Decimals

A decimal writes part of a whole with a decimal point, rather than a fraction bar — a different notation for the same idea as a fraction. In $3.45$, the digits after the point carry their own place value, just like the digits before it:

The first digit after the point is the tenths place ($\frac{1}{10}$), the second is hundredths ($\frac{1}{100}$), the third is thousandths ($\frac{1}{1000}$), and so on.

Converting a fraction to a decimal

When a fraction's denominator can be expanded to a power of $10$ ($10$, $100$, $1000$, …), expand it and read off the result:

In general: divide the numerator by the denominator. That always works — though sometimes the expansion never ends, like $\frac{1}{3} = 0.3333\ldots$

The denominator alone decides which it will be. Only a denominator that, in lowest terms, factors into nothing but twos and fives can be expanded to a power of ten — because $10 = 2 \cdot 5$ and a power of ten has no other prime factor. That is why $\frac{7}{20}$ terminates ($20 = 2^2 \cdot 5$) while $\frac{1}{3}$ and $\frac{5}{6}$ do not: there is a three in their denominators. The last section of this lesson is about those endless expansions — they do not stop, but they are not random either.

Converting a decimal to a fraction

The other direction is easier still, because the decimal notation states its own denominator: a power of ten with as many zeros as there are decimal places. The digits after the point go in the numerator, and then you reduce:

A number with a whole part works the same way — every digit goes into the numerator, point ignored:

Reducing at the end is not decoration: it is the step that answers which fraction this number really is. Writing $\frac{225}{100}$ is correct, but only $\frac{9}{4}$ shows that $2.25$ is two and a quarter.

Adding and subtracting decimals

Decimals add just like whole numbers — you only need to line up the decimal points (exactly like ones under ones), padding with zeros where needed:

Written in columns: $3.25$ and $1.40$ — the columns line up because both numbers have two decimal places.

Multiplying decimals

Multiplication works as if the decimal points were not there at all — multiply the whole numbers, then place the point at the end:

The number of decimal places in the result is the sum of the decimal places in both factors: $2.5$ has one place, $1.2$ has one place, so the result has two — $3.00$, that is $3$.

Dividing decimals

In a division the point is only in the way in the divisor — so that is where you get rid of it first. Shift the point in both numbers the same number of places to the right until the divisor becomes a whole number:

The result does not change, because both numbers are multiplied by the same power of ten and a quotient depends only on the ratio of dividend to divisor: $\frac{14.4}{1.2} = \frac{14.4 \cdot 10}{1.2 \cdot 10}$. It is the same move as expanding a fraction, written with a decimal point.

When the point stays in the dividend, divide as usual and carry it straight up into the answer:

Dividing by a number smaller than $1$ makes the result larger — $6 \div 0.5 = 60 \div 5 = 12$, because the question is how many halves fit into six. That is the same observation as dividing by a fraction.

Rounding

To round to a given number of decimal places, look at the first digit you drop: $5$ or more rounds the last kept digit up by $1$; less than $5$ leaves it unchanged.

Rounding to hundredths drops the third decimal digit ($5$), so the second digit ($6$) rounds up to $7$.

Infinite repeating expansions

The division $1 \div 3$ never ends: a remainder of $1$ comes back at every step, so a three repeats after the point without stopping. Such an expansion is called infinite repeating, and the block of digits that repeats is the period, written in brackets:

(Textbooks also draw a bar over the period, $0.\overline{3}$; the bracket is the notation used throughout this site and in the exercises below.)

The repetition is no coincidence. Dividing by $b$, every step leaves a remainder smaller than $b$, so there are at most $b - 1$ possible remainders; after that many steps some remainder must come round again, and from there the whole calculation repeats. Every fraction therefore has an expansion that either terminates or repeats — and conversely, every such expansion can be written as a fraction.

Converting back follows a rule you can read off $\frac{1}{3} = 0.(3)$ and $\frac{3}{11} = 0.(27)$: the period into the numerator, as many nines into the denominator as the period has digits.

When other digits stand before the period, the expansion is called mixed repeating. Add one zero after the nines for each of those digits, and subtract the non-repeating part from the numerator:

One repeating expansion is a special case: $0.(9)$ equals $1$, because the rule gives $\frac{9}{9}$. That is not an approximation or a slip of notation — the same number simply has two decimal spellings here, just as $\frac{1}{2}$ and $\frac{2}{4}$ are two spellings of the same fraction.

Expansions that are infinite and non-repeating cannot be written as a fraction at all — those are the irrational numbers, such as $\sqrt{2}$ and $\pi$; the topic on number sets takes them up.

Common mistakes

• Adding without lining up the points — computing $3.5 + 1.25$ as $35 + 125$ gives the wrong answer; the decimal points must sit on top of each other.
• Forgetting the decimal places when multiplying — $2.5 \cdot 1.2$ is not $3.0$ by luck, it is $300 \div 100 = 3.00$ from two combined decimal places.
• Misreading the rounding boundary — you only look at the first dropped digit, not the rest of the number.
• Shifting the point in the divisor only — in $14.4 \div 1.2$ the point moves in both numbers by the same amount; moving it in one of them changes the answer tenfold.
• Writing $0.(3)$ as $\frac{3}{10}$ — a repeating expansion sits over a nine, not a ten: $0.(3) = \frac{3}{9} = \frac{1}{3}$, while $\frac{3}{10}$ is $0.3$ and nothing more.
• Confusing $0.3$ with $0.(3)$ — the first stops at three tenths, the second has endlessly many threes and is the larger of the two.