Place Value and Rounding | Year 8 Mathematics (Cambridge) PDF Download

Understanding and Applying Place Value

Place Value Basics

Place value refers to the value of a digit depending on its position in a number. Each position represents a different power of 10.

• Units (Ones)
• Tens
• Hundreds
• Thousands
• Ten Thousands
• Hundred Thousands
• Millions

Example: Consider the number 45,382.

• The digit 4 is in the ten thousands place, so its value is 4 × 10,000 = 40,000.
• The digit 5 is in the thousands place, so its value is 5 × 1,000 = 5,000.
• The digit 3 is in the hundreds place, so its value is 3 × 100 = 300.
• The digit 8 is in the tens place, so its value is 8 × 10 = 80.
• The digit 2 is in the units place, so its value is 2 × 1 = 2.

Decimal Place Value

Decimal numbers have place values too, which extend to the right of the decimal point.

• Tenths
• Hundredths
• Thousandths
• Example: Consider the number 6.349.
  The digit 6 is in the units place, so its value is 6 × 1 = 6.
• The digit 3 is in the tenths place, so its value is 3 × 0.1 = 0.3.
• The digit 4 is in the hundredths place, so its value is 4 × 0.01 = 0.04.
• The digit 9 is in the thousandths place, so its value is 9 × 0.001 = 0.009.

Importance of Place Value

Understanding place value is crucial for performing arithmetic operations, comparing numbers, and rounding.

Rounding Numbers to Significant Figures and Decimal Places

Rounding to Significant Figures

Significant figures (sig figs) are the digits in a number that carry meaning contributing to its precision.

Rules for Rounding to Significant Figures:

• Identify the number of significant figures required.
• Count from the leftmost non-zero digit.
• Look at the next digit (one place to the right).
• If it is 5 or more, round up.
• If it is less than 5, leave the last significant digit as it is.

Example: Round 0.007845 to 3 significant figures.

• Identify the first three significant figures: 7, 8, and 4.
• The next digit is 5, so we round up.
• The result is 0.00785.

Rounding to Decimal Places

Rounding to decimal places means adjusting a number to keep a specified number of digits to the right of the decimal point.

Rules for Rounding to Decimal Places:

• Identify the number of decimal places required.
• Look at the next digit (one place to the right).
• If it is 5 or more, round up.
• If it is less than 5, leave the last decimal place as it is.

Example: Round 12.4673 to 2 decimal places.

• The second decimal place is occupied by 6.
• The next digit is 7, so we round up.
• The result is 12.47.

Why Rounding is Important

• Rounding makes numbers easier to work with and understand.
• It is useful in estimating and simplifying calculations.
• Rounding is often necessary when dealing with measurements and reporting data.

Examples in Real-Life Contexts:

• Money: When dealing with currency, you might round to the nearest cent.
  • Example: $12.345 becomes $12.35 when rounded to the nearest cent.
• Measurements: Scientists and engineers often round measurements to significant figures for precision.
  • Example: A length of 1.234 meters might be rounded to 1.23 meters if using three significant figures.
• Everyday Estimation: Estimating population or large quantities often involves rounding.
  • Example: The population of a city might be rounded from 1,234,567 to 1.23 million.

Solved Examples

Example 1: Round the number 0.004567 to 2 significant figures.
Solution: 0.0046

Example 2: Round the number 78.3421 to 1 decimal place.
Solution: 78.3

Example 3: Identify the place value of the digit 5 in the number 54,321.
Solution: Ten thousands

Example 4: What is the value of the digit 7 in the number 6.782?
Solution: 0.7 (tenths place)