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:

$$3.45 = 3 + \frac{4}{10} + \frac{5}{100}$$

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:

$$\frac{3}{4} = \frac{75}{100} = 0.75$$

In general: divide the numerator by the denominator. That always works — though sometimes the expansion never ends, like $\frac{1}{3} = 0.3333\ldots$ This lesson only covers fractions with a finite decimal expansion.

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:

$$3.25 + 1.40 = 4.65$$

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:

$$2.5 \cdot 1.2 = 3.00$$

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 $3$.

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.

$$3.465 \approx 3.47$$

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

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

as a decimal 2/5ResultCheck

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.