was pythagoras sarcaed of bean ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | so , there 's a whole number ratio in simplest form , where this square plus this square equals this square . '' `` yeah , that 's the pythagorean theorem , i made it . '' `` yeah , though for this triangle you do n't even need the full theorem . | what is the real pythagorean theorem ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | i mean maybe it was a long time ago and he was afraid of beans . as in beans , they just like freaked him out or something , i do n't know . but mostly i want to talk about the murdery part . | how do you know pythagoras was scared of beans ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | oh , and he killed a guy . i mean maybe it was a long time ago and he was afraid of beans . as in beans , they just like freaked him out or something , i do n't know . | why was pythagoras afraid of beans ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . | does pythagoras really have dirt on him ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | so , there 's a whole number ratio in simplest form , where this square plus this square equals this square . '' `` yeah , that 's the pythagorean theorem , i made it . '' `` yeah , though for this triangle you do n't even need the full theorem . | if the pythagorean theorem was disproved , how come vi hart is the only one that talks about it ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | so , there 's a whole number ratio in simplest form , where this square plus this square equals this square . '' `` yeah , that 's the pythagorean theorem , i made it . '' `` yeah , though for this triangle you do n't even need the full theorem . | can someone explain to me why vi hart maid a video proving and then disproving the pythagorean theorem ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | others say he ran off and starved himself to death . or just got caught by his enemies because he ran around the bean field instead of through it or who knows what happened . people claim pythagoras did n't like beans because he thought they were bad for digestion , or gave you bad dreams or reminded him of male genita... | then why did he proclaim `` i would rather get slaughtered by enemies than trample over a bean field '' ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | oh , and he killed a guy . i mean maybe it was a long time ago and he was afraid of beans . as in beans , they just like freaked him out or something , i do n't know . | why is pythagoras so afraid of beans ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | so , there 's a whole number ratio in simplest form , where this square plus this square equals this square . '' `` yeah , that 's the pythagorean theorem , i made it . '' `` yeah , though for this triangle you do n't even need the full theorem . | why does the pythagorean theorem work ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | so this proof stayed secret . until someone spilled the beans . according to some , it was all a guy named hippasus and pythagoras threw him of a boat to drown him as punishment for ruining what had been perfect . | was pythagoras really scared of beans ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | or maybe it was someone else who discovered it or hippasus or someone else who was killed by the pythagoreans long after pythagoras was dead or maybe they just got exiled , who knows ? and how did pythagoras die ? well , according to one guy some guys got mad because they did n't get into the cool kids ' club . | was pythagoras really a son of mercury ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | it 's not a simple proportion but hey , the box still has a whole number proportion , so pythagoras is happy . unless it 's a box of beans , then he freaks out . i 'd like to imagine what it would be like to think of numbers the way they did . maybe you think of numbers as being on a line . numbers one way , zero , and... | whay is pythagoras afriad of beans and way think of numbers of adjective ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | oh , and he killed a guy . i mean maybe it was a long time ago and he was afraid of beans . as in beans , they just like freaked him out or something , i do n't know . | pythagoras was afraid of beans ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | so this proof stayed secret . until someone spilled the beans . according to some , it was all a guy named hippasus and pythagoras threw him of a boat to drown him as punishment for ruining what had been perfect . | why did the beans have a unrealistic look ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | `` you just proved it 's not . it ca n't be both . '' `` unless it does n't exist ! | if such revered ideas of pythogoras were proved wrong for better than why ca n't these things be ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . | what income is reported on 1041 ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | that 's not a number ! get out of our club ! '' and then they 'd make the units half the length and call it four by seven , and everything was okay . | did pythagoras get arrested for killing a person ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | so they set pythagoras ' house on fire . and pythagoras was running away and they were chasing him , but then they came upon a field and not just any field , but a field of beans . and pythagoras turned around to face his pursuers and proclaimed : `` better to be slaughetered by enemies than to trample on beans ! '' | why a field of beans ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | here 's einstein , euler , newton and da vinci - - that sure was a while ago ! now let 's go all the way back to when arabic numerals were invented and brought to the west by fibbonacci . before that , arithmetic was nightmarishly hard , so if you can multiply multi-digit numbers together you can go back in time and im... | did the greeks use roman numerals ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | so , there 's a whole number ratio in simplest form , where this square plus this square equals this square . '' `` yeah , that 's the pythagorean theorem , i made it . '' `` yeah , though for this triangle you do n't even need the full theorem . | if the pythagorean theorem was invented , would n't we all divide wrong ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | well , according to one guy some guys got mad because they did n't get into the cool kids ' club . so they set pythagoras ' house on fire . and pythagoras was running away and they were chasing him , but then they came upon a field and not just any field , but a field of beans . and pythagoras turned around to face his... | why did n't pythagoras run right through the bean field ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | or maybe it was someone else who discovered it or hippasus or someone else who was killed by the pythagoreans long after pythagoras was dead or maybe they just got exiled , who knows ? and how did pythagoras die ? well , according to one guy some guys got mad because they did n't get into the cool kids ' club . | does anyone know which theory about how pythagoras died is correct ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | until someone spilled the beans . according to some , it was all a guy named hippasus and pythagoras threw him of a boat to drown him as punishment for ruining what had been perfect . or maybe it was someone else who discovered it or hippasus or someone else who was killed by the pythagoreans long after pythagoras was ... | did really pythagoras killed hippasus ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | or maybe it was someone else who discovered it or hippasus or someone else who was killed by the pythagoreans long after pythagoras was dead or maybe they just got exiled , who knows ? and how did pythagoras die ? well , according to one guy some guys got mad because they did n't get into the cool kids ' club . | why was pythagoras also scared of irrationality ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | that 's not a number ! get out of our club ! '' and then they 'd make the units half the length and call it four by seven , and everything was okay . | where did vi get those cube things ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | so this proof stayed secret . until someone spilled the beans . according to some , it was all a guy named hippasus and pythagoras threw him of a boat to drown him as punishment for ruining what had been perfect . | why was pythoragras scared of beans ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | numbers one way , zero , and negative numbers the other , and there are numbers between them : fractions , rationals , filling in the gaps . but pythagoras did n't think about numbers like this at all . they were n't points in a continuum they were each their own , separate being , which was still pretty modern because... | or pythagoras states that the triangle is n't right - so why did vi draw a perpendicular sign inside it ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | i 'd like to imagine what it would be like to think of numbers the way they did . maybe you think of numbers as being on a line . numbers one way , zero , and negative numbers the other , and there are numbers between them : fractions , rationals , filling in the gaps . but pythagoras did n't think about numbers like t... | is there a .5 in roman numerals or is it all whole numbers ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | oh , and he killed a guy . i mean maybe it was a long time ago and he was afraid of beans . as in beans , they just like freaked him out or something , i do n't know . | was pythagoras really afraid of beans ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | `` unless it does n't exist . '' `` but if they 're both even you can divide both by two and start again , but this still has to be even which means this still has to be even , which means you can divide by two again , but then it has to be even so everything is even forever and you never find the perfect ratio . aww ,... | define means and measures of an item ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . | how did cold war technology affect current technology ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | oh , and he killed a guy . i mean maybe it was a long time ago and he was afraid of beans . as in beans , they just like freaked him out or something , i do n't know . | why was pythagoras afraid of beans ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . | bc is before christ , but whats e ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | how far was he willing to go to protect the proportions he loved ? would he kill for them ? would he die for them ? | how would the pythagoreans know to divide units into smaller sections ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | but mostly i want to talk about the murdery part . see , pythagoras and his cult of pythagoreans had this cool-kids club where they 'd talk about proportions all day . they 'd be like , `` hey , i drew a two by three rectangle using a straight-edge and compass . | what exactly is a cult ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | so what 's the square root of two and how do we make it into a whole number ratio ? square root two is very close to 1.4 which would be a whole number ratio of 10:14 but 10 squared plus 10 squared is definitely not 14 squared , and a ratio of 1,000 to 1,414 is even closer , and a ratio of 100,000,000 to 141,421,356 is ... | what is greater than 0.9999999999999 ... but less than 1 ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . | what is the difference between theroem and theory ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | oh , and he killed a guy . i mean maybe it was a long time ago and he was afraid of beans . as in beans , they just like freaked him out or something , i do n't know . | was pythagoras really afraid of beans ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | so this proof stayed secret . until someone spilled the beans . according to some , it was all a guy named hippasus and pythagoras threw him of a boat to drown him as punishment for ruining what had been perfect . | beans beans beans is pythagoras scared of me or what ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | i 'd like to imagine what it would be like to think of numbers the way they did . maybe you think of numbers as being on a line . numbers one way , zero , and negative numbers the other , and there are numbers between them : fractions , rationals , filling in the gaps . but pythagoras did n't think about numbers like t... | has anyone ever heard of conspiracy theory 's involving numbers ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | and for a number of squares to be even what does the number have to be , pythagoras , oh my brother ? '' `` if leg squared is even then the leg is even . but it ca n't be even , because it 's already odd . '' | why does that mean that b is even ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | in pythagoras 's time there were no variables , no equations or formulas like we see today , pythagoras 's theorem was n't 'a squared plus b squared equals c squared , ' it was 'the squares of the legs of a right triangle have the same area as the square of the hypotenuse , ' all written out . and when he said 'square ... | so , i was right when i put a square around 'mc ' in elementary school ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | so this proof stayed secret . until someone spilled the beans . according to some , it was all a guy named hippasus and pythagoras threw him of a boat to drown him as punishment for ruining what had been perfect . | why does pythagoreas hate beans ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | but the trouble began with the simplest right triangle one where both the legs are the same length , one where both the legs ' squares are equal . if the legs are both one then the hypotenuse is something that , when squared , gives two . so what 's the square root of two and how do we make it into a whole number ratio... | at 09 , if the hypotenuse is irrational does this mean that the hypotenuse is a multipal of `` root 2 '' or can it also be another irrational number ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | and before that there was no concept of zero , except in india where zero was discovered around here . and if you keep going back you get to the year one , ( there 's no year zero , of course , because zero had n't been invented ) and back a bit more you get to folk like aristotle , euclid , archimedes and then finally... | vi sweetheart , can you speak a bit slower ? |
ok , so i 've been learning about pythagoras and the dirt on him is just too good . you 've probably heard of the pythagorean theorem but not the part where pythagoras was a crazy cult leader who thought he 'd made a deal with a god thousands of years ago and could remember all of his past lives . oh , and he killed a ... | oh , and he killed a guy . i mean maybe it was a long time ago and he was afraid of beans . as in beans , they just like freaked him out or something , i do n't know . | a long long time ago long before the superbowl and things liiike lemonade the hellenic republic was full of smarts and a question resting on the grecian hearts was what is the circumference of a circle ? |
so we have the graph of a function right over here , and we want to think about what does the limit of f of x as x approaches 3 appear to be ? and to do that , let 's just think about the limit as x approaches 3 from values less than 3 , and from values greater than 3 . so let 's first think about the limit of f of x a... | so this limit right over here does not exist . does not exist . the only way that this would have existed is if we got the same value for both of these , and then the limit would be that value . | is there any mathematical symbol for `` does not exist '' that can be used in these kinds of problems ? |
so we have the graph of a function right over here , and we want to think about what does the limit of f of x as x approaches 3 appear to be ? and to do that , let 's just think about the limit as x approaches 3 from values less than 3 , and from values greater than 3 . so let 's first think about the limit of f of x a... | so we have the graph of a function right over here , and we want to think about what does the limit of f of x as x approaches 3 appear to be ? and to do that , let 's just think about the limit as x approaches 3 from values less than 3 , and from values greater than 3 . | how are we so sure that the limits are whole numbers ? |
so we have the graph of a function right over here , and we want to think about what does the limit of f of x as x approaches 3 appear to be ? and to do that , let 's just think about the limit as x approaches 3 from values less than 3 , and from values greater than 3 . so let 's first think about the limit of f of x a... | so we have the graph of a function right over here , and we want to think about what does the limit of f of x as x approaches 3 appear to be ? and to do that , let 's just think about the limit as x approaches 3 from values less than 3 , and from values greater than 3 . | what does colored and empty dot stand for ? |
so we have the graph of a function right over here , and we want to think about what does the limit of f of x as x approaches 3 appear to be ? and to do that , let 's just think about the limit as x approaches 3 from values less than 3 , and from values greater than 3 . so let 's first think about the limit of f of x a... | does not exist . the only way that this would have existed is if we got the same value for both of these , and then the limit would be that value . but we 're clearly not getting the same value . | would the concept of a limit finally show the solution to zeno 's paradox , which states that you can get infinitely close to passing a turtle with a head start , but it will always surpass your distance ? |
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