0.4 Decimals

Learning Objectives

By the end of this section, you will be able to:

Almost every number you meet outside a math classroom is a decimal. Prices, batting averages, gas mileage, race times, interest rates — all decimals. They are so familiar that it is easy to treat them as their own species of number, with their own odd rules that you memorize and apply by reflex.

They are not a separate species. A decimal is a fraction whose denominator is a power of ten, written in a compressed form that reuses the place-value system from Section 0.1 instead of a fraction bar. Every rule in this section comes from that one fact. Line up the points to add, count the places to multiply, shift both points to divide — each of those is a consequence, not a separate law. Once you can see the fraction hiding behind the notation, you can rebuild a rule you have forgotten instead of guessing at it.

Students go wrong in two predictable places. The first is comparing: \(0.7\) and \(0.68\) look like a two-digit number losing to a three-digit number, and a habit built on whole numbers says the longer one wins. The second is bookkeeping in multiplication and division, where the digits come out right and the decimal point lands one spot off, turning $34.50 into $3.45. Both errors go away the same way — by thinking about what each place is worth instead of counting digits. The section ends with a bigger question that decimal notation finally lets us answer: exactly which numbers are out there, and how can we tell them apart?

0.4.1 Decimal Notation and Rounding

In Section 0.1 the place values ran ones, tens, hundreds, thousands, with each place worth ten times the place to its right. Nothing stops that pattern from running the other way. Moving right divides by ten each time, and the decimal point marks the spot where the whole-number part ends.

Here are the places on both sides of the point:

Two things in that table are worth naming. There is no "oneths" place, because the ones place is the pivot the pattern reflects around, so tenths sits opposite tens rather than opposite ones. And the "-th" ending is doing real work: ten thousand is a large number, but one ten-thousandth is a tiny one. Two letters are the entire difference between \(10{,}000\) and \(0.0001\).

Expanding a decimal makes the structure visible:

That zero in the hundredths place is a placeholder, doing the same job the zero in 4,072 did back in Section 0.1. Delete it and \(36.407\) collapses to \(36.47\), which is a different number.

Reading decimals aloud

There is a four-step routine for turning a decimal into words:

So \(36.407\) is "thirty-six and four hundred seven thousandths." The last digit sits in the thousandths place, so the whole fractional part gets counted in thousandths.

Here is the routine applied to several numbers at once. Watch what one extra placeholder zero does to the name.

The \(7.4\) and \(7.04\) rows differ by one syllable when you say them and by one zero when you write them, and that zero costs the 4 a factor of ten: it is worth four tenths in the first number and only four hundredths in the second.

Going the other way, from words to standard form, the named place tells you where the last digit lands, and you fill any gaps with zeros. "Five and eighty-one thousandths" puts the 1 in the thousandths place, so the digits 81 take up hundredths and thousandths and the tenths place needs a placeholder: \(5.081\). Writing \(5.81\) instead is a common slip, and it multiplies the fractional part by ten.

Comparing decimals

With whole numbers, more digits always means a bigger number. That rule is false for the fractional part of a decimal, and the false version is the most stubborn misconception in this section.

Compare \(0.7\) and \(0.68\). Counting digits suggests \(0.68\) should win. Place value says otherwise: \(0.7\) has 7 tenths while \(0.68\) has only 6 tenths, and no pile of hundredths can make up a whole missing tenth. So \(0.7 > 0.68\).

The reliable method is to compare place by place from the left and stop at the first place where the digits differ. Padding with trailing zeros makes the comparison easy to see, and padding is safe because adding zeros to the right of the last decimal digit does not change the value: \(0.7 = 0.70 = 0.700\), since \(\dfrac{7}{10} = \dfrac{70}{100} = \dfrac{700}{1{,}000}\).

That last row brings in Section 0.2. Among positives, \(0.40\) is larger than \(0.35\). Making both negative flips their order on the number line, so \(-0.4\) sits to the left of \(-0.35\) and is therefore the smaller of the two. The rule from Section 0.2 still runs everything: a number is greater than whatever sits to its left, no matter how many decimal places it carries.

Rounding

Rounding a decimal uses the same four steps you used on whole numbers, with one change at the end.

Step 4 is the change. When you rounded 6,473 to the nearest hundred, the zeros in 6,500 were required, because those places carry value. To the right of the decimal point a trailing zero carries no value, so you simply drop it.

