positive, and to assume that such expressions obey all the fundamental laws of algebra.

Since all squares, whether of positive or of negative quantities, are positive, it follows that a cannot represent any positive or negative quantity; it is on this account called an imaginary quantity. Also expressions of the form a+b√-1 where a and b are real, are called complex quantities.

181. The question now arises whether the meanings of the symbols of algebra can be so extended as to include these imaginary quantities. It is clear that nothing would be gained, and that very much would be lost, by extending the meanings of the symbols, except it be possible to do this consistently with all the fundamental laws remaining true.

Now we have not to determine all the possible systems of meanings which might be assigned to algebraical symbols, both to the symbols which have hitherto been regarded as symbols of quantity and to the symbols of operation, subject only to the restriction that the fundamental laws should be satisfied in appearance whatever the symbols may mean: our problem is the much simpler and more definite one of finding a meaning for the imaginary expression a which is consistent with the truth of all the fundamental laws.

r

182. We already know that -1 is an operation which performed upon any quantity changes it into a magnitude of a diametrically opposite kind. And, if we suppose that √-1 obeys the law expressed by 1x√-1×√-1=-1, it follows that I must be an operation which when repeated is equivalent to a reversal.

Now any species of magnitude whatever can be represented by lengths set off along a straight line; and, when a magnitude is so represented, we may consider the

operation 1 to be a revolution through a right angle, for a repetition of the process will turn the line in the same direction through a second right angle, and the line will then be directly opposite to its original direction.

Hence, when magnitudes are represented by lengths measured along a straight line, we see that √−1, regarded as a symbol of operation, has a perfectly definite meaning.

The symbol-1 is generally for shortness denoted by i, and the operation denoted by i is considered to be a revolution through a right angle counter-clockwise, i denoting revolution through a right angle in the opposite direction.

183. It is clear that to take a units of length and then rotate through a right angle counter-clockwise gives the same result as to rotate the unit through a right angle counter-clockwise and then multiply by a. Thus ai = га.

Again, to multiply ai by bi is to do to ai what is done to the unit to obtain bi, that is to say we must multiply by b and then rotate through a right angle; we thus obtain ab units rotated through two right angles, so that ai × bi = — ab = abii.

From the above we see that the symbol i is commutative with other symbols in a product.

Since (ai) × (ai) = aaii = a2 (− 1) = — a2, it follows that aai; it is therefore only necessary to use one imaginary expression, namely √1.

184. With the above definition of -1 or i, namely that it represents the operation of turning through a right angle counter-clockwise, magnitudes being represented by lengths measured along a straight line, the truth of the fundamental laws of algebra for imaginary and complex expressions can be proved. Some simple cases have been considered in the previous Article: for a full discussion see De Morgan's Double Algebra; see also Clifford's Common Sense of the Exact Sciences, Chapter IV. §§ 12

and 13.

185. If a + bi = 0, where a and b are real, we have a=-bi. But a real quantity cannot be equal to an imaginary one, unless they are both zero.

Hence, if a + bi= 0, we have both a = 0 and b = 0.

Note. In future, when an expression is written in the form a + bi, it will always be understood that a and b are both real.

c + (b − d) i = 0;

and hence, from Art. 185, ac=0 and b-d=0. Thus, two complex expressions cannot be equal to one another, unless the real and imaginary parts are separately equal.

187. The expressions a + bi and a- bi are said to be conjugate complex expressions. The sum of the two conjugate complex expressions a + bi and a bi is a + a + (b − b) i = 2a; also their product is aa + abi — abi — b2i2 = a2 + b2.

Hence the sum and the product of two conjugate complex expressions are both real.

Conversely, if the sum and the product of two complex expressions are both real, the expressions must be conjugate.

For let the expressions be a + bi and c+di. The sum is a + bi+c+ di = a + c + (b + d) i, which cannot be real unless b+d=0. Again,

(a + bi) (c+di) = ac+bci + adi + bdi2 = ac − bd + (bc + ad)i, which cannot be real unless bc + ad=0. Now, if b+d=0 and also bead = 0, we have b(c-a)=0; whence a=c or b=0. If b = 0, d is also zero, and both expressions are real; and, if b÷0, we have a=c, which with b=d, shew that the expressions are conjugate.

188. Definition. The positive value of the square root of a2+b is called the modulus of the complex

quantity a + bi, and is written mod (a + bi). Thus mod (a+bi) = + √a2+b2.

It is clear that two conjugate complex expressions have the same modulus; also, since (a + bi) (abi) = a2 + b2 [Art. 187], the modulus of either of two complex expressions is equal to the positive square root of their product.

Since a and b are both real, a2+b2 will be zero if, and cannot be zero unless, a and b are both zero. Thus the modulus of a complex expression vanishes if the expression vanishes, and conversely the expression will vanish if the modulus vanishes.

If in mod (a+bi) =+√aa +b2 we put b = 0, we have moda =+ √a2, so that the modulus of a real quantity is its absolute value.

189. The product of a + bi and c + di is

ac+bci + adi + bdi = ac - bd + (bc + ad) i.

Hence the modulus of the product of a + bi and c+ di is

√{(ac − bd)2 + (bc + ad)2} = √/{(a2 + b2) (c2 + d2)}

=

= √(a2 + b2) × √(c2 + ď2). Thus the modulus of the product of two complex expressions is equal to the product of their moduli.

The proposition can easily be extended to the case of the product of more than two complex expressions; and, since the modulus of a real quantity is its absolute value, we have the following

Theorem. The modulus of the product of any number of quantities whether real or complex, is equal to the product of their moduli.

190. Since the modulus of the product of two complex expressions is equal to the product of their moduli, it follows conversely that the modulus of the quotient of two expressions is the quotient of their moduli. This may also be proved directly as follows:

191. It is obvious that in order that the product of any number of real factors may vanish, it is necessary and sufficient that one of the factors should be zero; and, by means of the theorem of Art. 189, the proposition can be proved to be true when all or any of the factors are complex quantities.

For, since the modulus of a product of any number of factors is equal to the product of their moduli, and since the moduli are all real, it follows that the modulus of a product cannot vanish unless the modulus of one of its factors vanishes.

Now if the product of any number of factors vanishes its modulus must vanish [Art. 188]; therefore the modulus of one of the factors must vanish, and therefore that factor must itself vanish. Conversely, if one of the factors vanishes, its modulus will, vanish; and therefore the modulus of the product and hence the product itself must vanish.

where a, b, c,...k are all real, let a + Bi be substituted for x; and let P be the sum of all real terms in the result, and Qi the sum of all the imaginary terms. Then the given expression becomes P + Qi.

Since P and Q are both real, they can contain only

S. A.

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