=Multiplication.=--An operation which consists in repeating a number called =multiplicand= (M) as many times as there are units in another column called =multiplier= (m); the result is called the =product= (p) of the numbers, and the numbers themselves are called =factors= of the product. This definition may be extended to the case where the factors are not whole numbers. =Sign of Multiplication.=--The sign of multiplication is the oblique or St. Andrew’s cross ×, called multiplied by, and placed between the factors written one after the other. Thus: 35 × 7 = p; 35 × 7 = 245. Generally M × m = p. =To Prove a Multiplication.=--Multiplication may be proved by a second multiplication in which the factors are inverted. This is the surest but the longest method. =Another Proof of the Multiplication.=--Find the residue of the multiplicand and multiplier. Multiply them and find the residue of their product; this is equal to the residue of the product of the multiplication. 64327 4 Residue of the multiplicand. 781 7 Residue of the multiplier. -------- -- 28 1 Residue of the product of the residues 64327 514616 450289 -------- 50239387 1 Residue of the product of multiplication. =Proof Not Absolute.=--Practically a proof is not absolute, because an error may be committed in its use, and also it may not work well in all cases. =Power of a Number.=--When the factors of a product are equal, the product is called a power of the factor. =Square of a Number.=--A power is a square when it is the product of two (2) equal factors, as 7 × 7 = 49, in which 49 is the square of 7. The term square is derived from the fact that the area of a square is obtained by multiplying the length of its side by itself, or taking it twice as a factor. =Cube of a Number.=--A power is a cube when it is the product of three (3) equal factors, as 5 × 5 × 5 = 125, in which 125 is the cube of 5. The term cube is derived from the fact that the volume of a cube is obtained by multiplying the length of its side by itself and again by itself, or by taking it three times as a factor. A product, for instance, of 4, 9, etc., equal factors would be called the 4th or the 9th, etc., power of that number. Division. =Division.=--An operation by means of which we find one of two factors of a product when that product and the other factor are given. The given product is called =Dividend= (D) of the division; the known factor is called the =Divisor= (d), and the unknown factor which is sought is called =Quotient= (q). We know that a quotient is seldom exact and that there is generally a =Remainder= (r) or =Residue=. =Sign of Division.=--The sign of division is a small dash with a point above and one below ÷; it is read divided by, is placed after the dividend, and is followed by the divisor. For instance, to indicate the division of 72 by 8, which we know gives the quotient 9, we write 72 ÷ 8 = 9; generally D ÷ d = q.