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155. De Moivre's Theorem. Expressions of the form

cos x + i sin x, where i = V-1, play an important part in modern analysis. Since (cos x + i sin x) cos y + i sin y)

= COS X COS Y sin x sin y + i (cos w sin y + sin x cos y)

= cos (x + y) + i sin (x + y), we have

(cos x + i sin x)2 = cos 2 x + i sin 2 x; and again, (cos x + i sin x)=(cos x + i sin xocos x + i sin x)

(cos 2 x + i sin 2 x) cos x + i sin .)

= cos 3 x + i sin 3x. Similarly, (cos x + i sin x)"= cos nx + i sin nr.

To find the nth power of co8 x + i sin x, n being a positive integer, we have only to multiply the angle x by n in the expression.

This is known as De Moivre's Theorem, from the discoverer (c. 1725).

156. De Moivre's Theorem extended. Again, if n is a positive integer as before,

(cos. * + i sin = COS X + i sin x. .. (cos x + i sin x)

in ay However, x may be increased by any integral multiple of 2 7 without changing the value of cos x + i sin x. Therefore the following n expressions are the nth roots of cos x + i sin x :

X + 2

x + 2 + i sin

ti sin

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n

3

XC = COS

+ i sin n

n

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X
COS

n

COS

n

n

n

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(cos x + i sin x)"
X + 2 km

x + 2 km
tisin

(k=0,1,2, ...n-1). Similarly, it may be shown that

= COS

n

n

m

m

m

п

n

(cos x + i sin x) = cos(x + 2 kt) + i sin - (x + 2 kt). (k = 0,1,2, ..., n-1, m and n being integers, positive or negative.) 157. The Roots of Unity. If we have the binomial equation

20" --1=0, we see that

2c" = 1, and

x = the nth root of 1, of which the simplest positive root is V1 or 1. Since the equation is of the nth degree, there are n roots. In other words, 1 has n nth roots. These are easily found by De Moivre's Theorem.

There are no other roots than those in § 156. For, letting k = n, n + 1, and so on, we have

2 + n (27) X + n (27)

+ i sin

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n

-1.

= COS

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or

and so on, all of which we found when k = 0, 1, 2,

For example, required to find the three cube roots of 1.
If

cosø + i sin ø =1, the given number, then

$= 0, 2 , 47, :
Also
(coso + i sin o)t = 15

= the three cube roots of 1.
k (2 7T) +

k (2 t) +.
But
(cos¢ + i sin o)

+ i sin 3

3
where k = 0, 1, or 2, and p = 0, 2 77, 47,
Therefore

= cos 2 + + i sin 2 h = -1,
15
= cosa + i sin a = cos 120° + i sin 120°

- 1 + 13.1 =-0.5 + 0.8660 i,
1$
= cosa + i sin f t = cos 240° + i sin 240°
-}-}

- 0.5 -0.8660 i. The three cube roots of 1 are therefore

1, - + fv-3, -1-fv3. These roots could, of course, be obtained algebraically, thus :

28 – 1= 0, Whence (x - 1) (x3 + 2 + 1) = 0; and either

1-1=0, whence x = 1, 2a + 2+1= 0, whence x =

}+fv-3. Most equations like an a = 0 cannot, however, be solved algebraically

or

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or

- 1; that is, to solve the

Required to find the seven 7th roots of equation x= -1, or x? +1= 0.

If

coso + i sin p =-1, the given number, then

ф = 1, 377, 57, ....

k (2 7) + 0 Also (coso + i sino)} = cos

k (2T) + $.
+ i sin
7

7
where k 0, 1, 6, and $ = , 39,
That is, in this case

(2k + 1)
(coso + i sino)?
= COS

+ i sin
7

7 Hence the seven 7th roots of 1 are

(2k + 1).

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All these values may be found from the tables. For example,

cos 25° 42' 513" + i sin 25° 42' 513" = 0.9010 + 0.4339 V-1.

Exercise 81. Roots of Unity

1. Find by De Moivre's Theorem the two square roots of 1.
2. Find by De Moivre's Theorem the four 4th roots of 1.
3. Find three of the nine 9th roots of 1.
4. Find the five 5th roots of 1.
5. Find the six 6th roots of +1 and of – 1.
6. Find the four 4th roots of -1.
7. Show that the sum of the three cube roots of 1 is zero.
8. Show that the sum of the five 5th roots of 1 is zero.

