1.8 The real numbers  (Page 6/13)

	0.31	0.308
Convert to fractions.	$\frac{31}{100}$	$\frac{308}{1000}$
We need a common denominator to compare them.
	$\frac{310}{1000}$	$\frac{308}{1000}$

Because 310>308, we know that $\frac{310}{1000}>\frac{308}{1000}.$ Therefore, 0.31>0.308.

Notice what we did in converting 0.31 to a fraction—we started with the fraction $\frac{31}{100}$ and ended with the equivalent fraction $\frac{310}{1000}.$ Converting $\frac{310}{1000}$ back to a decimal gives 0.310. So 0.31 is equivalent to 0.310. Writing zeros at the end of a decimal does not change its value!

We say 0.31 and 0.310 are equivalent decimals    .

We use equivalent decimals when we order decimals.

The steps we take to order decimals are summarized here.

When we order negative decimals, it is important to remember how to order negative integers. Recall that larger numbers are to the right on the number line. For example, because $\mathrm{-2}$ lies to the right of $\mathrm{-3}$ on the number line, we know that $\mathrm{-2}>\mathrm{-3}.$ Similarly, smaller numbers lie to the left on the number line. For example, because $\mathrm{-9}$ lies to the left of $\mathrm{-6}$ on the number line, we know that $\mathrm{-9}<\mathrm{-6}.$ See [link] .

If we zoomed in on the interval between 0 and $\mathrm{-1},$ as shown in [link] , we would see in the same way that $\mathrm{-0.2}>\mathrm{-0.3}\text{and}-0.9<\mathrm{-0.6}.$

Key concepts

• Square Root Notation
  $\sqrt{m}$ is read ‘the square root of m .’ If $m=n^2,$ then $\sqrt{m}=n,$ for $n\ge 0.$
• Order Decimals
  1. Write the numbers one under the other, lining up the decimal points.
  2. Check to see if both numbers have the same number of digits. If not, write zeros at the end of the one with fewer digits to make them match.
  3. Compare the numbers as if they were whole numbers.
  4. Order the numbers using the appropriate inequality sign.