Reference · module 3
Decimals and powers
Everything this module explains, on one page.
Place value
Read, compare and round decimal numbers.
- In 3.47 the digit after the point counts tenths and the next counts hundredths: 3.47 is 3 + 4/10 + 7/100. Each place to the right is worth ten times less.
- To compare decimals, line up the points and compare digit by digit from the left. 0.5 is bigger than 0.45, because 5 tenths beat 4 tenths — the extra digit does not matter.
- To round to one decimal place, look at the next digit. 5 or more rounds up, less than 5 rounds down: 2.46 becomes 2.5, and 2.43 becomes 2.4.
The mistake you are about to make
0.45 > 0.5
0.5 > 0.45
45 is bigger than 5, so 0.45 looks bigger too. After the point it is the places that count: 0.5 is 0.50, and 50 hundredths beat 45.
Calculating with decimals
Add decimals, shift the point and switch between decimals and fractions.
- Add decimals by lining up the points, not the last digits. 2.5 + 0.35 is 2.50 + 0.35, which is 2.85. Writing the missing zero keeps the places in their columns.
- Multiplying by 10 moves the point one place right, by 100 two places: 3.14 × 100 is 314. Dividing by 10 moves it left: 3.14 ÷ 10 is 0.314.
- A decimal is a fraction with 10, 100 or 1000 below: 0.25 is 25/100, which is 1/4. Going the other way, divide top by bottom: 3/5 is 3 ÷ 5, which is 0.6.
The mistake you are about to make
2.5 + 0.35 = 0.60
2.5 + 0.35 = 2.85
Lining numbers up on the right is how whole numbers are added, so the 5s end up together. With decimals, the points line up, not the ends.
Powers and roots
Use powers and square roots, and place them in the order of operations.
- A power is repeated multiplication: 2³ means 2 × 2 × 2, which is 8. The small number, the exponent, says how many times the base is used — not what to multiply by.
- A square root undoes squaring: √49 is 7, because 7² is 49. Perfect squares worth knowing are 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.
- Powers come right after brackets in the order of operations, before multiplication. So 3 × 2² is 3 × 4, which is 12, not 6² which would be 36.
The mistake you are about to make
2³ = 6
2³ = 8
The 2 and the 3 sit side by side, so it is tempting to multiply them. The exponent counts copies of the base: 2 × 2 × 2.