The Number System Chapter Notes | Year 6 Mathematics IGCSE (Cambridge) PDF Download

Place value

• The objective is to explain the value of each digit in numbers with up to 3 decimal places.
• Multiply and divide whole numbers and decimals by 10, 100, 1000.
• Compose, decompose, and regroup numbers with up to 3 decimal places.
• Numbers are used daily for tasks like telephoning (digits), calculating prices (decimals), and measuring temperatures (positive and negative numbers).
• Each digit’s value depends on its position in a number.
• Whole number positions (from right): ones, tens, hundreds, thousands, etc.
• Decimal positions (from decimal point): tenths (1/10), hundredths (1/100), thousandths (1/1000).
• Example: In 809.46, the digit 9 is in the ones place, worth 9; in 2021.89, 9 is in the hundredths place, worth 0.09.

Composing numbers

• Combine place values to form a number, e.g., 10 + 3 + 1/10 + 3/100 + 6/1000 = 13.136.
• Write digits in a place value grid, aligning them by position (thousands, hundreds, tens, ones, tenths, hundredths, thousandths).

Decomposing numbers

• Break a number into its place value components, e.g., 37.844 = 30 + 7 + 0.8 + 0.04 + 0.004.

Regrouping numbers

• Rearrange digits to represent the same number differently, e.g., 0.546 = 0.4 + 0.14 + 0.006.

Multiplying by 10, 100, 1000

• Shifts digits left (decimal point moves right): 7.2 × 1000 = 7200, 0.85 × 100 = 85, 4.28 × 10 = 42.8.
• Example: 0.01 × 10 = 0.1 (ten times bigger than 0.01).

Dividing by 10, 100, 1000

• Shifts digits right (decimal point moves left): 670 ÷ 100 = 6.7, 151 ÷ 1000 = 0.151, 5.5 ÷ 10 = 0.55.
• Example: 555 ÷ 100 = 5.55 (one hundred times smaller than 555).
• Reverse process: If dividing by 10 three times yields 0.005, the original number is 0.005 × 10 × 10 × 10 = 5.

Rounding decimal numbers

• The objective is to round a number with 2 decimal places to the nearest whole number.
• Round a number with 2 decimal places to the nearest tenth.
• Rounding simplifies numbers for easier understanding, e.g., Usain Bolt’s 100 m time of 9.63 seconds is rounded to 10 seconds.

Rounding to the nearest whole number

• If the tenths digit is 0, 1, 2, 3, 4, round down (keep the whole number part), e.g., 6.2 ≈ 6, 4.09 ≈ 4.
• If the tenths digit is 5, 6, 7, 8, 9, round up (increase the whole number by 1), e.g., 36.5 ≈ 37, 7.89 ≈ 8.
• Example: Smallest number with 1 decimal place rounding to 100 is 99.5; largest rounding to 10 is 10.4.

Rounding to the nearest tenth

• If the hundredths digit is 0, 1, 2, 3, 4, round down (keep the tenths digit), e.g., 8.80 ≈ 8.8, 7.81 ≈ 7.8.
• If the hundredths digit is 5, 6, 7, 8, 9, round up (increase the tenths digit by 1), e.g., 6.45 ≈ 6.5, 3.95 ≈ 4.0.
• Answer always has 1 decimal place, even if it is zero, e.g., 4.0.
• Example: Numbers rounding to 7.5 include 7.45, 7.49, 7.50, 7.51, 7.53, 7.55, 7.56.