An Elementary Treatise on Algebra

1. Algebra is a general method of computation, in which abstract quantities and their several relations are made the subject of calculation, by means of alphabetical letters and other signs.

2. The letters of the alphabet may be employed at pleasure for denoting any quantities, as algebraical symbols or abbreviations; but, in general, quantities whose values are known or determined, are expressed by the first letters, a, b, c, &c.; and unknown or undetermined quantities are denoted by the last or final ones, u, v, w, x, &c.

3. Quantities are equal when they are of the same magnitude. The abbreviation a=b implies that the quantity denoted by a is equal to the quantity denoted by b, and is read a equal to b; a>b or a greater than b, that the quantity a is greater than the quantity b; and ab or a less than b, that the quantity a is less than the quantity b.

4. Addition is the joining of magnitudes into one sum. The sign of addition is an erect cross; thus, a+b implies the sum of a and b, and is called a plus b, if a represent 8 and b, 4; then, a+b represents 12, or a+b=8+4=12.

5. Subtraction is the taking as much from one quantity as is equal to another. Subtraction is denoted by a single line; as a-b, or a minus b, which is the part of a remaining, when a part equal to b has been taken from it; if a=9, and b=5; a—b expresses 9 diminished by 5, which is equal to 4, or a-b-9-5=4.

6. Also, the difference of two quantities a and b ; when it is not known which of them is the greater, is represented by the sign; thus, ab is a-b, or b-a; and ab.signifies the sum or difference of a and b.

7. Multiplication is the adding together so many numbers or quantities equal to the multiplicand as there are units in the multiplier, into one sum called the product. Multiplication is expressed by an oblique cross, by a point, or by simple apposition; thus, a ×b, a. b, or ab, signifies the quantity denoted by a, is to be multiplied by the quantity denoted by b; if a=5 and b=7; then a Xb=5×7 = 35, or a . b=5.7=35, or ab=5×7=35.

Scholium. The multiplication of numbers cannot be expressed by simple apposition. A unit is a magnitude considered as a whole complete within itself. And a whole number is composed of units by continued additions; thus, one plus one composes two, 2+1=3, 3+1=4, &c.

8. Division is the subtraction of one quantity from another as often as it is contained in it; or the finding of that quotient, which, when multiplied by a given divisor, produces a given dividend."

Division is denoted by placing the dividend before the sign, and the divisor after it; thus a-÷b, implies that the quantity a is to be divided by the quantity b. Also, it is frequently denoted by placing

Βιβλία

An Elementary Treatise on Algebra: Theoretical and Practical