Ordering 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 won the lottery and they offered you either \$1.9 million or \$4.1 million, you'd take the \$4.1 even though the .9 is higher than the .1, the 4 is the more important digit in that number!). So, whenever you want to compare decimals, line your numbers up along the decimal (as if you were adding up money), and compare from left to right.  

Also, remember, if a number does not include a decimal, it's a whole number.  The decimal goes just to the right.  So, 4 is 4.0 and 123 is 123.0.

So, let's say you were given the numbers: $1.35$, $.345$, $.98$, $4.61$, $.34$, $.982$ and you were asked to put them in order from greatest to least.

First, line them up by the decimal. 

$$\eqalign{1&.35\\&.345\\&.98\\4&.61\\&.34\\&.982}$$

Next, you want to compare the numbers that are the farthest to the left.  The number with the highest "far-left" digit, is the highest number.

$$\eqalign{1&.35\\&.345\\&.98\\4&.61&& \text{Greatest number}\\&.34\\&.982}$$

To find the next highest number, find the number that is next greatest in that far left digit.

$$\eqalign{1&.35&&\text{2nd greatest number}\\&.345\\&.98\\4&.61&& \text{Greatest number}\\&.34\\&.982}$$

Keep going until you run out of numbers with digits in that far-left position.  Then move to the next left-most position (in this case, the tenths place).  From now on, you will ignore the numbers that you have already sorted.

Now, $.98$ and $.982$ are tied at the tenths and the hundreds place.  So, you look to the thousandths place.  Sometimes, it's easier to add zeros into empty place values to help with comparison.

$$\eqalign{1&.35&&\text{2nd greatest number}\\&.345\\&.980\\4&.61&& \text{Greatest number}\\&.340\\&.982}$$

$2 \text{thousandths}$ is greater than $0 \text{thousandths}$ so, $.982$ is greater than $.98$.

$$\eqalign{1&.35&&\text{2nd greatest number}\\&.345\\&.980&& \text{4th greatest number}\\4&.61&& \text{Greatest number}\\&.340\\&.982&&\text{3rd greatest number}}$$

Now you are just left to compare $.345$ and $.340$ (as you can see, we already added that helpful zero).  They are the same at the tenths place and the hundredths place, but $.345$ is higher at the thousands place.

$$\eqalign{1&.35&&\text{2nd greatest number}\\&.345&&\text{5th greatest number}\\&.980&& \text{4th greatest number}\\4&.61&& \text{Greatest number}\\&.340&&\text{Least number}\\&.982&&\text{3rd greatest number}}$$

From there, you can order the numbers from greatest to least:

$$\eqalign{4&.61\\1&.35\\&.982\\&.98\\&.345\\&.34}$$

Order the numbers from least to greatest:

$$\eqalign{&.34\\&.345\\&.98\\&.982\\1&.35\\4&.61}$$

Find the least number: $.34$

Find the greatest number: $4.61$