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0 1.00000 + 0.00000¿
1 0.95882 + 0.28402¿
2 0.83867 + 0.54465:
3 0.64944 + 0.76042/
4 0.40672 + 0.91356:
b) 0.13050 + 0.99146
6 —0.15647 + 0.9877
t —0.43055 + 0.90260%
8 —0.66917 + 0.74315
9 —0.85268 + 0.52249;
10 —0.96596 + 0.25880;
11 —0.99969 — 0.02620¿
12 —0.95104 — 0.30905
14 —0.62928 — 0.7771 7
16 —0.10447 — 0.99453¿
18 +0.45454 — 0.89098¿
20 +0.86648 — 0.49967/
22 +0.99884 + 0.05287/
24 +0.80890 + 0.58836/
multiplying it. We see that + decreases, passes through zero, swings aÌlmost to —1
(ïf we could get in between ø = 10 and p = I1 it would obviously swing to —T),
and swings back. 'Phe -value is going back and forth too.
In Eig. 22-1 the dots represent the numbers that appear in Table 22-4, and
the lines are Just drawn to help you visually. So we see that the numbers ø and
oscillate; 107% repea#s ifself, ït is a periodie thing, and as such, it is easy enough
to explain, because 1Í a certain power is ¿, then the fourth power of that would
be 72 squared. It would be +1 again, and therefore, since 100:588 ¡s equal to ¡, by
taking the fourth power we diseover that 10272? is equal to +1. Therefore, if we
wanted 103%, for instance, we could write it as 1027?! times 10:28, In other
words, it has a period, it repeats. Of course, we recognize what the curves look
liket They look like the sine and cosine, and we shall call them, for a while, the
algebraic sine and algebraic cosine. However, instead of using the base 10, we
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" 10® =x + íy
MÀ 15 20 25 /30
Figure 22-1
shall put them into our natural base, which only changes the horizontal scale;
so we denote 2.3025s by , and write 107% = e#, where # is a real number. NÑow
cï = ø-+iụ, and we shall write this as the algebraie cosine of plus 2 tỉimes the
algebraic sine of ý. Thus
©“ = cosf + isin f. (22.8)
What are the properties of cosf and sin? Eirst, we know, for instance, that
#2 + 2 must be 1; we have proved that before, and it is just as true for base e
as for base 10. 'Therefore cos2f + sin2£ = 1. We also know that, for small
t, e# = 1+ it, and therefore cos is nearly 1, and sin is nearly , and so it
goes, that øÏÏ oƒ the 0uarious propertlics oƒ these remarkable ƒunctions, which
come from taking imaginary powers, ør© the same œs the sine and costne oj
trigonometrg.
ls the period the same? Let us fnd out. e to what power is equal to 2? What
1s the logarithm of ¿ to the base c? We worked ¡% out before, in the base 10
it was 0.68184/, but when we change our logarithmic scale to e, we have to
multiply by 2.3025, and if we do that it comes out 1.570. 5o this will be called
“algebraic z/2” But, we see, ¡9 differs from the regular z/2 by only one place
in the last point, and that, of course, is the result of errors in our arithmetiel
So we have created two new functions in a purely algebraic manner, the cosine
and the sine, which belong to algebra, and only to algebra. We wake up at the
end to discover the very functions that are natural to geometry. 5o there is a
connection, ultimately, between algebra and geometry.
We summarize with this, the most remarkable formula in mathematics:
c'? = cosØ + ¿ sỉn 6. (22.9)
'This is our jewel.
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W©e may relate the geometry to the algebra by representing complex numbers
in a plane; the horizontal position of a point is ø, the vertical position of a point
1s (Eig. 22-2). We represent every complex number, # + 2. Then ïf the radial
distance to this poïint is called z and the angle is called Ø, the algebraic law is that
øœ + # is written in the form re”, where the geometrical relationships between
z, , r, and Ø are as shown. This, then, is the unifñcation of algebra and geometry.
Fig. 22-2. x + iy = re.
'When we began this chapter, armed only with the basic notions oŸ integers
and counting, we had little idea of the power of the processes of abstraction
and generalization. sing the set of algebraic “laws,” or properties of numbers,
q. (22.1), and the definitions of inverse operations (22.2), we have been able
here, ourselves, to manufacture not only numbers but useful things like tables of
logarithms, powers, and trigonometric functions (for these are what the Imaginary
powers of real numbers are), all merely by extracting ten successive square roos
of tenl
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Tồosortdrree©
23-1 Complex numbers and harmonic motion
In the present chapter we shall continue our discussion of the harmonic
oscillator and, in particular, the forced harmonic oscillator, using a new technique
in the analysis. In the preceding chapter we introduced the idea of complex
numbers, which have real and imaginary parts and which can be represented
on a diagram in which the ordinate represents the imaginary part and the
abscissa represents the real part. lÝ ø is a complex number, we may write it as
gœ = đ; + ?d¿, where the subscript rz means the real part of ø, and the subscript
means the imaginary part of ø. Referring to Fig. 23-1, we see that we may
also write a complex number ø = # +? in the form z + i = re”, where
r2 = z2 + 9Ÿ = (# + i0)(+ — iụ) = aa*. (The complex conjugate of a, written
đ*, is obtained by reversing the sign of ? in a.) So we shall represent a complex
number in either of two forms, a real plus an imaginary part, or a magnitude z and
a phase angle Ø, so-called. Given z and Ø, z and are clearly r cos Ø and r sin 8
and, in reverse, given a complex number # -Ƒ 2, = v⁄/#2 + 2 and tan 0 = 0/z,
the ratio of the imaginary to the real part.
IMAGINARY
x REAL AXIS
Fig. 23-1. A complex number may be represented by a point in the
“complex plane.”
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