Rounding Numbers (Edexcel GCSE Maths): Revision Notes
Rounding numbers
Rounding numbers is an essential mathematical skill that allows us to simplify calculations and express answers to an appropriate level of accuracy. There are two main ways to specify where a number should be rounded: to a certain number of decimal places or to a certain number of significant figures.
Understanding rounding is crucial for mathematics, science, and everyday life. It helps us work with manageable numbers while maintaining appropriate precision for our needs.
Decimal places rounding
When rounding to decimal places, you need to focus on the position after the decimal point. The basic method remains consistent regardless of how many decimal places you're rounding to.
The step-by-step method
To round any number to a specific number of decimal places, you need to follow a systematic approach that ensures accuracy every time.
Worked Example: Rounding to Decimal Places
Let's round 7.45839 to 2 decimal places:
Step 1: Identify the last digit - Find the position of the digit that will be your final decimal place in the rounded answer (the 5 in 7.45839)
Step 2: Find the decider - Look at the digit immediately to the right of your last digit (the 8)
Step 3: Apply the rounding rule - If the decider is 5 or more, round the last digit up. If the decider is 4 or less, keep the last digit unchanged (since 8 ≥ 5, we round up from 5 to 6)
Step 4: Remove extra digits - Delete all digits after your last digit (the answer is 7.46)
The decider is the key to this process - it's the digit that determines whether you round up or down. Remember the rule: 5 or more rounds up, 4 or less stays the same.
Dealing with tricky nines
Sometimes rounding becomes more complex when you encounter nines. These situations require special attention to avoid common mistakes.
Critical Rules for Nines:
- When rounding up a 9: If you need to round up a 9 to 10, replace the 9 with 0 and carry 1 to the left
- Maintaining decimal places: Remember to include enough zeros to show the correct number of decimal places
For example, when rounding 45.699 to 2 decimal places:
- The last digit is 9 (second decimal place)
- The decider is 9 (third decimal place)
- Since 9 ≥ 5, we round up, but 9 becomes 10
- This means we write 0 and carry 1 to make it 45.70
Another example with 64.996 to 2 decimal places:
- The carrying effect continues: 64.996 becomes 65.00
- Notice how the carrying affects multiple digits
The carrying effect can cascade through multiple digits. Always work carefully through each step when dealing with nines to avoid errors.
Significant figures rounding
Significant figures work differently from decimal places because they focus on the overall precision of a number rather than its position relative to the decimal point.
Understanding significant figures
The first significant figure of any number is simply the first digit that isn't zero. All subsequent significant figures follow immediately after the first one, regardless of whether they are zeros or not.
Worked Example: Identifying Significant Figures
- In 0.002309, the significant figures are 2, 3, 0, 9
- In 2.03070, the significant figures are 2, 0, 3, 0, 7, 0
Notice how zeros can be significant figures when they appear after the first non-zero digit.
Rounding rules for significant figures
The method for rounding to significant figures follows the same basic principle as decimal places, but with an important additional rule about zeros.
Essential Rule for Significant Figures:
After rounding the last digit, you must fill in zeros up to (but not beyond) the decimal point. You should never add extra zeros after the decimal point.
The process involves:
- Identify your last significant figure based on how many you need
- Look at the decider (the digit immediately to the right)
- Round up or down using the same 5-or-more rule
- Fill in zeros up to the decimal point if necessary, but never add extra zeros after the decimal point
EXAMPLES:
| to 3 s.f. | to 2 s.f. | to 1 s.f. | |
|---|---|---|---|
| 1) 54.7651 | 54.8 | 55 | 50 |
| 2) 17.0067 | 17.0 | 17 | 20 |
| 3) 0.0045902 | 0.00459 | 0.0046 | 0.005 |
| 4) 30895.4 | 30900 | 31000 | 30000 |
Estimating
Estimating is a valuable technique that helps you quickly approximate the answer to complex calculations. This skill is particularly useful when you need to check if your calculator answer is reasonable.
The estimation method
Estimating involves a simple two-step process that makes complex calculations manageable:
Worked Example: Estimation Process
To estimate (127.8 + 41.9)/(56.5 × 3.2):
Step 1: Round everything to convenient numbers (usually 1 or 2 significant figures)
- Round 127.8 to 130
- Round 41.9 to 40
- Round 56.5 to 60
- Round 3.2 to 3
Step 2: Work out the answer using these simplified numbers
- Calculate: (130 + 40)/(60 × 3) = 170/180 ≈ 1
The key is to show all your working steps, especially in exams, to demonstrate that you haven't used a calculator.
When to use estimation
Estimation is particularly useful when:
- You need to check calculator answers quickly
- You're asked to give an answer "to an appropriate degree of accuracy"
- You want to get a rough idea of the answer before doing detailed calculations
- You're working without a calculator and need approximate results
Remember!
Key Points to Remember:
- For decimal places: Count positions after the decimal point, use the decider rule (5 or more rounds up), and remove all digits after your last digit
- For significant figures: Find the first non-zero digit, count from there, and fill zeros up to the decimal point only
- The rounding rule is universal: 5 or more rounds up, 4 or less stays the same
- Show your working: Especially important for estimation questions to prove you haven't used a calculator
- Watch out for nines: They can cause a carrying effect that changes multiple digits when rounding up