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Order decimals

Ordering decimals involves comparing nonnegative numbers through the thousandths place by aligning place values, interpreting each digit’s value, and using equivalent forms such as trailing zeros (for example, 0.5 = 0.500) or positions on a number line. It addresses the misconception that more digits always means a greater number and is limited to finite decimals through thousandths, not negative decimals or more advanced generalizations.

Detailed Explanation: Order decimals

To order decimals, compare their digits by place value:

  1. Line up the decimal points.
  2. Add trailing zeros so every number has the same number of decimal places.
  3. Compare digits from left to right: tenths, then hundredths, then thousandths.
  4. The first different digit tells which number is greater.

Example: Order these decimals from least to greatest:

0.7,0.605,0.65,0.560.7,\quad 0.605,\quad 0.65,\quad 0.56

First, write each decimal through the thousandths place by adding zeros when needed:

0.7000.6050.6500.560\begin{array}{r} 0.700\\ 0.605\\ 0.650\\ 0.560 \end{array}

Now compare the tenths digits:

  • 0.5600.560 has 55 tenths.
  • 0.6050.605 has 66 tenths.
  • 0.6500.650 has 66 tenths.
  • 0.7000.700 has 77 tenths.

So 0.5600.560 is least and 0.7000.700 is greatest. To compare 0.6050.605 and 0.6500.650, look at the hundredths digits:

  • 0.6050.605 has 00 hundredths.
  • 0.6500.650 has 55 hundredths.

Therefore,

0.56<0.605<0.65<0.7\boxed{0.56<0.605<0.65<0.7}

Notice that 0.6050.605 has more digits than 0.650.65, but it is not greater. Since 0.65=0.6500.65=0.650, comparing 0.6050.605 and 0.6500.650 shows that 0.6050.605 is smaller.

Learn by doing: Order decimals

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    Decimals Comparisons - Thousandths

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