21. A courier proceeds from one place P to another place Q in 14 hours; a second courier starts at the same time as the first from a place 10 miles behind P, and arrives at Q at the same time as the first courier. The second courier finds that he takes half an hour less than the first to accomplish 20 miles. Find the distance of from P.

22. Two travellers A and B set out at the same time from two places P and Q respectively, and travel so as to meet. When they meet it is found that A has travelled 30 miles more than B, and that A will reach Q in 4 days, and B will reach P in 9 days, after they meet. Find the distance between P and Q.

23. A vessel can be filled with water by two pipes; by one of these pipes alone the vessel would be filled 2 hours sooner than by the other; also the vessel can be filled by both together in 13 hours. Find the time which each pipe alone would take to fill the vessel.

24. A vessel is to be filled with water by two pipes. The first pipe is kept open during of the time which the second would take to fill the vessel; then the first pipe is closed and the second is opened. If the two pipes had both been kept open together the vessel would have been filled 6 hours sooner, and the first pipe would have brought in 3 of the quantity of water which the second pipe really brought in. How long would each pipe take to fill the vessel ?

25. A certain number of workmen can move a heap of stones in 8 hours from one place to another. If there had been 8 more workmen, and each workman had carried 5 lbs. less at a time, the whole work would have been completed in 7 hours. If however there had been 8 fewer workmen, and each had carried 11 lbs. more at a time, the work would have occupied 9 hours. Find the number of workmen and the weight which each carried at a time.

XXV. IMAGINARY EXPRESSIONS.

354. Although the square root of a negative quantity is the symbol of an impossible operation, yet these roots are frequently of use in Mathematical investigations in consequence of a few conventions which we shall now explain.

355. Let a denote any real quantity; then the square roots of the negative quantity – a3 are expressed in ordinary notation by√(-a). Now -a may be considered as the product of a2 and -1; so if we suppose that the square roots of this product can be formed, in the same manner as if both factors were positive, by multiplying together the square roots of the factors, the square roots of a will be expressed by a (-1). We may therefore agree that the expressions + √(-a2) and a √(-1) shall be considered equivalent. Thus we shall only have to use one imaginary expression in our investigations, namely, √(−1).

356. Suppose we have such an expression as a +ẞ√(-1), where a and ẞ are real quantities. This expression may be said to consist of a real part a and an imaginary part ẞ(-1); or on account of the presence of the latter term we may speak of the whole expression as imaginary. When ẞ is zero, the term B√(-1) is considered to vanish; this may be regarded then as another convention. If a and ẞ are both zero, the whole expression vanishes, and not otherwise.

357. By means of the conventions already made, and the additional convention that such terms as B√(-1) shall be subject to the ordinary rules which hold in Algebraical transformations, we may establish some propositions, as will now be seen.

358. In order that two imaginary expressions may be equal, it is necessary and sufficient that the real parts should be equal, and that the coefficients of √(-1) should be equal.

may be considered as a symbolical mode of asserting the two equalities a = y and ẞ=8 in one statement.

359. Consider now two imaginary expressions a + ẞ √(−1) and y+(-1), and form their sum, difference, product, and quotient.

Their sum is

a+y+ (B+8) √(−1).

If the second be taken from the first, the remainder is

{a + B √(− 1)} {y + 8 √(− 1)} = ay — ßd + (ad + By) √(− 1);

for √(−1) × √(−1) is, by supposition, — 1.

The quotient obtained by dividing the first by the second is

This may be put in another form by multiplying both numerator and denominator by y-d√(-1). The new numerator is thus

2

and the new denominator is y2 + 82; therefore

360. We will now give an example of the way in which imaginary expressions occur in Algebra. Suppose we have to solve the equation 1. We may write the equation thus,

Thus we satisfy the proposed equation either by putting x-1=0, or by putting x2+x+1=0. The first gives x=1; the second may be written

Thus we conclude that if either of the imaginary expressions last written be cubed, the result will be unity. This we may verify; take the upper sign for example, then

If ∞3 = 1, we have x=(1); it appears then that there are

three cube roots of unity, namely, 1 and ㄓ

1 /3
2

√(−1).

361. We have seen in Art. 337, that the quadratic expression ax2 + bx + c is always identical with a (x-p) (x-q), where p and q are the roots of the equation ax2 + bx+c=0. If the roots are imaginary, p and q will be of the forms aẞ(-1); thus we have then

ax2 + bx + c = a {x — a − ẞ √(− 1)} {x − a + ẞ √(− 1)}.

This will present no difficulty when we remember the convention that the usual algebraical operations are to be applicable to the term ẞ√(-1). For the second side of the asserted identity is

362. Two imaginary expressions are said to be conjugate when they only differ in the sign of the coefficient of (-1). Thus a+B√(-1) and a-ẞ(-1) are conjugate.

Hence the sum of two conjugate imaginary expressions is real, and so also is their product. In the above example the sum is 2a, and the product is a3 + ß3.

363.

The positive value of the square root of a2 + ẞ2 is called

the modulus of each of the expressions

a+ẞ(-1) and a-ẞ(-1).

From this definition it follows that the modulus of a real quantity is the numerical value of that quantity taken positively.