Compare, Order and Round Decimals

To compare two numbers is to decide which is greater, or whether they are equal. The symbol $<$ means is less than and $>$ means is greater than, so $2 < 7$ and $7 > 2$. To order numbers is to list them from least to greatest, or from greatest to least. Two facts about place value let you do both with decimals.

Zeros at the right end. A zero added at the right end of a decimal, after the decimal point, does not change its value: $0.4 = 0.40 = 0.400$, because $\frac{4}{10}$, $\frac{40}{100}$ and $\frac{400}{1000}$ are equivalent fractions (fractions with the same value). A zero in front of the $4$ is different. It moves the $4$ to the hundredths place, so $0.04$ is not $0.4$.

Smaller places. In a decimal that ends, the digits to the right of a place are together worth less than a $1$ in that place. For example, in a decimal that ends at the thousandths place, the hundredths and thousandths digits are worth at most $0.099$, or $99$ thousandths. One tenth is $100$ thousandths, so those digits are always worth less than one tenth. The same reasoning works at every place.

Now for the second job. To round a decimal to a place is to replace it with the closest number that ends at that place.

Round $4.37$ to the nearest tenth, which means to the tenths place. The numbers just below and just above it that end at the tenths place are $4.3$ and $4.4$. To see which is closer, count all three in hundredths: $4.30$ is $430$ hundredths, $4.37$ is $437$ hundredths and $4.40$ is $440$ hundredths. Now subtract the counts: $437 - 430 = 7$ and $440 - 437 = 3$. So $4.37$ is $7$ hundredths above $4.3$ but only $3$ hundredths below $4.4$. It is closer to $4.4$, so $4.37$ rounded to the nearest tenth is $4.4$.

The number exactly halfway between $4.3$ and $4.4$ is $4.35$. It is $435$ hundredths, which is $5$ hundredths from each of them, so neither one is closer. For a halfway number like this, the agreed rule for the positive decimals in this lesson is to round up to the greater of the two.

You do not need to count every time. To round a positive decimal that ends, to the ones place or a place to its right, use two digits:

1. Find the rounding digit, the digit in the place you are rounding to.
2. Find the deciding digit, the digit just to the right of the rounding digit in the number you were given.
3. If the deciding digit is $4$ or less, keep the rounding digit. If it is $5$ or more, add $1$ to the rounding digit (a $9$ becomes ten, as the last example shows).
4. Leave out every digit to the right of the rounding digit.

In $4.37$, keeping the rounding digit $3$ gives $4.3$ and adding $1$ gives $4.4$, the same two numbers as before. The deciding digit counts hundredths, and ten hundredths make $1$ tenth, so $5$ hundredths are exactly half of a tenth. That is why the change comes at $5$, and the same holds at every place. A deciding digit of $5$ or more puts the decimal at halfway or past it. A deciding digit of $4$ or less leaves it below halfway, because any digits after the deciding digit are together worth less than a $1$ in its place.

Here is the idea to keep: each place in a decimal is worth ten of the place to its right, so you work from the largest place. To compare or order positive decimals that end, match up the places and let the first place where the digits differ decide. To round one of these decimals, let the digit just to the right of the rounding place decide: $4$ or less keeps the rounding digit, and $5$ or more adds $1$ to it. Next, you will use the same matched-up places to add and subtract decimals.