Round \(8.4763\) to four different places and the answers land all over the map:

Every one of those answers came from the original \(8.4763\), and that matters more than it looks. If you round to hundredths first and then round that result to tenths, you are rounding a rounded number, and the small error from the first step gets carried into the second. Sometimes you get lucky and the two routes agree; often you do not. Start over from the original number every time.

Rounding money

Money is the rounding job you will do most often, and it comes with a fixed convention: round to the nearest cent, which is the hundredths place. Amounts with more than two decimal places show up constantly. Sales tax on a purchase, a per-unit price at the grocery store, an interest calculation, and the price of gasoline are all computed to three, four, or five places before anyone rounds them.

Carry the exact value all the way through a calculation and round only at the very end. Rounding partway through is the same chained-rounding mistake as before, and on a long receipt those small errors add up into a total that visibly disagrees with the register.

0.4.2 Operations with Decimals

You can only add quantities of the same kind. Tenths add to tenths and hundredths add to hundredths, but tenths never add directly to hundredths, any more than 3 hours can be added straight to 8 minutes. Lining up the decimal points is exactly what forces matching places into the same column.

Definition 0.4.2 — Appending zeros re-slices the same amount into finer pieces without moving its edge.

Subtraction works the same way. To subtract \(6.28\) from \(40\), remember that a whole number has an invisible decimal point at its right end, so \(40\) is \(40.00\) and the columns line up: \(40.00 - 6.28 = 33.72\).

The mistake this whole procedure exists to prevent is right-aligning the digits instead of the points. Right-aligning \(12.4\) and \(3.75\) would stack the 4 over the 5, adding tenths to hundredths, and the answer would be nonsense. Line up the points, not the edges.

Multiplying: count the decimal places

Multiplication does not need the points lined up. Multiply as though the points were not there, then place the point in the answer so that

Multiply \(1.6 \times 0.45\). Ignore the points and compute \(16 \times 45 = 720\). The factors carry 1 and 2 decimal places, for a total of 3, so the product needs three decimal places: \(0.720\), which we write as \(0.72\).

That rule looks like arbitrary bookkeeping until you rewrite the decimals as fractions, and then it turns into the fraction multiplication from Section 0.3:

Multiplying the denominators takes \(10\) times \(100\) and gives \(1{,}000\), and the number of zeros in that denominator is exactly the number of decimal places. One place plus two places gives three places because \(10^1 \cdot 10^2 = 10^3\). The counting rule is a statement about exponents wearing a disguise. It also explains the placeholder in \(0.021\): the digits 21 have to be pushed out to the third decimal place, so the tenths place needs a zero to hold it open.

The fourth row deserves a second look, because \(0.5 \times 0.5 = 0.25\) is smaller than either factor. Multiplying by a number between 0 and 1 shrinks things. That runs against an instinct built entirely on whole numbers, where multiplying never made anything smaller, and the instinct has to go before it causes trouble in Chapter 2.

Dividing by a whole number

Put the decimal point in the quotient directly above the point in the dividend, then divide the way you would with whole numbers:

If the division does not come out even, append trailing zeros to the dividend and keep going, since \(19.5\) and \(19.500\) are the same number. Check any division by multiplying back: \(6 \times 3.25 = 19.5\).

Dividing by a decimal

A decimal divisor is awkward, so we get rid of it. Write the division as a fraction and multiply the top and the bottom by whatever power of ten turns the divisor into a whole number. Multiplying both parts by the same thing leaves the value unchanged — that is the equivalent-fractions idea from Section 0.3:

In practice people describe this as "move both decimal points the same number of places to the right, enough places to make the divisor whole." The word doing the work is both. Moving only the divisor's point changes the answer by a factor of ten.

The third row shows the other instinct that needs retraining. Dividing 7 by 0.25 gives 28, which is much larger than the number we started with. That is not a mistake. Dividing by a number between 0 and 1 makes things bigger, because the question "how many quarter-units fit inside 7 units?" has a large answer.

Multiplying and dividing by powers of ten

This case gets its own rule because it comes up constantly and takes no computation at all. Multiplying by 10 bumps every digit up one place, which is the same as sliding the decimal point one place to the right. Dividing by 10 slides it one place to the left.

Count the zeros in the power of ten and that is how many places the point moves, appending placeholder zeros whenever the digits run out. In \(4.62 \times 1{,}000\) the point has to move three places, but only two digits follow it, so a zero fills the ones place: \(4{,}620\). This shortcut is behind every metric-unit conversion you will ever do, and Section 1.3 comes back to it as a statement about exponents.

