CONTENTS. PAGB I I. DEFINITIONS and Explanations of Signs II. Change of the order of Terms. Reduction of like Terms. Addition, Subtraction, use of Brackets IX. Equations of the First Degree X. Problems which lead to Simple Equations with one Un- XI. Simultaneous Equations of the First Degree with two Un- XII. Simultaneous Equations of the First Degree with more than XIII. Problems which lead to Simple Equations with more than XIV. Discussion of some Problems which lead to Simple Equa- XV. Anomalous forms which occur in the solution of Simple XXI. Equations which may be solved like Quadratics XXII. Theory of Quadratic Equations and Quadratic Expressions 187 . PAGE 2II . XXXIII. Mathematical Induction XXXIV. Permutations and Combinations XXXV. Binomial Theorem. Positive Integral Exponent XXXVI. Binomial Theorem. Any Exponent XXXIX. Exponential and Logarithmic Series XL. Convergency and Divergency of Series XLV: Reduction of a Quadratic Surd to a continued Fraction 369 XLVI. Indeterminate Equations of the First Degree XLVII. Indeterminate Equations of a Degree higher than the first . 385 ALGEBRA. I. DEFINITIONS AND EXPLANATIONS OF SIGNS. The method of reasoning about numbers, by means of letters which are employed to represent the numbers, and signs which are employed to represent their relations, is called Algebra. 2. Letters of the alphabet are used to represent numbers, which may be either known numbers, or numbers which have to be found and which are therefore called unknown numbers. It is usual to represent known numbers by the first letters of the alphabet a, b, c, &c., and unknown numbers by the last letters X, Y, z; this is not however a necessary rule, and so need not be strictly obeyed. Numbers may The word quantity is frequently used as synonymous with number. The word integer is often used instead of whole number. 3. The sign + signifies that the number to which it is prefixed is to be added. Thus a+b signifies that the number represented by b is to be added to the number represented by a. sent 9, and b represent 3, then a+b represents 12. The sign + is called the plus sign, and a +b is read thus “a plus b.” 4. The sign – signifies that the number to which it is prefixed is to be subtracted. Thus a – b signifies that the number represented by b is to be subtracted from the number represented by a. If a represent 9, and b represent 3, then a- - 6 represents 6. The sign – is called the minus sign, and a -6 is read thus “a minus b.” 5. The sign signifies that the numbers between which it nds are to be multiplied together. Thus a < b signifies that the T. A. 1 If a repreor ab. number represented by a is to be multiplied by the number represented by b. If a represent 9, and b represent 3, then a x b represents 27. The sign x is called the sign of multiplication, and a xb is read thus “ a into b.” Similarly a xb xc denotes the product of the numbers represented by a, b and c. It should be observed that the sign of multiplication is often omitted for the sake of brevity; thus ab is used instead of a xb, and has the same meaning; so abc is used for a xbx c. Sometimes a point is used instead of the sign x; thus a.b is used for a xb But the point is here superfluous, because, as we have said, ab is used instead of a xb. Nor is the point, nor the sign x, necessary between a number expressed in the ordinary way by a figure and a number represented by a letter; so that, for example, 3a is used instead of 3 x a, and has the same meaning. The sign of multiplication must not be omitted when numbers are expressed by figures in the ordinary way. Thus 45 cannot be used to express the product of 4 and 5, because a different meaning has already been appropriated to 45, namely forty-five. We must therefore express the product of 4 and 5 thus 4 x 5, or thus 4.5. To prevent any confusion between the point thus used as a sign of multiplication and the point as used in the notation for decimal fractions, it is advisable to write the latter higher up; thus 4.5 may be kept to denote 4 + 1o. -b sig 6. The sign - signifies that the number which precedes it is to be divided by the number which follows it. Thus a: nifies that the number represented by a is to be divided by the number represented by b. If a represent 9, and b represent 3, then a + b represents 3. The sign is called the sign of division, and a = b is read thus “a by b.” There is also another way of denoting that one number is to be divided by another; the dividend is placed over the divisor with a line between them. Thus 7 is used instead of a = b and has the same meaning. a 7. The sign signifies that the numbers between which it is placed are equal. Thus a = b signifies that the number repre sented by a is equal to the number represented by b, that is, a and b represent the same number. The sign = is called the sign of equality, and a=b is read thus “a equals b” or “a is equal to b." the sign 8. The difference of two numbers is sometimes denoted by i thus amb denotes the difference of the numbers denoted by a and b, and is equal to a-b or to b-a, according as a is greater than b or less than b. 9. The sign > denotes greater than, and the sign < denotes less than; thus a >b denotes that the number represented by a is greater than the number represented by b, and b<a denotes that the number represented by b is less than the number represented by a. Thus in both signs the opening of the angle is turned towards the greater number. 10. The sign .: denotes then or therefore; the sign :: denotes since or because, 11. When several numbers are to be taken collectively they are enclosed by brackets. Thus (a-+c) (d+e) signifies that the number represented by a-b+c is to be multiplied by the number represented by d + e. This may also be written thus (a – b+c) (d'+e). The use of the brackets will be seen by comparing the above expressions with (a-b+c) d+e; the latter denotes that the number represented by a-b+c is to be multiplied by d, and then e is to be added to the product. Sometimes instead of using brackets a line called a vinculum is drawn over the numbers which are to be taken collectively. b + cxd te is used with the same meaning as (a – b+c) * (d+e). Thus a 12. The letters of the alphabet, and the signs or marks which we have already introduced and explained, together with those which may occur hereafter, are called algebraical symbols, since they are used to represent the things about which we may be reasoning. Any collection of algebraical symbols is called an algebraical expression or a formula. |