Decimals

The decimal point separates whole-number places from tenths, hundredths, thousandths, and smaller powers of ten.

Place value

In

$4$ represents tens, $2$ ones, $3$ tenths, $0$ hundredths, and $7$ thousandths:

Every move one place right divides place value by $10$.

Reading and writing

Read the whole-number part, say “and” for the decimal point, then name the final decimal place. Thus $5.042$ is “five and forty-two thousandths.”

Trailing zeros do not change value: $3.5=3.50$, though they can communicate measurement precision.

Comparing decimals

Align decimal points and add placeholder zeros when helpful:

because $0.700>0.659$. Compare from the greatest place value to the first place that differs.

Adding and subtracting

Align decimal points so equal place values line up. Then calculate as with whole numbers and bring the decimal point straight down.

Multiplying decimals

Multiply as whole numbers, then place the decimal so the product has the total number of decimal places from both factors.

For example,

Estimation explains the placement: $12.5$ times roughly one third should be near $4$, not $40$ or $0.4$.

Dividing decimals

Make the divisor a whole number by multiplying both dividend and divisor by the same power of ten. This preserves the quotient:

Place-value reasoning is safer than moving decimal points without tracking why.

Powers of ten

Multiplying by $10^n$ shifts place values $n$ positions left relative to the decimal point; dividing shifts them right.

For example,

Digits do not literally move on paper—their place values change.

Fractions and percents

A terminating decimal can be written over a power of ten:

To convert a decimal to percent, multiply by $100\%$:

Repeating decimals represent rational numbers too, but may need algebra or known patterns for exact conversion.

Rounding and measurement

To round to a place, inspect the digit immediately to its right. If it is $5$ or greater, increase the target digit by one; otherwise keep it.

Round only after calculating when possible. In measurement, do not report more precision than the instrument or context supports.

Money and rates

Money is usually recorded to the nearest cent, but intermediate tax, interest, or unit-price calculations should keep extra digits before final rounding.

Label dollars, metres, seconds, or other units to avoid treating every decimal as unitless.

Common mistakes

Aligning final digits instead of decimal points for addition. Align place values.

Thinking $0.9<0.12$ because $9<12$. Compare $0.90$ with $0.12$.

Placing a product decimal without estimating. Use magnitude as a check.

Changing only the divisor when clearing a decimal division. Scale dividend and divisor equally.

Rounding every intermediate line. This accumulates error.

Quick self-check

• Can I name every digit's place value?
• Are decimal points aligned for addition/subtraction?
• Does multiplication/division preserve reasonable magnitude?
• Were both parts of a quotient scaled equally?
• Can the number be connected to a fraction and percent?
• Is rounding delayed and matched to contextual precision?

Related topics

• Fractions
• Percent
• Place Value
• Estimation