Nothing about the order of operations from Section 0.1 changes when decimals show up. Grouping symbols still go first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. The only difference is that each individual computation now needs the decimal point tracked, and the point is where the errors hide. In a problem like \(0.6(4.5 - 1.25) + 0.8\), almost everyone gets the digits right; what separates a correct answer from a wrong one is whether the point in the product landed in the right spot. Count the places before you write the product down, every time.

0.4.3 Decimals, Fractions, and Percents

Decimals, fractions, and percents are three notations for the same quantities. Each one is convenient somewhere and clumsy somewhere else. Fractions are exact and are what algebra prefers. Decimals are easiest to compare and are what a calculator hands you. Percents are the clearest way to report a proportion to another person. Being fluent means converting between them without stopping to think, because a problem almost never arrives in the notation you want to work in.

Turning a decimal into a fraction

Read the decimal aloud. The name is the fraction. Since \(0.36\) is "thirty-six hundredths," it is \(\dfrac{36}{100}\), which reduces to \(\dfrac{9}{25}\).

The denominator is always the place value of the last digit, so the number of decimal places tells you the power of ten before you write anything down. Reducing afterward uses the greatest-common-factor method from Section 0.3.

Turning a fraction into a decimal

A fraction bar means division, so \(\dfrac{a}{b}\) is \(a \div b\). Divide the numerator by the denominator, appending trailing zeros to the numerator as needed:

Every fraction converts, but the division can end in one of two ways, and each way has a name.

Definition 0.4.3 — A terminating decimal stops because the division reaches a remainder of zero.

We write a repeating decimal with a bar over the repeating block. The bar is not decoration. It states that the block goes on without end, which writing \(0.8333\ldots\) only hints at.

Look closely at where each bar starts. In \(0.8\overline{3}\) only the 3 repeats and the 8 appears once, so the number is \(0.8333\ldots\). In \(0.\overline{36}\) the bar covers both digits, so the pair repeats as a block: \(0.363636\ldots\).

Predicting which fractions terminate

You can tell which ending you will get before you divide. Reduce the fraction to lowest terms and factor the denominator. The decimal terminates exactly when that denominator's only prime factors are 2 and 5, the two primes that divide 10. Any other prime factor forces the division to repeat.

Reducing first is not optional here. The fraction \(\dfrac{6}{15}\) has a denominator with a factor of 3, which predicts a repeating decimal, but in lowest terms it is \(\dfrac{2}{5} = 0.4\), which terminates. Test the fraction in lowest terms, not whatever form you were handed.

Percent

Definition 0.4.5 — The percent sign stands in for a denominator of 100.

So \(42\%\) means \(\dfrac{42}{100}\), which is \(0.42\). Since dividing by 100 moves the decimal point two places to the left, both conversions are mechanical. To go from a percent to a decimal, drop the \(\%\) and move the point two places left. To go from a decimal to a percent, move the point two places right and attach the \(\%\).

Two of those rows contradict a widespread belief that a percent has to land between 0 and 100. It does not. A percent of \(125\%\) is a perfectly good number — it is \(1.25\), more than the whole — and it turns up any time something grows past its original size. At the other end, \(0.8\%\) is less than one percent, and the temptation to write it as \(0.8\) is an error by a factor of one hundred. Read the symbol as "per hundred" every time and \(0.8\%\) becomes \(\dfrac{0.8}{100} = 0.008\) without any drama.

Some equivalents come up so often that recognizing them on sight saves real time, especially in the data work of Chapter 4 where the same handful of proportions keeps reappearing. Table 0.4.1 collects them.

The \(\dfrac{1}{3}\) row explains why you usually see \(33\dfrac{1}{3}\%\) rather than \(33.3\%\). The decimal repeats forever, so any decimal percent you write down is rounded, while the fraction form is exact. When exactness matters, keep the fraction.

0.4.4 Square Roots and the Real Numbers

Squaring a number multiplies it by itself, so \(9^2 = 81\). A square root asks the reverse question: given 81, what number was squared to produce it?

Definition 0.4.6 — Squaring asks what 9 rows of 9 come to; the square root asks the reverse.

There is a subtlety hiding in that line. Both 9 and \(-9\) square to 81, since like signs multiply to a positive back in Section 0.2. If \(\sqrt{81}\) meant "either one," it would not name a single number, and every expression containing it would be ambiguous. So the radical sign is defined to hand back only the nonnegative one.

