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or a total of 26. 'Phus we have only 26 more to go, and so we fñnd that the true
number is 2.3025. (Actually, we shall later see that the ezøc£ number should
be 2.3026, but to keep it realistic, we shall not alter anything in the arithmetic.)
trom this table we can now calculate any power of 10, by compounding the power
out of 1024ths.
Let us now actually calculate a logarithm, because the process we sha]l use is
where logarithm tables actually come from. The procedure is shown in Table 22-2,
and the numerical values are shown in Table 22-1 (columns 2 and 3).
Table 22-2
Calculation of a logarithm: log 2
2~ 1.77828 = 1.124682
1.124682 ~ 1.074607 = 1.046598, etc.
-2 = (1.77828)(1.074607)(1.036633)(1.0090350)(1.000573)
Ị 308.254
S73
— 1030103 =
= 10 (mñ 0254)
.l0g+o 2 = 0.30103
Suppose we want the logarithm of 2. That is, we want to know to what power
we Imust raise 10 to get 2. Can we raise 10 to the 1/2 power? No; that is too bịg.
In other words, we can see that the answer is goïng to be bigger than 1/4, and
less than 1/2. Let us take the factor 101/4 out; we divide 2 by 1.778..., and get
1.124..., and so on, and now we know that we have taken away 0.250000 from
the logarithm. The number 1.124..., is now the number whose logarithm we
necd. When we are finished we shall add back the 1/4, or 256/1024. Ñow we
--- Trang 398 ---
look in the table for the next number just below 1.124..., and that is 1.074607.
We© therefore divide by 1.074607 and get 1.046598. Erom that we discover that 2
can be made up of a product of numbers that are in Table 22-1, as follows:
2 = (1.77828)(1.074607) (1.036633)(1.0090350)(1.000573).
There was one factor (1.000573) left over, naturally, which is beyond the range of
our table. To get the logarithm of this factor, we use our result that 10/1924 œ
1+ 2.3025A/1024. We fnd A = 0.254. Therefore our answer is 10 to the
following power: (256 + 32 + 16 + 4+ 0.254)/1024. Adding those together, we geb
308.254/1024. Dividing, we get 0.30108, so we know that the logo 2 = 0.30103,
which happens to be right to 5 ñguresl
This is how logarithms were originally computed by Mr. Briggs of Halifax,
in 1620. He said, “[ computed successively 54 square roots of 10” We know he
really computed only the first 27, because the rest of them can be obtained by this
trick with A. His work involved calculating the square root of 10 twenty-seven
times, which is not mụuch more than the ten times we did; however, it was more
work because he calculated to sixteen decimal places, and then reduced his answer
to fourteen when he published it, so that there were no rounding errors. He
made tables of logarithms to fourteen decimal places by this method, which 1s
quite tedious. But all logarithm tables for three hundred years were borrowed
trom Mr. Briggs' tables by reducing the number of decimal places. Only in
modern times, with the WPA and computing machines, have new tables been
independently computed. 'Phere are much more efficient methods of computing
logarithms today, using certain series expansions.
In the above process, we discovered something rather interesting, and that
1s that for very small powers e we can calculate 10“ easily; we have discovered
that 10° = 1 + 2.3025, by sheer numerical analysis. Of course this also means
that 10923925 — ] + n iŸ n is very small. Now logarithms to any other base
are merely multiples of logarithms to the base 10. The base 10 was used only
because we have 10 fngers, and the arithmetic of it is easy, but if we ask for a
mathematically natural base, one that has nothing to do with the number of
ñngers on human beings, we might try to change our scale of logarithms in some
convenient and natural manner, and the method which people have chosen 1s
to redefne the logarithms by multiplying all the logarithms to the base 10 by
2.3025... This then corresponds to using some other base, and this ¡is called the
nakural base, or base e. Note that log„(1 + n) 3m, or eƒ” + as n — 0.
--- Trang 399 ---
It is easy enough to ñnd out what e is: e = 101/23925 or 109434294. an
Irrational power. Our table of the successive square roots of 10 can be used
to compute, not just logarithms, but also 10 to any power, so let us use it tO
calculate this natural base e. Eor convenience we transform 0.434294... into
444.73/1024. Now, 444.73 is 256 + 128 + 32+ 16+ 8+ 4+ 0.73. Therefore e,
since it is an exponent of a sum, will be a product of the numbers
(1.77828X1.33352(1.0746071.036633(1.018152)1.009035X1.001643) = 2.7184.
(The only problem is the last one, which is 0.73, and which is not in the table,
but we know that if A is small enough, the answer is 1 + 2.3025 A.) When we
multiply all these togebher, we get 2.7184 (it should be 2.7183, but it is good
enough). The use of such tables, then, is the way in which irrational powers and
the logarithms of irrational numbers are all calculated. 'Phat takes care of the
irrationals.
22-5 Complex numbers
Now it turns out that after all that work we s2 cannot solve every equationl
Eor instance, what is the square root of —1? Suppose we have to fnd z2 = —1.
'The square of no rational, of no irrational, of nothøng that we have discovered so
far, is equal to —1. 5o we again have to generalize our numbers to a still wider
class. Let us suppose that a speeific solution of z2 = —1 is called something, we
shall call it ¿; ¿ has the property, by defnition, that is square is —1. Thhat is
about all we are going to say about it; oŸ course, there is more than one root
of the equation #? = —1. Someone could write ¡, but another could say, “No,
l prefer —¿. My ¿ is minus your 2.” lt is jus as good a solution, and since the
only defnition that ¿ has is that 72 = —1, it must be true that any equation we
can write is equally true if the sign of ¿ is changed everywhere. This ¡is called
taking the cormplez conƒugate. Ñow we are goïng to make up numbers by adding
successive 7's, and multiplying ?'s by numbers, and adding other numbers, and
So on, according to all of our rules. In this way we fnd that numbers can all be
written in the form ø-+ ?g, where ø and g are what we call real numbers, i.e., the
numbers we have been defning up until now. The number ¿ is called the n¿£
#maginar number. Any real multiple of ¿ is called pure ïmaginaru. The most
general number, ø, is of the form ø + ;q and is called a complez number. 'Things
do not get any worse IÍ, for instance, we multiply two such numbers, let us say
--- Trang 400 ---
(r-+2s)(p+ 4). Then, using the rules, we get
(+ is) + 14) = rp + rũ) + (4s)p + (15)(44)
= rp + i(rq) + (sp) + (1/)(s9)
= p~ sq) + lírg + sp), (22.4)