Factors. Coefficients. Powers.

17. Factors.

When a number consists of the product of two or more numbers, each of these numbers is called a factor of the product.

The sign $\times$ is generally omitted between a figure and a letter, or between letters; thus, instead of $63 \times a \times b$ , we write $63 a b$ ; instead of $a \times b \times c$ , we write $a b c$ .

The expression $a b c$ must not be confounded with $a + b + c$ . $a b c$ is a product; $a + b + c$ is a sum.

When a sign of operation is omitted in the notation of Arithmetic, it is always the sign of addition; but when a sign of operation is omitted in the notation of Algebra, it is always the sign of multiplication. Thus, $456$ means $400 + 50 + 6$ , but $4 a b$ means $4 \times a \times b$ .

Factors expressed by letters are called literal factors; factors expressed by figures are called numerical factors.

If one factor of a product is equal to $0$ , the product is equal to $0$ , whatever the values of the other factors. Such a factor is called a zero factor.

20. Coefficients.

A known factor of a product which is prefixed to another factor, to show the number of times that factor is taken, is called a coefficient.

Thus, in $7 c$ , $7$ is the coefficient of $c$ ; in $7 a x$ , $7$ is the coefficient of $a x$ , or, if $a$ is known, $7 a$ is the coefficient of $x$ .

By coefficient, we generally mean the numerical coefficient with its sign. If no numerical coefficient is written, $1$ is understood. Thus, $a x$ means the same as $1 a x$ .

21. Powers and Roots.

A product consisting of two or more equal factors is called a power of that factor, and one of the equal factors is called a root of the number.

Thus, $9 = 3 \times 3$ ; that is, $9$ is a power of $3$ , and $3$ is a root of $9$ .

22. Indices or Exponents.

An index or exponent is a number-symbol written at the right of, and a little above, a number.

If the index is a whole number, it shows the number of times the given number is taken as a factor.