Rounding is the most common way to make approximations. We round numbers to a specific place value.

Steps for Rounding

1. Find the digit at the place value you want to round to
2. Look at the digit immediately to its right
3. If that digit is 5 or more, round up (add 1 to the target digit)
4. If that digit is less than 5, keep the target digit the same
5. Replace all digits to the right with zeros (or remove them for decimals)

---

Example 1: Round 42,850,671 to the nearest million

• The million digit is 2
• The next digit is 8 (8 ≥ 5, so round up)
• 2 becomes 3, and all digits to the right become zeros

$42,850,671 \approx 43,000,000$

Example 2: Round 678,912 to the nearest thousand

• The thousand digit is 8
• The next digit is 9 (9 ≥ 5, so round up)
• 8 becomes 9

$678,912 \approx 679,000$

Example 3: Round 34,649 to the nearest hundred

• The hundred digit is 6
• The next digit is 4 (4 < 5, so keep it the same)

$34,649 \approx 34,600$

---

When rounding decimals, we round to a certain number of decimal places (the digits after the decimal point).

Example 1: Round 0.24736 to 2 decimal places

• Second decimal place is 4
• Next digit is 7 (7 ≥ 5, so round up)
• 4 becomes 5

$0.24736 \approx 0.25$

Example 2: Round 3.14159 to 3 decimal places

• Third decimal place is 1
• Next digit is 5 (5 ≥ 5, so round up)
• 1 becomes 2

$3.14159 \approx 3.142$

Example 3: Round 5.9999 to 2 decimal places

• Second decimal is 9
• Next digit is 9 (9 ≥ 5, so round up)
• This causes 9 + 1 = 10, so we write 0 and carry 1

$5.9999 \approx 6.00$

---

Significant figures (sig figs) are the digits in a number that carry meaning. This method is especially useful in science and measurements.

Key Rules for Identifying Significant Figures

1. All non-zero digits are significant
2. Zeros between non-zero digits are significant
3. Leading zeros (before the first non-zero digit) are NOT significant
4. Trailing zeros in a decimal are significant
5. Trailing zeros in a whole number may or may not be significant

Rounding to Significant Figures

Follow the same rounding rules, but count the required number of significant figures from the first non-zero digit.

Examples

Round 37,846 to 2 significant figures:

• First two significant figures: 37
• Next digit is 8 (8 ≥ 5, so round up)
• 37 becomes 38

$37,846 \approx 38,000$

Round 0.004786 to 2 significant figures:

• First two significant figures: 47
• Next digit is 8 (8 ≥ 5, so round up)
• 47 becomes 48

$0.004786 \approx 0.0048$

Round 560.089 to 3 significant figures:

• First three significant figures: 560
• Next digit is 0 (0 < 5, so keep it)
• Note: 560 already has 3 significant figures

$560.089 \approx 560$

---

When we measure quantities, we often cannot get exact values. Approximations help us report reasonable measurements.

Example: Measuring Length

A student measures a pencil and records 14.7 cm. If we round to 1 decimal place:

• The digit in the first decimal place is 7
• There is no second decimal digit (consider it 0)
• 0 < 5, so we keep 7

$14.7 \text{ cm} \approx 14.7 \text{ cm}$

If the measurement was 14.75 cm and we round to 1 decimal place:

• First decimal is 7
• Second decimal is 5 (5 ≥ 5, so round up)
• 7 becomes 8

$14.75 \text{ cm} \approx 14.8 \text{ cm}$

Example: Rounding in Calculations

When adding or multiplying measured numbers, the result should be rounded to match the least precise measurement.

Addition: 12.5 L + 7.26 L = 19.76 L

• 12.5 has 1 decimal place (least precise)
• Round to 1 decimal place: 19.8 L

$12.5 + 7.26 \approx 19.8 \text{ L}$

---

In Tanzania, approximations are used every day. For example, when shopping at a local market in Dar es Salaam, a vendor might price tomatoes at TSh 500 per kilogram. If you want to buy 3.7 kg, you can estimate the cost by rounding:

• Round 3.7 kg to the nearest kilogram: 4 kg
• Estimated cost: 4 × 500 = TSh 2,000

This quick approximation helps you check if you have enough money before the vendor weighs the exact amount. Approximations are also used when reading electricity meters, estimating travel distances between cities, and calculating household budgets.