ALGEBRA. CHAPTER I. DEFINITIONS AND NOTATION. 1 Algebra is the science which treats of numbers. In Arithmetic the numbers are positive, and represented by figures; while in Algebra numbers may be positive or negative, real or imaginary, and represented by figures or letters. In most algebraic operations these numbers have an abstract signification; that is, they represent the measure, absolute or approximate, of some quantity referred to a unit of its own kind, arbitrarily selected as the standard. Through the annexation of the name of the measuring unit they become concrete. NOTE 1. Quantity, the subject of all mathematical investigation, has been variously used to mean not only anything which can be measured, as time, distance, etc., but also the abstract number arising from its measurement, and the concrete number representing its measurement, together with their various combinations. "It is any symbol which results from the rules of calculation." (De Morgan.) In this work the word "6 'quantity" will be used to mean the thing measured, or the concrete number representing its measurement. When the abstract number arising from its measurement is clearly meant, the word "number" will be used. 2. The first letters of the alphabet, a, b, c, etc., generally stand for what are called known numbers, that is, those whose values are given; and the last letters, x, y, z, etc., for unknown numbers, that is, those whose values are to be determined. Accented and subscript letters, as r', r'', etc. (read, r prime, r second, etc.), or r1, r2, etc. (read, r sub one, r sub two, etc.), are often used to represent numbers of the same kind, but of different values; as the terms of a continued proportion, or different rates of interest. Each letter may represent any number whatever, but throughout the same investigation the same letter is supposed to stand for the same number. 3. Addition is denoted by the sign + (read, plus), which indicates that the number following is to be added to that which precedes; thus, 3 + 2 (that is, 3 plus 2) signifies that 2 is to be added to 3. 4. Subtraction is denoted by the sign (read, minus), which indicates that the number following is to be subtracted from that which precedes; thus, 74 (that is, 7 minus 4) signifies that 4 is to be subtracted from 7. These signs also denote the character, or quality, of numbers. Thus, those before which the plus (+) sign stands are called positive numbers; and those before which the minus (-) sign stands, negative numbers. This extension of meaning will be fully explained later. 5. Multiplication is denoted by the sign X (read, into, times, or multiplied by), which indicates that the number preceding is to be multiplied by that which follows; thus, 6 x 5 (that is, 5 times 6) signifies that 6 is to be multiplied by 5. Between a figure and a letter, or between letters, the sign is usually omitted; thus, 6 a b is the same as 6 хахъ. Sometimes the sign × is replaced by a point above the line; thus, 8 64 is the same as 8 × 6 × 4. When figures are to be multiplied together, some sign for multiplication must always be employed; thus, 23 has already a meaning assigned to it,-namely, the sum of two tens and three units, or twenty-three, and hence caunot stand for 2 × 3, or 23. 6. Division is denoted by the sign÷ (read, divided by), which indicates that the number preceding is to be divided by that which follows; thus, 93 (that is, 9 divided by 3) signifies that 9 is to be divided by 3. Division is also indicated by the sign, and by the fractional form; thus, 9:3, 3, and 93, all have for their value 3. FACTORS AND POWERS. 7. A Factor is any one of several numbers, integral or fractional, which are to be multiplied together to form a product. 8. Any one or more of the factors which go to make up the product is called the Coefficient of the remaining factors. Thus, in 3 abc, 3 is the coefficient of a be, or be the coefficient of 3 a, or 3 a b the coefficient of c, and so on. A coefficient is called numerical, literal, or mixed, according as it is a numeral, a letter or letters, or a numeral and letters combined. The three cases above, taken in their order, illustrate this. By coefficient, the numerical coefficient, together with the sign of the expression, is usually meant. If no figure is expressed, a unit is understood; thus, x is the same as 1 x. 9. The Reciprocal of a number is a unit divided by that number; thus, the reciprocal of 5 is }; of x, 1. 10. A Power is the product obtained by repeating a number a given number of times. 11. An Index, or Exponent, is some number symbol, either positive or negative, integral or fractional, placed at the right, and a little above the number. If the index is positive and integral, it indicates how many times the number enters as a factor into the power. 2 × 2 × 2 × 2 = 16; read, 2 fourth power, or the fourth power of 2. Thus, 24 = a2 = axa; read, a second power, or a square. a3 = a × a × a; read, a third power, or a cube. a" = a xaxa. 12. A Root is one of the equal factors into which a number may be resolved. A root is indicated by the radical sign, the initial letter of the word radix. The root index is written at the top of the sign, though the index denoting the second, or square, root is generally omitted. Thus, V; read, the second root, or the square root, of x. ; read, the nth root of x. |