2. Interpret ii. and iii. of 1 as theorems in numbers. 3. By means of v. of 1 find two numbers such that the sum of their squares shall be a square. Make a table of such numbers. 4. Find the value of x in each of the following i. x + { x + } x = 2x-2. ii. 3(1 2.3x)=2(1 + 2(x − 2)}. 5. Simplify 1-1−(1−x)}+1+{1−(1+x)}+x− {x-(x-1)}. 6. Find x in the equation (x − 4) (x + 6) = (x + 8) (x + 2). 7. Find a number whose half exceeds the sum of its fourth and fifth parts by 40. 8. Find a number such that if it be increased by a, and if it be diminished by b, one third of the first result is equal to one half the second. 9. The sum of the ages of A, B, and C is 108 years. A is twice as old as B, and twice C's age is equal to A's and B's together. Find their ages. 10. After paying 2% taxes on my income I have $1078 left. What is my income? 11. I pay 331% duty on the cost price of a horse. I keep him 2 months at an expense of $16, and I then sell him for $200, making 20% profit on the cost price. What did the horse cost? 12. In a certain school of the pupils are in the first form, }} in the second, in the third, and 14 in the fourth. 3 20 pupils are in the school? How many 13. A market-woman sells to A half an egg more than half she has, to B half an egg more than C, and she then has 6 eggs left. half she has left, and 10 eggs to How many had she at first? CHAPTER II. THE FOUR ELEMENTARY OPERATIONS. ADDITION AND SUBTRACTION. 24. The addition of a and b is denoted by a+b, where a and b stand for any quantities whatever. If, however, a 5 and b=-3, the expression becomes 53, and we have a case of subtraction. Thus, symbolically, addition and subtraction are one and the same; for in the expression a+b we cannot know whether an addition or a subtraction is to be performed, until we know something about the quantities for which the letters stand. Moreover, any subtraction may be put into the form of an addition, and vice versa; for a b is the same as a +(-b), and a +b is the same as a -(—b). - Thus the subtraction of one expression from another may be expressed as an addition, by changing all the signs of the subtrahend. Hence the rule for Algebraic Subtraction: Change the signs of the subtrahend, and then perform addition. Ex. To subtract 3a-2b+3 from 6a+3b-4 is the same as to add -3a+2b-3 to 6a+3b-4; and the result is 3 a +5b-7. 25. Since a series of terms connected by + and — signs may be written as one connected by + signs only, such a series is called an Algebraic Sum. Thus 5-4 + 3−1 has 3 as its algebraic sum, and may be written 5 + (−4) + 3 + ( − 1). We are not justified in speaking of this as an Arithmetic Sum, for 4 and 1 have no meaning in pure arithmetic, and if we put it into the form 8−(+5), it becomes an arithmetic difference. 26. Symmetry. When the interchanging of two letters of an expression leaves the expression unchanged, except as to the order of the letters in a term, or the order of the terms in the expression, the expression is symmetrical in the two letters. Thus, by interchanging a and b in ac+ad - bc + bd — cd, ab ac + ad - cd, an expression the same as the former, except as to the order of the terms and of the letters in some of the terms. Hence the expression is symmetrical in a and b. It is readily seen that the expression is not symmetrical in c and d. An expression which is symmetrical in every pair of two letters is symmetrical in all the letters. Thus ab+be+ca and abc + abd + acd + bcd are each symmetrical in all the letters which they contain. Some special kinds of symmetry will be considered in a more advanced stage of the work. 27. When we know the letters which enter into an expression symmetrical in them all, and we are given a type-term, we can write the full expression by building up the form of the type in every possible way from the given letters, and taking the algebraic sum of all the terms so produced. Thus from the letters a, b, c : i. with type ab2 we have ab2+ bc2 + ca2 + ba2 + cb2 + ac2. ii. with type a (bc − a) : a (bc - a) + b (ca − b) + c (ab — c). iii. with type abc2: abc2+bca2+ cab2. iv. with type (b − c) (a2 — bc): (b − c) (a2 − bc) + (c − a) (b2 — ca) + (a − b) (c2 — ab). v. with a, b, c, and d, and type ab2: ab2 + ba2 + ac2 + ca2 + ad2 + da2 + bc2 + cb2 + bd2 +db2 + cd2 + dc2. a It will be noticed that in examples ii. and iii. each term is formed from the preceding one by changing a to b, b to c, and c to a. This is called a cyclic or circular substitution; for if we write the letters in a circle, as in the margin, we pass from one term to the next by commencing with a letter one step further around the circle until the whole is completed. Many substitutions, where three letters are concerned, are of this character, the distinctive feature being that we do not interchange any two letters without, at the same time, interchanging every two in circular order. A cyclic change with 4 letters and type ab gives ab+be+cd+ da. This lacks the terms ac and bd to make it completely symmetrical. In examples i. and iv. a circular change is not sufficient; for from the, type ab2 we must have a term ba2, which is not given by a mere circular substitution. In other words, we must interchange two letters without affecting the third. A little care and observation are all that are required in writing out such expressions from a given type. 28. The symbol (sigma), amongst other uses, is conveniently employed to denote expressions consisting of algebraic sums, written from a type. These are symmetrical in all the letters employed, and when written out are frequently of inconvenient length. The notation Σab, with three letters involved, stands for a2b+b2a + b2c + c2b + c2a + a2c. With four letters involved it stands for v. of the preceding article. (b-ca2-be) stands for iv. of the preceding article. As employed hereafter, 3 letters will be understood unless a different number is indicated, or in cases where misunderstanding is not possible. 4 will serve to indicate 4 letters, and generally Σ, to indicate n letters. This is known as the Sigma Notation. vii. (a)2 + (a + b − c)2 − 4 Za2 + 2 Zab. |