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1. Varieties of Numbers.—The numbers occurring in elementary mathematics are:
The whole number, also called integer or natural number.
The indicated imperfect quotient of two integers, or common fraction.
The indicated imperfect root of an integer, or surd.
The indicated even root of a negative number, or an imaginary.
Integers and fractions are classed together as rational numbers, all others being irrational.
Besides imperfect roots, irrational numbers include such numbers as , the ratio of the circumference of a circle to its diameter.
Rational and irrational numbers are classed together as real numbers. We can imagine all the real numbers arranged in magnitude on a scale, ranging from zero up through the positive numbers and down through the negative numbers; on this scale we can fix the position of a real number with any required degree of accuracy. Thus we can say that V6 lies between 2.44949 and 2.44948 because (2.44949)2 is more than 6, and (2.44948)2 is less than 6; it can be shown, too, that is more than 3.14159 and less than 3.14160, and so on: any irrational number can be expressed approximately by a rational number, and so used in arithmetic processes.
Imaginary numbers, on the other hand, have no place in the scale of real numbers, but are of an entirely different kind.
They are useful because operations can be performed with them that lead to results concerning real numbers, and because in some branches of science, meaning may be attached to the imaginary number itself.
When no meaning is attached to imaginary numbers their occurrence as answers to problems indicates impossibility of solution. This of course is equally true of real numbers; in a problem which asks for a number of men, such answers as į or – 1 are meaningless.
2. Operations with Surds.-There are two principles in accordance with which all simplification of surds must be performed:
I. Only integral multiples of the same root of the same number can be added or subtracted. II. In multiplication or division,
VT b For instance, V3+V6 cannot be simplified without evaluation, but
(3+V6)-(5+ V54)=–2+V6- V9V6=-2–2V6,
3. Operations with Imaginaries. The simplest imaginary is V-1, denoted by i. It is defined by the relation
v-11-1=-1, or =-1. If a is a positive number, V-a is an imaginary, defined by the relation
Var Vav-1, or Variva.
It is important to note that this does not come under the law for the multiplication of surds; i. e., V-av-b is not equal to V(-a)(-6), which would be + Vab, the symbol V always indicating the positive square root. Until this distinction becomes familiar, the imaginary unit i should be kept separate in all operations. For example, to multiply
(V-2+V-3) (V-2-2V-3), rewrite the factors:
(iV2+iV3) (iV2- 2iV3), and multiply, remembering that i2= -1:
4+ v6. Ans. In operations calling for higher powers of imaginaries, it is well to notice that
j? = -1, 3= -i, it=1,
jb=i, 26=-1, 7=-i, 8=1, etc., the powers repeating in cycles of 4.
4. If a+bi=c+di, then a=c and b=d.
For the equation may be written, a-c=i(d—); and since the difference between two real numbers can not, by the nature of their definitions, be equal to an imaginary number, this equation can be true only when (a-c)=0 and (d—b)=0, or a=c, b=d. This same principle applies also to surds; thus, if
we should have a=c and b=d, if only Vī and Vd are irrational numbers. For the equation may be written (a-c)=(Vd-Vā), in which form (a -c) must be rational or zero, while Vd-V7 can only be irrational or zero; hence (a-c) and Vd-V6 must each equal zero to satisfy the equation.
5. When two surds, or two imaginary expressions, differ only in the sign of the surd or imaginary part, such quantities are said to be conjugate. Thus,
a-Vb is conjugate to a+V6 and
a-bi is conjugate to a+bi. 6. The sum and product of two conjugate surds are rational; for
(a+ V7) +(a-V5)=2a,
(a+V%) x (a-V%)=ao –b. 7. The sum and product of two conjugate imaginary expressions are real; for
(a+bi) + (a - bi) = 2a,
(a+bi) x (a-bi) =a? +62. 8. Fractions involving surds in the denominator should generally be simplified, by making the denominator rational by multiplying both numerator and denominator by the conjugate of the denominator. Thus,
9. To Find the Square Root of at.-Assume
Va+Vo=Vx+ Vy. Then
a+ Vī=x+y+2Vxy and (Art. 4) x+y=a,
(1) 2 Vxy=V6.
(2) Taking the difference of the squares of (1) and (2), and extracting the square root of the difference, X-y= Va? - 7.
(3) If Va? – is rational, the value of Væ+ Vy is obtained from (1) and (3); if Va? — b is not rational, the expression Va+ Vo can not be simplified.
10. To Find the Square Root of a+bi.-Assume
a+bi= x2 - y2 + 2i • xy. Equating the real and imaginary parts, x2 - y2=a,
(2) Adding the squares of (1) and (2) and extracting the square root of the sum, * + y2= Va? +62.
From (1) and (3) the values of x? and yż are easily obtained; and if Va? +62 is rational, the expression may be simplified.