Example 1.18. Repeated Addition.
In this case, there are 5 terms of 2 resulting in 10.
\begin{equation*}
\underbrace{2+2+2+2+2}_{\text{$5$ terms of $2$}} = \underbrace{5\cdot 2}_{\text{A multiple of $2$}} = 10.
\end{equation*}
Section 1.5 Terms versus Factors
This section emphasizes the importance of correctly identifying the parts of an algebraic expression. Carefully use the words "term" and "factor" accurately to avoid many pitfalls in algebra.
Repeated addition of a number gives a multiple of that number. Each occurrence of the number is a term in the sum.
Repeated multiplication of a number gives a power of that number. Each occurrence of the number is a factor in the product.
Example 1.19. Repeated Multiplication.
In this case, there are 5 factors of 2 resulting in 32.
\begin{equation*}
\underbrace{2\cdot 2\cdot 2\cdot 2\cdot 2}_{\text{$5$ factors of $2$}} = \underbrace{2^5}_{\text{a power of $2$}} = 32.
\end{equation*}
The number of factors, \(5\text{,}\) in the power \(2^5\) is called its exponent. The number \(2\) in the power is called the base of the power.
Definition 1.20. Terms and Factors.
Whenever we add (or subtract) real numbers
\begin{equation*}
a+b+c+\cdots
\end{equation*}
the individual numbers \(a,b,c,\ldots\) are called terms in the sum. Whenever we multiply (or divide) real numbers
\begin{equation*}
a\times b\times c\times \cdots
\end{equation*}
the numbers \(a,b,c,\ldots\) are called factors in the product. An expression of the form \(a^b\) is called a power with base \(a\) and exponent \(b\text{.}\)
Example 1.21. Canceling.
It’s crucial that you distinguish between terms in a sum versus factors in a product. For instance, you may cancel the common factors of \(2\) in the ratio:
\begin{equation*}
\frac{3\cdot 2}{2} = 3.
\end{equation*}
However, you must never cancel common terms in a ratio. For instance, do not cancel the common terms of \(2\) in the ratio \(\frac{3+2}{2}\text{.}\) Instead,
\begin{equation*}
\frac{3+2}{2} = \frac{5}{2}.
\end{equation*}
If \(a\) is a real number, then \(-a\) is its opposite value and appears at the same, but opposite location on the real line. I recommend you resist the urge to refer to \(-a\) as “negative \(a\)” as it is unclear if the result is positive or negative. For instance, the opposite of \(-4\) is \(-(-4) = 4\text{,}\) a positive integer. Adding a number with its opposite results in zero:
\begin{equation*}
a + (-a) = 0.
\end{equation*}
It is for this reason that \(-a\) is sometimes called the additive inverse of \(a\text{.}\)
If \(a\) is non-zero (see Warning 1.2 ) real number, then \(1/a\) is called its reciprocal. Multiplying \(a\) by its reciprocal results in one:
\begin{equation*}
a \times \frac{1}{a} = 1.
\end{equation*}
It is for this reason that \(1/a\) is sometimes called the multiplicative inverse of \(a\text{.}\) For instance, \(2\) is the multiplicative inverse of \(1/2\) because \(2\times (1/2) = 1\text{.}\)
Example 1.22. Inverses of Rational Numbers.
The reciprocal of a rational number interchanges the numerator and denominator:
\begin{equation*}
\frac{1}{\frac{2}{3}} = \frac{1}{2/3} = \frac{3}{2}.
\end{equation*}
The opposite of a rational number can be formed with the opposite of the numerator or denominator. That is, the following are all equal:
\begin{equation*}
- \left(\frac{2}{3}\right) = -\frac{2}{3} = \frac{-2}{3} = \frac{2}{-3}.
\end{equation*}
Checkpoint 1.23.
- What real numbers are factors in the expression \((2+3)\times 4\text{?}\)
- What real numbers are terms in the expression \(2\times 4+3\times 4\text{?}\)
- The word "inverse" appears frequently in mathematics. What were the two kinds of inverses you saw above? Give an example of each.
