OCR GCSE Maths: Approximations and Significant Figures

Approximations and Significant Figures

Introduction

In everyday life, we often encounter numbers that are too long or complex to work with easily. This is where approximations and significant figures come in. These techniques help us simplify numbers while maintaining a reasonable level of accuracy.

Rounding to the Nearest Whole Number

• Rule: Look at the digit in the units place.
  • If the digit is 5 or greater, round up the tens digit by 1.
  • If the digit is less than 5, keep the tens digit the same.
• Example: Round 3.7 to the nearest whole number.
  • The units digit is 7, which is greater than 5.
  • Rounding up the tens digit gives us 4.
  • Therefore, 3.7 rounded to the nearest whole number is 4.

Rounding to the Nearest Decimal Place

• Rule: Focus on the digit one place value to the right of the desired decimal place.
  • If the digit is 5 or greater, round up the digit at the desired decimal place by 1.
  • If the digit is less than 5, keep the digit at the desired decimal place the same.
• Example: Round 2.345 to the nearest hundredth.
  • The digit in the thousandths place is 5.
  • Rounding up the digit in the hundredths place gives us 2.35.
  • Therefore, 2.345 rounded to the nearest hundredth is 2.35.

Significant Figures

Significant figures represent the reliable digits in a number. They tell us how precise a measurement is.

• Rules:

  • All non-zero digits are significant.
  • Zeros between non-zero digits are significant.
  • Leading zeros (zeros before non-zero digits) are not significant.
  • Trailing zeros (zeros after non-zero digits) are significant only if there is a decimal point.

• Example: Identify the number of significant figures in 0.02305.

  • The leading zeros (0.0) are not significant.
  • The digits 2, 3, and 5 are significant.
  • The zero between 3 and 5 is significant.
  • Therefore, 0.02305 has 4 significant figures.

Rounding to Significant Figures

• Rule: Follow the rounding rules based on the desired number of significant figures.
  • If the digit to the right of the last significant digit is 5 or greater, round up the last significant digit by 1.
  • If the digit to the right of the last significant digit is less than 5, keep the last significant digit the same.
• Example: Round 12.345 to 3 significant figures.
  • The first three digits (12.3) are significant.
  • The digit to the right of the last significant digit (4) is greater than 5.
  • Rounding up the last significant digit (3) gives us 12.4.
  • Therefore, 12.345 rounded to 3 significant figures is 12.4.

Real-Life Applications

• Measurements: Measurements are often approximate, and rounding them to appropriate significant figures ensures a realistic representation of the precision.
• Estimates: We use approximations to estimate quantities, like the number of people at a gathering.
• Calculations: Rounding numbers before calculations can simplify them and provide a quick estimate of the result.

Remember: The choice of rounding or using significant figures depends on the context and the level of accuracy required.