Mada za sehemu hiiUse geometry, approximations, relations, and functions in various contextsMada 4
- Explain the concept of approximations (rounding off, significant figures, and decimal places)
- Round off numbers and estimate values of expressions
- Approximate numbers to the required significant figures and decimal places
- Use approximations in computations and measurements of quantities in various contexts
Approximating Numbers
When we approximate a number, we write it in a simpler form that is easier to use while still keeping its value close to the original. We round numbers to a certain number of significant figures or decimal places depending on what accuracy is needed.
Significant figures (s.f.) are the digits in a number that carry meaning. They tell us how precise a number is.
Key Rules for Identifying Significant Figures
- All non-zero digits (1–9) are significant.
- Zeros between two significant digits are significant.
- Leading zeros (zeros before the first non-zero digit) are NOT significant.
- Trailing zeros in a decimal number ARE significant.
- Trailing zeros in a whole number without a decimal point are NOT significant.
Examples
- 378 has 3 significant figures.
- 0.00478 has 3 significant figures (the leading zeros don't count).
- 560.0 has 4 significant figures (the zero after the decimal point is significant).
- 980500 has 3 significant figures (trailing zeros don't count).
Steps
- Identify the first non-zero digit — this is the first significant figure.
- Count the required number of significant figures from the left.
- Look at the digit immediately after the last significant figure.
- If that digit is 5 or more, round up (add 1 to the last significant figure).
- If it is less than 5, keep the last significant figure unchanged.
- Replace all remaining digits with zeros (or remove them if decimal), keeping the decimal point in its original position.
Example 1: Round 37,846 to 2 significant figures
- First significant figure: 3
- Second significant figure: 7
- The digit after 7 is 8 (5 or more), so round up: 7 → 8
- Replace remaining digits with zeros
Answer: 38,000
Example 2: Round 0.004786 to 3 significant figures
- First significant figure: 4
- Second: 7 | Third: 8
- The digit after 8 is 6 (5 or more), so round up: 8 → 9
- Keep the leading zeros and remove the rest
Answer: 0.00479
Example 3: Round 560.089 to 4 significant figures
- First: 5 | Second: 6 | Third: 0 | Fourth: 0
- The digit after the fourth significant figure is 8 (5 or more), so round up: 0 → 1
- Keep the decimal point and remove the remaining digits
Answer: 560.1
Sometimes we need to round a decimal number to a specific number of decimal places (d.p.) — that is, to a certain number of digits after the decimal point.
Steps
- Identify the digit at the required decimal place.
- Look at the digit immediately to the right of it.
- If that digit is 5 or more, round up.
- If it is less than 5, keep the digit unchanged.
- Drop all digits to the right.
Example 1: Round 8.7659 to 2 decimal places
- The second decimal place is 6.
- The next digit is 5 (5 or more), so round up: 6 → 7.
- Drop the remaining digits.
Answer: 8.77
Example 2: Round 3.14159 to 3 decimal places
- The third decimal place is 1.
- The next digit is 5 (5 or more), so round up: 1 → 2.
Answer: 3.142
Example 3: Round 0.24736 to 1 decimal place
- The first decimal place is 2.
- The next digit is 4 (less than 5), so keep 2 unchanged.
Answer: 0.2
| Term | What it means | Example |
|---|---|---|
| 2 s.f. | Round to 2 significant figures | 37,846 → 38,000 |
| 3 s.f. | Round to 3 significant figures | 0.004786 → 0.00479 |
| 2 d.p. | Round to 2 decimal places | 8.7659 → 8.77 |
| 3 d.p. | Round to 3 decimal places | 3.14159 → 3.142 |
In Tanzania, shopkeepers and traders frequently round prices when giving change or calculating totals. For example, if a customer buys items totalling TSh 2,847 and pays with TSh 3,000, the seller might round the change to the nearest hundred shillings (TSh 200) for simplicity. Similarly, when reporting monthly sales or calculating profit, a business owner may round financial figures to 2 significant figures to make the numbers easier to communicate and compare.
Swali
Round 37846 to 2 significant figures.
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