21. Although different numbers in arithmetic are represented by different definite number symbols, each number may be represented in an unlimited number of ways by combinations of other numbers. Thus, the number 24 may be represented by 12 x 2, 6 × 4, 8 × 3, 20 + 4, 100 ÷ 5 + 4, etc.

22. Fixing our attention upon the results of indicated operations, as in the illustration above, we commonly speak of such expressions as 12 × 2, 6 × 4, 20 + 4, etc., as numbers, meaning thereby the numbers resulting when we perform these operations.

We say that the number 12 × 2 is equal to the number 20+ 4, since each represents 24 × 1..

23. As a symbol of relation we have in algebra, as in arithmetic, the sign of equality, which may be read "equals," "is equal to," "is replaceable by," etc.

Thus, 8 + 2 = 10.

24. The statement in symbols that two expressions represent the same number is called an equation.

E. g. 5+ 1 = 4+2.

The part at the left of the equality sign is called the first member, and the part at the right the second member, of the equation. E. g. In 5+1 = 4 + 2, 5 + 1 is the first member, and 4+ 2 is the second member of the equation.

25. The sign = should never be used except to connect numbers or expressions which are equal, that is, which stand for the same number. It should never be used in place of any form of the verb "to be." E. g. We should write, “Ans. is 8,” never "Ans.

- 8."

26. Two algebraic expressions are equivalent when they represent the same numerical value, no matter what particular values may be assigned to the letters appearing in them.

E. g. 3xx is equivalent to 4x; 5y-2y is equivalent to 3 y.

27. An equality whose members are equivalent expressions is called an identity.

In general any values may be assigned at will to the letters appearing in an identity. Such restrictions as may be necessary in certain cases will be explained in a later chapter.

28. An equality which is true for particular values only of certain of the letters appearing in it is called a conditional equation.

E. g. The equation x + 1 = 5 is a conditional equation, for the first member is equivalent to the second only on condition that x be given the particular value 4.

We shall use the word equation to mean conditional equation, that is, an equality which is not an identity.

29. In order to distinguish identical from conditional equations we shall use the triple sign of equality, = (read "is the same as," "is identical with," "stands for,") for identical equations, and the double sign of equality for conditional equations.

=

To conform to the usage in arithmetic we shall commonly use the double sign of equality instead of the triple sign when writing identities in which arithmetic numbers only appear.

=

Other reasons for this use of the sign will be given in a later chapter.

Thus, instead of 6+2 = 8, we shall write 6 + 2 = 8.

30. The signs > and <, read "is greater than," and "is less than," respectively, are used as symbols of inequality.

E. g. We may denote that 10 is greater than 8 by writing 10 > 8; 7 is less than 9 by writing 7 < 9.

that

It should be noted that the larger end of each symbol is directed toward the greater quantity.

31. The signs of equality or of inequality, when crossed by lines, are understood as meaning "not equal to," "not greater than," and "not less than."

E. g.

23 means that 2 is not equal to 3; 69 means that 6 is not greater than 9; 87 means that 8 is not less than 7.

EXERCISE I. 2

Find the values represented by the following expressions when the given numerical values are substituted for the letters:

If a = 4, b = 2, c = 5, and d = 1, find the value of

19.

20.

(a+c)(b+d)+(a+d)(b+c).

21. (a+b)(b-c)+(b+c) (c-d).

18. abc + bed + cda + abd. 22. (a-b)(c-d)+(b−c)(d—a). (a+b)(b+c)+(b+c)(c+d). 23. (a+b)c+(b+c)d+(c+d)a. If a = 6, b = 3, x = 7, and y = 1, find the values represented by the following expressions :

24. a2 + b2.

25. a2 - b2.

28. (a2+b2)x + (x2 + y2)b.
29. a2 + ab + b2.

30. a2bab2 -X- y.
31. ax2 - by2

x + y.

