Decimal Place Value

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:

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:

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.

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

Ten tenths make one whole, the same way ten ones make one ten. So one tenth is one whole split into ten equal parts, and one hundredth is one tenth split into ten equal parts again. The “one tenth” rule to the right of the point is not a separate rule for decimals. It is the exact same bundling rule that builds the whole-number places, simply run in reverse.

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:

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 when a zero sits before a later nonzero digit, it is still essential: it holds that digit’s place open so the digit lands where it belongs.

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

Decimals can also name 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 $1$.

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:

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.

The same rule works when the whole-number part is not zero. Keep it in front and write the part after the point as a fraction over the matching power of ten, or fold everything into one fraction:

A fraction read straight off the digits this way is not always in lowest terms; the lesson on converting between fractions and decimals, later in this chapter, covers reducing it in full.

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.

Whatever count from $0$ to $99$ you stop on, the picture is some number of complete rows plus some number of leftover squares smaller than a row. Those two counts are exactly what the tenths digit and the hundredths digit of the decimal record. Shade all $100$ squares and the ten rows regroup into the single whole $1.00$, so that one count sits outside the pattern rather than breaking it.

Reading a decimal aloud

A decimal that ends is said aloud by its places, and its 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.

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.

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.

Comparing decimals

To compare two decimals that end, like the ones in this lesson, line up their decimal points and compare digit by digit from the left, exactly as you compare whole numbers.

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 a comparison between two decimals that end. Add zeros until both numbers have the same number of places, and the columns line up on their own.

Common mistakes