Decimal Place Value

Learning goals

Read a decimal by its places, tenths, hundredths and thousandths
Explain why each place right of the point is a tenth of its neighbor
Write $3.42$ as the sum of its place values
Convert a terminating decimal to a fraction over ten, a hundred, or a thousand
Distinguish a trailing zero, which changes nothing, from a placeholder zero, which does
Compare two decimals place by place from the left, not by counting digits

Place value does not stop at the ones place

You already know how whole-number places work. Reading a number from right to left, each place is worth ten times the place before it: ones, tens, hundreds, thousands. Read that same row the other way, from left to right, and each place is worth one tenth of the place before it. A thousand becomes a hundred, a hundred becomes ten, and ten becomes one. Nothing forces that dividing-by-ten pattern to stop once you reach the ones place. Keep dividing by ten and you get brand-new, smaller places:

\ldots,\quad 100,\quad 10,\quad 1,\quad \tfrac{1}{10},\quad \tfrac{1}{100},\quad \tfrac{1}{1000},\quad \ldots

The decimal point marks the boundary between the two regions. Immediately to its left is the ones place; immediately to its right the places continue as tenths, hundredths, and thousandths. The point itself is not a place; it is just a marker. In the number $3.42$ , each digit sits in a place that fixes what it is worth:

3.42 \;=\; \underbrace{3}_{\text{ones}}\, . \,\underbrace{4}_{\text{tenths}}\;\underbrace{2}_{\text{hundredths}}

Notice that the place names to the right mirror the place names to the left, with “-ths” tacked on. The pairs run tens and tenths, hundreds and hundredths, thousands and thousandths.

The decimal point divides the columns. To its left the places grow by ten; to its right they shrink by ten. The digit 4 is worth 4/10 here because it sits one place right of the point.

Why each place is one tenth of the place to its left

The “one tenth” rule to the right of the point is not a separate rule that someone bolted on for decimals. It is the exact same bundling rule that builds the whole-number places, simply run in reverse.

Each decimal place is one tenth of the place to its left#

The whole-number places are built by bundling in tens: ten ones make one ten, ten tens make one hundred, ten hundreds make one thousand. Stepping one place to the left therefore multiplies the place value by $10$ . Going one place to the right is the inverse of that step. So a step to the right must divide the place value by $10$ .

Now apply that same one-step-right rule starting from the ones place, and do not stop. One step right of the ones place divides $1$ by $10$ , giving $\tfrac{1}{10}$ , the tenths place. One step further right divides $\tfrac{1}{10}$ by $10$ again, giving $\tfrac{1}{100}$ , the hundredths place. One more step gives $\tfrac{1}{1000}$ , the thousandths, and the pattern never breaks because nothing about the bundling rule changes when you cross the decimal point.

So every place, on either side of the point, is exactly ten times the place to its right. That relation reads equally well in the other direction. Every place is one tenth of the place to its left.

This is why the names mirror. The hundredths place is to the right of the ones place by the same two steps that carry the hundreds place to its left. So one is $100$ and the other is $\tfrac{1}{100}$ , a matched pair built by the same rule pointing in opposite directions.

A decimal is the sum of its place values

Reading $3.42$ by its places, it is three ones, four tenths, and two hundredths. Writing that out as a sum is called expanded form, and it lays bare exactly what each digit contributes:

3.42 = 3 + \frac{4}{10} + \frac{2}{100}.

This is the same move you already make for whole numbers, where $342 = 300 + 40 + 2$ . The only new part is that some of the places are fractions instead of whole numbers. As with whole numbers, a zero digit contributes nothing to the sum, but it is still essential. That zero holds a place open so the other digits land where they belong.

Worked example 1 Write $5.806$ in expanded form

Read off the place of each digit, working left to right from the point. Here $5$ is ones, $8$ is tenths, $0$ is hundredths, and $6$ is thousandths.

Write each digit times its place value, then drop the term that is zero:

5.806 = 5 + \frac{8}{10} + \frac{0}{100} + \frac{6}{1000} = 5 + \frac{8}{10} + \frac{6}{1000}.

The $0$ in the hundredths place adds nothing to the sum, but it cannot be erased. That zero is the placeholder that keeps the $6$ in the thousandths place. Remove it and the $6$ would slide left into the hundredths place, turning the number into the much larger $5.86$ .

The first place after the point is tenths, so a single decimal digit is simply that many tenths.

The decimal 0.3 is three tenths: a whole cut into ten equal parts, three of them shaded.

