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Comparing Decimal Numbers

Is the chocolate bar for $ 0.89 or the one for $ 0.95 cheaper? To find this out, you need to compare two decimal numbers.

There are two different strategies here. One is the comparison of decimal places and the other is the comparison by shifting the decimal point.

I. Comparison of decimal places

  1. First, we look at the "pre-decimal numbers". These are the numbers that come before the decimal point. If they differ, the number whose pre-decimal point number is already larger is the larger one. In that case, you are done. If both pre-decimal numbers are identical, then you need to go to step two.

  2. Next, compare the tenths digits. The tenth digit is the first number that comes after the decimal point. If a number has a tenth digit which is larger then the tenth digit of the other number, then the total number is larger then the other number. If both tenth digits are the same, go to step three.

  3. Next, compare the hundredths digits of the two numbers, i.e. the respective second digits after the decimal point. Proceed as in step two for the comparison. If this is also the same, then compare the thousandths digits, then the ten-thousandths digits, and so on.

  4. If all digits are identical, then both numbers are the same.

Examples

Compare each pair of numbers with each other. Use the above procedure to decide which of the two numbers is greater.

  • 23.6 and 24.6

Here the two pre-decimal numbers are 23 and 24. Since 24 is greater than 23, you already know that 24.6 is greater than 23.6.

  • 23.6 and 23.7

Here, the pre-decimal number is 23 for both numbers. So you have to go to step 2, i.e. comparing the tenths digits. These are 6 and 7, respectively. Since 7 is greater than 6, you finally know that 23.7 is greater than 23.6.

  • 23.026 and 23.0265

Now it gets a little more difficult: The pre-decimal number is 23 in both numbers. Also the tenths digits agree (both are 0), and so do the hundredths digits (both are 2) and the thousandths digits (both are 6). Now, in 23.026 there is no further digit coming. But remember that adding zeros in the end of a number behind the decimal point does not change the value of the number (e.g., 1=1.0=1.00).

So you may as well compare 23.0260 and 23.0265

If you now look at the ten-thousandths digit, you finally see that 5 is greater than 0. Thus, 23.0265 is greater than 23.0260=23.026.

Common mistakes

  • Omitting the decimal point

  • Pretending, there was a seperate number after the decimal point

II. Comparison by shift of the decimal point

If we compare two numbers with each other, the relation (greater/smaller) does not change if we multiply both numbers by 10.

For example, instead of comparing 0.3 with 0.4, we could also compare 3 with 4.

Now, multiplying by 10 just shifts the decimal point one place to the right. We may do this arbitrarily often. That is, we shift the decimal point in both numbers simultaneously to the right, until both numbers are natural numbers:

Multiply two numbers to be compared by 10 (i.e., simultaneously shift the decimal place to the right) until there are no digits beyond the decimal point.

Then, compare the two resulting natural numbers.

Examples

  • 0.035 and 0.056

Multiply by 10:

You obtain 0.35 and 0.56

Multiply by 10:

You obtain 3.5 and 5.6

Multiply by 10:

You obtain 35 and 56

35 is smaller than 56

Accordingly, 0.035 is smaller than 0.056.

Note: With a bit of practice, you may also directly multiply by 1000 instead of multiplying by 10 three times.

Common mistakes


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