CHAP. XIII.

OF SQUARE NUMBERS.

136. THE product of any number multiplied into itself is called a square; and in like manner, the number considered in relation to such product is termed a square root.

For example, if we multiply 12 by 12, the product 144 is a square, of which 12 is the square root.

The origin of this term is taken from geometry, where the contents of a square are found by multiplying its side by itself.

137. All square numbers are therefore found by multiplication, that is to say, by multiplying the square root into itself.

Thus 1 is the square of 1, because 1 multiplied into itself gives 1. In like manner 4 is the square of 2; and 9 is the square of 3; and 2 and 3 are the square roots of 4 and 9 respectively.

We shall consider in the first place the squares of the natural numbers, and we subjoin the following table in which will be found several numbers or roots in the first line, with their respective squares in the second.

Numbers 1 2 3 4 5 6 7 8 9 10 11 12 13 Square 1 4 9 16 25 36 49 64 81 100 121 144 169

138. It will here be easily seen, that in the squares thus arranged there is this singular property, viz., that if each of them be subtracted from that which immediately follows, the

remainders always increase by 2, and form the following series:

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, &c.

which is that of odd numbers.

139. The squares of fractions are in like manner found by multiplying each fraction into itself. For example,

TH

៖ > is

&c,

We have only therefore to divide the square of the numerator by the square of the denominator, and the fraction which expresses that division is the square of the given fraction. Thus is the square of, and reciprocally is the square root of ft.

140. If we wish to find the square of a mixed number, or a number composed of an integer and a fraction, we have only to reduce the mixed number to an improper fraction and take the square of it. For example, if it were required to find the square of 24, we must express that quantity by the fraction, and taking the square of that fraction we shall have 25 for the square of 24. So also the square of 3 or 4 is 18=10%.

The following is a table of the squares of the numbers between 3 and 4, increasing by one fourth.

From an inspection of this table, we may conclude, that if a root contain a fraction, its square will contain one also. Take 'for instance the root 1, the square is 4 or 21

141. In

141. In general, then, when the root is a, the square will be aa or a2; if the root be 2a the square is 4 a a or 4a2; from which we learn that if the root be doubled, the square will be four times as great. Thus also if the root be 3a the square will be 9 a2, &c.; but if the root be ab the square will be a abb or a2 b2, and if the root be abc, the square will be a2 b2 c2.

142. Thus when the root is composed of two or more factors, we multiply the squares of those factors together, and reciprocally if a square is composed of two or more factors, each of which is a square, we have only to multiply together the roots of those squares to obtain the complete root of the square proposed. Thus as 2304 is composed of the factors 4 × 16 × 56, the square root will be 2 × 4 × 6 = 48, which will be found to be the square root of 2304, for 48 × 48 = 2304 .

also pro

143. Let us therefore consider what is necessary to be observed with regard to the signs and It is evident, first, that if the root have the sign +, i. e. if it be a positive number, the square must necessarily be positive also, because + by + gives +, therefore the square of + a will be + a. But if the root be a negative number, as a, the square must still be positive, for it has already been shown that by duces + ; therefore the square of- a is also + a2. Hence it follows, that +a2 is the square of the two roots +a and and consequently that every square number must have two roots, the one positive and the other negative. The square root of 25 for instance, is both + 5 and - 5, for - 5 × - 5, and + 5 x + 5 equally produce 25.

CHAP. XIV.

OF SQUARE ROOTS AND IRRATIONAL NUMBERS RESULTING FROM THEM.

144. WHAT has been said in the preceding chapter amounts principally to this, that the square root of a given number is a number such that its square is equal to the number proposed, and that we may prefix to such roots either a positive or a negative sign.

145. Thus, when a square number is given, and we can retain in our memory a sufficient number of square numbers, it is easy to find the root of that number. If, for example, the number proposed be 196, we know that its square root is 14.

We are enabled to consider fractions with the same facility. For example, it is evident that is the square root of 25, for we have only to take the roots of the numerator and denominator.

4

If the number proposed be a mixed number, as 124, by reducing it to an improper fraction, it becomes 9, and we see at once that or 34 is the square root of 124.

146. But when the number proposed is not a square number, as for example 12, we are no longer enabled actually to extract its square root, or in other words to find such a number, as when multiplied into itself, will produce exactly 12. All that we know is, that the square root of 12 must be a number greater than 3, for 3 x 3 only produces 9; and less than 4, for 4×4 produces 16. We know also that the root must be less than 34, for we have already seen that the square of 3 is 124. We may however approach this root somewhat nearer, by comparing it with 3, for the square of 3, or of 3,