Definition 0.4.8 — A perfect square fills a complete square of dots with nothing left over.

Knowing the first fifteen perfect squares on sight is worth the small effort. It makes simplifying radicals fast, and it is the recognition step behind factoring quadratics in Chapter 7.

When a square root is not a real number

What would \(\sqrt{-49}\) be? It would have to be a number whose square is \(-49\). But a positive squared is positive, a negative squared is positive, and zero squared is zero, so no real number squares to a negative.

This is a real gap in our number system rather than an oversight, and it is the reason a later course introduces imaginary numbers. For now, treat the square root of a negative the way you treat division by zero from Section 0.2: an expression with no value among the numbers we have.

Do not confuse it with \(-\sqrt{49} = -7\), which is perfectly fine. The negative sits inside the radical in the first case and outside it in the second, and the two mean completely different things.

Estimating a root that is not a perfect square

Most numbers are not perfect squares, and their roots are not integers. You can still pin such a root between two consecutive integers by finding the perfect squares on either side of it. To estimate \(\sqrt{55}\), notice that the nearest perfect squares are 49 and 64:

So \(\sqrt{55}\) lands between 7 and 8. Since 55 is closer to 49 than to 64, the root is closer to 7. Squaring candidates closes in on it: \(7.4^2 = 54.76\) is too small, \(7.41^2 = 54.9081\) is still too small, and \(7.42^2 = 55.0564\) overshoots. The root therefore sits between 7.41 and 7.42, nearer the upper end, so \(\sqrt{55} \approx 7.42\) to two decimal places.

Estimating is not a lesser substitute for a calculator; it is how you catch a mistyped key. If a calculator reports \(\sqrt{55} \approx 24.1\), the estimate tells you instantly that something went in wrong.

Sorting numbers into sets

The decimal work from Section 0.4.3 turns out to answer a structural question about which numbers exist and how they relate.

Definition 0.4.9 — Five familiar numbers, each rewritten as a ratio of two integers.

Definition 0.4.10 — One decimal stops, one repeats, and one does neither.

Two families supply most of the examples. The first is roots of numbers that are not perfect squares: \(\sqrt{55} = 7.4161984\ldots\) runs forever with no repeating block, and in fact the square root of any whole number that is not a perfect square is irrational. The second is \(\pi\), the ratio of a circle's circumference to its diameter, which is \(3.14159265\ldots\) and also irrational. The familiar \(3.14\) and \(\dfrac{22}{7}\) are approximations, and both of them are rational, which is exactly why neither one equals \(\pi\).

Notice the asymmetry: \(\sqrt{36} = 6\) is rational, while \(\sqrt{35}\) is irrational. The radical sign is not what makes a number irrational. Whether the number underneath it is a perfect square is.

Definition 0.4.11 — The sets nest outward, the irrationals sit beside them, and together they fill the line.

These sets nest inside one another, so a number in an inner set automatically belongs to every set outside it:

The counting numbers are \(1, 2, 3, 4, \dots\). The whole numbers add zero to that list. The integers add the opposites, giving \(\dots, -2, -1, 0, 1, 2, \dots\) as in Section 0.2. The rational numbers add every ratio of integers, which is every terminating or repeating decimal. The irrational numbers are everything real that is left over, and this is the one set in the list that does not nest — it sits beside the rationals rather than inside them. Together the rationals and irrationals make up the real numbers.

Table 0.4.2 sorts ten sample numbers into those sets.

Three rows of that table carry the whole idea. The \(\sqrt{9}\) row looks exotic and is simply 3, a counting number, so always simplify before you classify — the notation can disguise an ordinary number. The \(\sqrt{35}\) row sits one line away and belongs to none of the inner sets. And \(\sqrt{-16}\) belongs to nothing on the chart at all, because it is outside the real numbers entirely.

The number line from Section 0.2 held the integers as evenly spaced ticks. Fractions and decimals fill the space between those ticks, so \(2.6\) sits six tenths of the way from 2 to 3, and \(-\dfrac{3}{4}\) sits three quarters of the way from 0 toward \(-1\). The irrationals take the positions that are left over, and \(\sqrt{55}\) has a definite spot just past 7.4 even though no finite decimal names it exactly. The result is that the real numbers fill the line completely, with no gaps anywhere. That completeness is what lets us talk about the graph of a line in Chapter 5 as one solid unbroken object instead of a dotted trail of points, and it is why the properties collected in Section 0.5 can be stated for all real numbers at once.

Problem Set 0.4

Key Terms