Number system and decimals
Strengthen the number foundations that support percentages, ratios, algebra, and measurement.
Place value#
In 4,507.36, the digit 5 represents five hundreds, while 3 represents three tenths. Moving one place left multiplies a digit's value by 10; moving right divides it by 10.
Trailing zeros after a decimal do not change value: 2.5 = 2.50. Zeros between other digits do matter.
Signed numbers#
Positive and negative numbers describe direction from zero. Adding a negative is equivalent to subtracting its positive magnitude. When multiplying or dividing, equal signs give a positive result and different signs give a negative result.
Factors and multiples#
A factor divides a number without remainder. A multiple is a result of multiplying that number by an integer. Prime factorization helps find greatest common factors and least common multiples.
Decimal operations#
Align decimal points for addition and subtraction. For multiplication, multiply as whole numbers and then place the decimal according to the total decimal places in the factors. For division, scale both dividend and divisor by the same power of 10 until the divisor is convenient.
Common mistakes#
Do not align decimals by the final digit. In 12.4 + 0.75, align units with units and tenths with tenths. Also remember that -8 is smaller than -3 because it lies farther left on the number line.
Estimate#
Round values before calculating. Since 19.8 × 5.1 is close to 20 × 5 = 100, an answer of 10 or 1,000 suggests a decimal error.
Key points
- Place value determines the contribution of each digit.
- Factors divide a number exactly, while multiples are produced by multiplication.
- Align place values when adding or subtracting decimals.
- Estimate first to detect misplaced decimal points and sign errors.