5.1 Introduction and Key Insights

As the prefix "dec" suggests, writing fractions in this way favors the number 10. If you really think about it, our standard way of writing whole numbers also favors the number 10. In fact, every whole number we have ever written down has secretly used 10s. For example, consider the whole number 135. The digits 1, 3, and 5 are arranged in a certain position or place that indicates how much they are worth. Their positions indicate "how many 10s" they are worth:

$135 = (1 \text{ one hundred}) + (3 \text{ tens }) +( 5 \text{ ones})$.

Each of these place values can be expressed using powers of 10 since $100 = 10^{2}$, $10 = 10^{1}$, and $1 = 10^{0}$. In other words:

$135 = (1\times 10^{2}) +( 3\times 10^{1}) +( 5 \times 10^{0})$.

If we add a tiny amount to the number 135, say $1/10$, we obtain the mixed number $135\,\frac{1}{10}$. Wouldn't it be nice to also express this number using 10s? That is what decimals do!

Indeed, you can think of decimal notation as extending our regular place value notation to the right. We can express the mixed number $135\,\frac{1}{10}$ using nothing but powers of 10s. But for the fractional amount, we need to use a negative power of 10, or you could call it a fraction with 10 in the denominator:

$135\,\frac{1}{10} =( 1\times 10^2) + (3\times 10^1) + (5 \times 10^0) + (1 \times 10^{-1})$.

A decimal point lets us rewrite this expression in a way that "hides" all of these powers of 10 and their various exponents:

$135\,\frac{1}{10} = 135.1$

The digit to the right of the decimal point is the "1/10's" place. Digits after that one occupy the "1/100's" and "1/1000's" places, respectively. For example:

$135.149 = 135 + \frac{1}{10} + \frac{4}{100} + \frac{9}{1000}$.

Before we perform any actual calculations with decimals, we want you to develop a solid understanding or familiarity with exactly how and what decimal notation expresses.