Every terminating decimal is a fraction over ten, a hundred, a thousand, and so on

Because each decimal place is built by dividing by ten, the digits after the point are tenths, hundredths, thousandths, and so on. That means any terminating decimal can be rewritten as a single fraction. Its bottom number is $10$ , or $100$ , or $1000$ , and the list keeps going one zero at a time.

Take $0.42$ . The $4$ is four tenths and the $2$ is two hundredths, so in expanded form $0.42 = \tfrac{4}{10} + \tfrac{2}{100}$ . Rewrite the tenths over a hundred, since $\tfrac{4}{10} = \tfrac{40}{100}$ , and add:

0.42 = \frac{40}{100} + \frac{2}{100} = \frac{42}{100}.

The two-digit string “42” lands directly on top of $100$ . Every place can be rewritten over the smallest place present, exactly as the tenths were rewritten over a hundred here. So the digits after the point always collect into a single count on top of that smallest place. The general rule is quick to apply: the number of digits after the point is the number of zeros in the denominator.

0.7 = \frac{7}{10}, \qquad 0.42 = \frac{42}{100}, \qquad 0.305 = \frac{305}{1000}.

Once a decimal is written as a fraction, you reduce it to lowest terms the same way you learned in the fractions chapter. Divide the top and bottom by their greatest common factor.

The grid below is one whole cut into $100$ equal squares. So a single square is one hundredth, and a complete row of ten squares is one tenth. Change how many squares are shaded, and the readout names the shaded amount as a count of squares. The readout also names that amount as a fraction over $100$ and as a decimal, all at the same time.

Shade exactly three complete rows and the same amount answers to two names: $\tfrac{30}{100} = 0.30$ counted in squares, and three tenths, $0.3$ , counted in whole rows. Then set the count to $5$ , and again to $50$ . Five squares is half of a single row, while fifty squares is half the whole grid. So $0.05$ and $0.5$ are nowhere near the same size, and the zero that separates them is doing real work.

Why $0.5$ is ten times $0.05$

Squares shaded

42 squares of 100 shaded. That is 4 complete rows and 2 spare squares. As a decimal, 0.42, which is 4 tenths and 2 hundredths.

A 10 by 10 grid is one whole. One square is one hundredth, and one complete row of ten squares is one tenth.

Whatever count you stop on, the picture is some number of complete rows plus some number of leftover squares. Those two counts are exactly what the tenths digit and the hundredths digit of the decimal record.

Worked example 2 Write $0.25$ as a fraction in lowest terms

There are two digits after the point, so the denominator is $100$ :

0.25 = \frac{25}{100}.

Now reduce. The greatest common factor of $25$ and $100$ is $25$ , so divide the top and bottom by it:

\frac{25}{100} = \frac{25 \div 25}{100 \div 25} = \frac{1}{4}.

So $0.25 = \frac{1}{4}$ , which matches the everyday fact that a quarter of a dollar is $25$ cents.

Check your understanding

In the decimal $7.36$ , what is the value of the digit $6$ ?

\tfrac{6}{100}

\tfrac{6}{10}

\tfrac{6}{1000}

6

Reading a decimal aloud

A decimal is said aloud by its places, and the last place is what names the whole reading. Out loud, $3.42$ is “three and forty-two hundredths.” The $3$ is the whole-number part, “and” stands for the point, and “forty-two” reads the two digits after the point as a whole number. The final word names the place where the last digit, the $2$ , lands. The name of that last place behaves like the denominator for the whole string of digits after the point. That behavior fits what you just saw: $0.42 = \tfrac{42}{100}$ , “forty-two hundredths.” So for any terminating decimal: read the whole-number part, then say “and” for the point. Read the rest as a whole number, and finish with the name of its last place.

Worked example 3 Write "two and seven hundredths" as a decimal

The whole-number part is two, so start with $2$ and a point. The fractional part is “seven hundredths,” and the word hundredths tells you the last digit must land in the hundredths place. The hundredths place is the second place after the point.

To put the $7$ in the second place, the tenths place needs a $0$ to hold its spot open:

\text{two and seven hundredths} = 2.07.

Writing $2.7$ instead would put the $7$ in the tenths place, naming seven tenths, which is ten times too big. The placeholder zero is doing real work here.

Trailing zeros change nothing, placeholder zeros change everything

A zero tacked onto the far right end of a decimal does not change its value: $0.3$ , $0.30$ , and $0.300$ are all the same number. This is genuinely useful, because it lets you give two decimals the same number of places. The decimal columns then line up when you compare or add them.

Why $0.3 = 0.30$ #

Write each one as a fraction. The single place in $0.3$ is tenths, so $0.3 = \tfrac{3}{10}$ . The two places in $0.30$ are hundredths, so $0.30 = \tfrac{30}{100}$ . Now reduce the second fraction by dividing the top and bottom by $10$ :

\frac{30}{100} = \frac{30 \div 10}{100 \div 10} = \frac{3}{10}.

