Thus by reference to Art. 97, we see that the sum, difference, product, and quotient of two imaginary expressions is in each case an imaginary expression of the same form.

105. To find the square root of a + b/-1.

Assume

√a+b√ −1 = x + y √ − 1,

where x and y are real quantities.

− √−1

By squaring, a+b√−1 = x2 - y2 + 2xy √-1; therefore, by equating real and imaginary parts,

Thus the required root is obtained.

Since x and y are real quantities, x2+ y2 is positive, and therefore in (3) the positive sign must be prefixed before the quantity √a2+b2.

Also from (2) we see that the product xy must have the same sign as b; hence x and y must have like signs if b is positive, and unlike signs if b is negative.

Since the product xy is negative, we must take

x=3, y=-4; or x=-3, y=4.

.(1),

...(2).

106. The symbol-1 is often represented by the letter i; but until the student has had a little practice in the use of imaginary quantities he will find it easier to retain the symbol -1. It is useful to notice the successive powers of √1 or i; thus

and since each power is obtained by multiplying the one before it by-1, or i, we see that the results must now recur.

107. We shall now investigate the properties of certain imaginary quantities which are of very frequent occurrence.

Suppose

that is,
.. either

x = 3/1; then 3 = 1, or ï3 − 1 = 0 ;

(x − 1) (x2 + x + 1) = 0.

x − 1 = 0, or x2 + x + 1

=

0;

x = 1, or X =

-1±√3
2

whence

It may be shewn by actual involution that each of these values when cubed is equal to unity. Thus unity has three cube roots,

two of which are imaginary expressions.

Let us denote these by a and ẞ; then since they are the roots

of the equation

their product is equal to unity;

x2 + x + 1 = 0,

108. Since each of the imaginary roots is the square of the other, it is usual to denote the three cube roots of unity by 1, w, w3.

Also o satisfies the equation a2 + x + 1 = 0 ;

... 1 + w + w2 = 0;

that is, the sum of the three cube roots of unity is zero.

therefore (1) the product of the two imaginary roots is unity;

(2) every integral power of w3 is unity.

109. It is useful to notice that the successive positive integral powers of w are 1, w, and w2; for, if n be a multiple of 3, it must be of the form 3m; and w" = @ 1.

3m

If ʼn be not a multiple of 3, it must be of the form 3m + 1 or 3m + 2.

110. We now see that every quantity has three cube roots, two of which are imaginary. For the cube roots of a3 are those of a3× 1, and therefore are a, aw, aw3. Similarly the cube roots of 9 are 3/9, w 3/9, w3 3/9, where 3/9 is the cube root found by the ordinary arithmetical rule. In future, unless otherwise stated, the symbol a will always be taken to denote the arithmetical cube root of a.

Example 3. Shew that

(a+wb+w2c) (a + w2b+ wc) = a2 + b2 + c2 − be − ca - ab.

In the product of

the coefficients of b2 and c2 are w3, or 1;

the coefficient of bc

a+wb+w2c and a+w2b+ wc,

=w2+w1=w2+w= −1 ;

1. Multiply 2-3+3-2 by 4-3-5-2.
2. Multiply 3-7-5-2 by 3-7+5/2.
3. Multiply e1+e-√-1 by e√-1-e-√=1.

11. Find the value of (−√√/− 1)4n+3, when n is a positive integer.

-A

12. Find the square of √9+40 √1+√9-40√1.

H. H. A.

6