Examples may be obtained from (143) and (150).
SECTION VII.
ON THE EXTRACTION OF
THE SQUARE ROOT.
156. We have already remarked (66), that a number multiplied by itself produces what is called the square of that number. Thus, 169, or 13 × 13, is the square of 13. Conversely, 13 is called the square root of 169, and 5 is the square root of 25; and any number is the square root of another, which when multiplied by itself will produce that other. The square root is signified by the sign
√ or √ ; thus, √25 means the square root of 25, or 5; √16 + 9
means the square root of 16 + 9, and is 5, and must not be confounded with √16 + √9, which is 4 + 3, or 7.
157. The following equations are evident from the definition:
√a × √a = a
√aa = a
√ab × √ab = ab
(√a × √b) × (√a × √b) = √a × √a × √b × √b = ab
whence
√a × √b = √ab
158. It does not follow that a number has a square root because it has a square; thus, though 5 can be multiplied by itself, there is no number which multiplied by itself will produce 5. It is proved in algebra, that no fraction[22] multiplied by itself can produce a whole number, which may be found true in any number of instances; therefore 5 has neither a whole nor a fractional square root; that is, it has no square root at all. Nevertheless, there are methods of finding fractions whose squares shall be as near to 5 as we please, though not exactly equal to it. One of these methods gives ¹⁵¹²⁷/₆₇₆₅, whose square, viz.