That is the very same number as $0.3$ . Tacking on the extra zero only added $\tfrac{0}{100}$ , which is nothing at all, so the value cannot change. The same reasoning works for any number of trailing zeros: each new zero opens one more place to the right and then leaves it empty.

Be careful, though, because this is only true for zeros on the far right. A zero that sits between the point and a nonzero digit is not optional filler; it is a placeholder doing real work. In $0.05$ the zero pushes the $5$ out of the tenths place and into the hundredths place. That shift makes the $5$ one tenth as large as the $5$ in $0.5$ . Erase that zero and you change the value entirely.

Decimals also fill in all the points that sit between the whole numbers on the number line. The decimal $0.7$ , for instance, sits seven tenths of the way from $0$ to $1$ .

The decimal 0.7 sits seven tenths of the way from 0 to 1, filling in a point between the whole numbers.

Comparing decimals

To compare two decimals, line up their decimal points and compare digit by digit from the left, exactly as you compare whole numbers.

Worked example 4 Which is larger, $0.45$ or $0.5$ ?

Give them the same number of decimal places by padding the shorter one with a trailing zero. You already showed that this padding does not change the shorter number’s value:

0.45 \quad \text{versus} \quad 0.50.

Now compare from the left, starting at the tenths place: four tenths against five tenths. Since $4 < 5$ , the tenths place settles it:

0.45 < 0.50, \quad \text{so } 0.5 \text{ is larger.}

It does not matter that $0.45$ shows more digits. A single higher place to the left outweighs everything to its right.

The first place where the digits differ decides the whole comparison, exactly as the tenths place settled $0.45$ against $0.5$ . What does not work is counting how many digits each number has. More digits do not make a larger number, because the places to the right keep shrinking.

Padding the shorter decimal with trailing zeros is the reliable way to prepare any comparison. Add zeros until both numbers have the same number of places, and the columns line up on their own.

Check your understanding

Which symbol makes a true statement: $0.6 \;\square\; 0.58$ ?

>

<

=

D none of these

Worked example 5 Order $0.4$ , $0.39$ , and $0.405$ from least to greatest

Pad every decimal to three places so the columns line up:

0.400, \qquad 0.390, \qquad 0.405.

Compare the tenths place first. Two of them have $4$ tenths ( $0.400$ and $0.405$ ) and one has only $3$ tenths ( $0.390$ ), so $0.39$ is the smallest of the three. Now settle the remaining pair by moving right to the hundredths place: $0.400$ has $0$ hundredths while $0.405$ has $0$ as well. So move once more to the thousandths place, where $0$ meets $5$ . Since $0 < 5$ :

0.39 < 0.4 < 0.405.

The longest-looking number, $0.405$ , is the largest of the three only because its first differing place is larger, not because it has the most digits.

Common mistakes

Thinking more digits means larger. $0.45 < 0.5$ , because the tenths place is compared first. Line decimals up by the point and compare from the left, never by counting digits.
Treating a zero right after the point as nothing. $0.05$ is only one tenth of $0.5$ . That placeholder zero shifts the $5$ into the hundredths place and changes the value.
Mixing up the last-place name. $0.3$ is three tenths, but $0.03$ is three hundredths, ten times smaller. The final digit’s place names the fraction.
Confusing place with value. In $7.36$ the place of the $6$ is the hundredths, but its value is $\tfrac{6}{100}$ . The place is the column’s name; the value is what the digit contributes.
Erasing the wrong zeros. Trailing zeros on the right are harmless ( $0.3 = 0.30$ ). A zero that sits between the point and another digit is not harmless ( $0.3$ and $0.03$ are very different).

Key takeaways

Place value continues past the decimal point: tenths, hundredths, thousandths. Each of those is one tenth of the place to its left, because the same bundle-by-ten rule runs in reverse.
A decimal is the sum of its place values, for example $3.42 = 3 + \tfrac{4}{10} + \tfrac{2}{100}$ .
Every terminating decimal is a fraction over $10$ , $100$ , $1000$ , and so on (the digits after the point match the zeros in the denominator). Then reduce the fraction, for example $0.25 = \tfrac{1}{4}$ .
Read with “and” at the point, and the final place names the denominator, so $3.42$ is “three and forty-two hundredths.”
Trailing zeros on the right do not change the value ( $0.3 = 0.30$ ), but placeholder zeros that shift a digit’s place do.
Compare decimals by padding to equal length and reading place by place from the left, not by counting digits.