26. a2 + b + x2 + y. 27. (a + b)2 + a + b. Verify the following algebraic identities for particular values of the letters appearing in them, by assigning values to the letters: Ex. 32. (a+1)(a + 2) = a2 + 3 a + 2.

If the members are identical they must represent equal numbers for all values which may be assigned to a.

Accordingly, letting a = 4, we obtain, substituting 4 for a,

(4 + 1)(4 + 2) = 42 + 3 x 4 + 2

5 x 6 16 + 12 + 2

30 = 30.

Accordingly the identity is true for a 4. By substitution it will be found to be true for all other values which may be assigned to a.

33. (y+5)(y+3)=y(y+8)+15. 35. (y+5)2=y2 + 5 (2 y + 5). 34. (x+3)= x(x+6) + 9. 36. m (m+4)+ 4 = m2 + 4 (m + 1).

2

Ex. 37. (a + b)2 = a2 + 2 ab + b2.

Assigning to a and b the values 8 and 2, we obtain by substitution,

Ex. 38.

(x + a) (x + b) = x2 + (a + b)x + ab.

Substituting 4, 3, and 2 for x, a, and b respectively, we obtain

43. a* + (a2 + ab + b2)2 =

(a2 + b2) [a2 + (a + b)2].

44. (x + y)* = 2 (x2 + y2)(x + y)2 — (x2 — y2)2.

45. (a2 + b2)(c2 + d2) = (ac + bd)2 + (ad — bc)2.

46. a3 + b3 = (a + b)(a2 — ab + b2).

47. (ab)3 +3 ab(a - b) = (a + b)3 — 3 ab(a + b) — 2 b3. 48. (x + y)3 — (x − y)2(x + y) = 4 xy(x + y).

49. (a2)24 (a + 1)2 + 6 a2 - 4 (a− 1)2 + (a− 2)2 = 0. 50. (x + y)3 = x23 + 3 x2y + 3 xy2 + y3.

32. An axiom is the statement of a truth which may be inferred directly from our experience, or from the nature of the things considered.

To be regarded as an axiom, a truth must be such that it is incapable of proof further than its mere statement.

As axioms common to mathematics, we may state the following, which were called by an early writer on mathematics Common Truths about Things:

1. Any number is equal to itself.

E. g. 4 = 4.

2. The Principle of Substitution. The numerical value of a mathematical expression is not altered when for any number or expression in it we substitute an equal number or expression.

That is, the "form" of an expression may be changed without altering its value.

E. g. The value of 4 + 3 + 10 remains unaltered if for

we substitute

its equal value 2; or again, if we substitute 7 for the sum of 4 and 3 and write 7+ 2.

3. Numbers equal to the same number are equal.

E. g. If 2xa and 10 = a, then 2 x = 10.

4. If equal numbers be added to equal numbers the resulting numbers will be equal.

5. If equal numbers be subtracted from equal numbers the resulting numbers will be equal.

6. If equal numbers be multiplied by equal numbers the resulting numbers will be equal.

E. g. If x = 5, then two times equals two times 5, that is, x = 10. 7. If equal numbers be divided by equal numbers (except zero) the resulting numbers will be equal.

E. g. If 3 x = 12, then 3x divided by 3 equals 12 divided by 3, that is,

x = 4.

There are two square roots, three cube roots, four fourth roots, etc., of any number, so that when applying this axiom it is necessary to distinguish carefully between these roots. (See Chapter XVIII, Principal Values of Roots.)

33. Substituting the word "identical" for the word "equal" in each of the statements above, we have corresponding axiomatic principles governing identical expressions.

34. If AB we may immediately write B A, since this is only another way of saying the same thing.

Identities such as those above, which are formed by interchanging the members, are said to be one the converse of the other.

Ex. 1. Find the value which must be assigned to a in order that the conditional equation 2a + 3 = 15 may be true.

Subtracting 3 from each member we obtain

Dividing both members of the last equation by 2, we obtain finally a = 6. This value is found to satisfy the original conditional equation.