Comparing Decimals

A decimal is a period in a number that separates whole numbers from fractions of numbers.  In the number $3.21$, $3$ represents 3 wholes (the portion of the number on the left side of the decimal represents whole numbers) and $.21$ represents 21 hundredths (the numbers on the right side of the decimals represent a portion of a whole).

To understand and work with decimals, you must understand place value and the fact that every digit in a number has a place value and represents a certain quantity (see Place Value lesson for more). Also see the lesson on Writing Numerals as Numbers (Decimals) for a detailed lesson on how to look at decimal place value and name numbers with decimals. 

Once you know place value, it’s easy to compare numbers with and without decimals.  Remember, the most important digit in any number is the digit farthest to the left. (Think about it, if you went on a game show and they said that they would give you \$111.11 but you could swap one digit for a nine, you'd swap the hundreds digit in order to get \$911.11, right?)  So, line your numbers up along the decimal (as if you were adding up money), and compare from left to right.

Since the place value of a digit is what determines how much any numeral in that place is worth, it's important to think about place value when you work with numbers with decimals. 

Again, think about it. Would you prefer .9 or 1?  The digit 9 is higher, but because it is behind the decimal, it is only worth 9 tenths or $\dfrac{9}{10}$ while the 1 is in the ones place, so it is worth one whole.  1 is definitely greater than .9.  If you think about this in terms of money, 1=1.=1.00=$1 while .9=.90=90 cents.  Which is more?  A dollar or 90 cents?

So, when you want to order or compare decimals, you always line them up by their decimals, so when you compare digits, you care comparing digits of equivalent place value!

Example:

Which number is greater: $1.5673$ or $1.589732$?

Line the numbers up by their decimal points:

$$\eqalign{1&.5673\\1&.589732}$$

When the numbers are lined up like this, you can see that:

• Their ones digits are the same (1).
• Their tenths digits are the same (5).
• The hundredths digit of the second number (8) is greater than the hundredths digit of the first number (6).

Stop there.  The second number is greater.  

What about the other digits?  They don't matter.  The place value of every digit farther to the right is smaller than the hundredths digit. The fact that the hundredths digit of the second number is greater means that none of the other digits matter in this comparison.

$1.5673<1.589732$

You can use the same process if you want to order a whole set of numbers.  Line them up by their decimals and it's easy to tell which numbers are greater and less.

Example:

Put the numbers in order from least to greatest: $12.3$, $52.13$, $8.99$, $4.12$, $4.13$, $4.44$, $12.33$.

Line the numbers up by their decimal points:

$$\eqalign{12&.3\\52&.13\\8&.99\\4&.12\\4&.13\\4&.44\\12&.33}$$

When the numbers are lined up like this, you can see that:

• One three numbers have a digit in the tens place.  Those are are largest numbers.

$$\eqalign{\underline{1}2&.3\\\underline{5}2&.13\\8&.99\\4&.12\\4&.13\\4&.44\\\underline{1}2&.33}$$

Of those, 52 is the greatest (5 in the tens place compared to 1).

12.3 and 12.33 seem very close, but when you line them up by their decimals (and fill in with zeros to make them the same size), you can see where one number has a higher digit:

$$\eqalign{12&.3\underline{0}\\12&.3\underline{3}}$$

12.33 is greater than 12.30 by 3 hundredths.

So far we know:

$$\eqalign{52&.13\\12&.33\\12&.3}$$

• Next, look at the ones digits of the remaining numbers:

$$\eqalign{12&.3\\52&.13\\\underline{8}&.99\\\underline{4}&.12\\\underline{4}&.13\\\underline{4}&.44\\12&.33}$$

The 8 is the highest digit in the ones column, so that's the next greatest number.

Next we have to compare all of the numbers with a 4 in the ones place.  Let's look at the tenths place:

$$\eqalign{4&.\underline{1}2\\4&.\underline{1}3\\4&.\underline{4}4}$$

The 4 is the highest digit in the tenths place, so that's the next greatest number.

To decide which of the last two is higher, we need to look at the hundredths place:

$$\eqalign{4&.1\underline{2}\\4&.1\underline{3}}$$

4.13 has a 3 in the hundredths place, compared to 4.12's 2.  4.13 is greater.

You now how have final order!

$$\eqalign{52&.13\\12&.33\\12&.3\\8&.99\\4&.44\\4&.13\\4&.12}$$

Overall, it's not difficult to order decimals, just line them up and remember that the value of the place values decreases from left to right, so always compare from left to right.