9. From Exs. 7 and 8 infer the law as to the sum of the nth roots of 1 and prove this law.

10. From Ex. 9 infer the law as to the sum of the nth roots of k and prove this law.

11. Show that any power of any one of the three cube roots of 1 is one of these three roots.

12. Investigate the law implied in the statement of Ex. 11 for the four 4th roots and the five 5th roots of 1.

158. Roots of Numbers. We have seen that the three cube roots

of 1 are

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x, = cos 120° + i sin 120° =- £+-3,
X2 = cos 240° + i sin 240°

- } V-3,
and xg = cos 360° + i sin 360o = cos 0° + i sin 0o =1.
Furthermore, x, is the square of x, because

(cos 120° + i sin 120°)2 = cos (2 · 120°) + i sin (2 · 120°), by De Moivre's Theorem. We may therefore represent the three cube roots by w, w?, and either ws or 1.

In the same way we may represent all n of the nth roots of 1 by w, w?, ws, w" or 1.

If we have to extract the three cube roots of 8 we can see at once that they are

2, 2w, and 2 w", because

28 = 8, (2 w) 8 = 28 w = 8.1= 8, and

(2 w2)8 = 28 w = 28 (w8)2 = 28 12 = 8. In general, to find the three cube roots of any number we may take the arithmetical cube root for one of them and multiply this by w for the second and by wo for the third.

The same is true for any root. For example, if w, wo, w8, wt, and ws or 1 are the five 5th roots of 1, the five 5th roots of 32 are 2 w, 2 w2, 2w8, 2w, and 2 wó or 2.

Exercise 82. Roots of Numbers

1. Find the three cube roots of 125. 2. Find the four 4th roots of 81 and verify the results. 3. Find three of the 6th roots of 729 and verify the results. 4. Find three of the 10th roots of 1024 and verify the results. 5. Find three of the 100th roots of 1.

6. Show that, if 2 w is one of the complex 7th roots of 128, two of the other roots are 2 w2 and 2 we.

7. Show that either of the two complex cube roots of 1 is at the same time the square and the square root of the other.

8. Show that a result similar to the one stated in Ex. 7 can be found with respect to the four 4th roots of 1.

9. Show that the sum of all the nth roots of 1 is zero. 10. Show that the sum of the products of all the nth roots of 1, taken two by two, is zero.

159. Properties of Logarithms. The properties of logarithms have already been studied in Chapter III. These properties hold true whatever base is taken. They are as follows :

1. The logarithm of 1 is 0.
2. The logarithm of the base itself is 1.

3. The logarithm of the reciprocal of a positive number is the negative of the logarithm of the number.

4. The logarithm of the product of two or more positive numbers is found by adding the logarithms of the several factors.

5. The logarithm of the quotient of two positive numbers is found by subtracting the logarithm of the divisor from the logarithm of the dividend.

6. The logarithm of a power of a positive number is found by multiplying the logarithm of the number by the exponent of the power.

7. The logarithm of the real positive value of a root of a positive number is found by dividing the logarithm of the number by the index of the root.

160. Two Important Systems. Although the number of different systems of logarithms is unlimited, there are but two systems which are in common use. These are

1. The common system, also called the Briggs, denary, or decimal system, of which the base is 10.

2. The natural system, of which the base is the fixed value which the sum of the series

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approaches as the number of terms is indefinitely increased. This base, correct to seven places of decimals, is 2.7182818, and is denoted by the letter e.

Instead of writing 1.2, 1.2.3, 1.2.3 · 4, and so on, we may write either 2., 3., 4!, and so on, or 12, 13, 14, and so on. The expression 2! is used on the continent of Europe, 12 being formerly used in America and England. At present the expression 2! is coming to be preferred to 12 in these two countries.

The common system of logarithms is used in actual calculation; the natural system is used in higher mathematics.

The natural logarithms are also known as Naperian logarithms, in honor of the inventor of logarithms, John Napier (1614), although these are not the ones used by him. They are also known as hyper bolic logarithms.

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