Indices and Surds
Senior 2 · Mathematics
Indices and Surds
Lesson 3 of 5
20 min
Rwanda's 2022 population and housing census counted 13,246,394 people. The country covers about 26,338 km2. A red blood cell is only about 0.000 007 m across. Numbers this large or this small are hard to read and easy to copy wrongly, because it is so easy to miss a zero.
Standard form (also called scientific notation) writes every number as a short number multiplied by a power of 10. Scientists, engineers and statisticians use it every day, and your calculator switches to it automatically when numbers become very large or very small.
103 = 1000, 106 = 1,000,000 and 100 = 1. Negative powers give decimals: 10−1 = 0.1, 10−2 = 0.01 and 10−3 = 0.001. Multiplying by 10n moves the decimal point n places to the right; multiplying by 10−n moves it n places to the left.
A number is in standard form when it is written as
A × 10n, where 1 ≤ A < 10 and n is an integer.
So 4.5 × 107 is in standard form, but 45 × 106 is not (45 is not less than 10), and 0.45 × 108 is not (0.45 is less than 1).
Large number. 45,000,000: place the decimal point after the first non-zero digit to get 4.5. The point moved 7 places to the left, so 45,000,000 = 4.5 × 107.
Small number. 0.000 32: the first non-zero digit is 3, giving 3.2. The point moved 4 places to the right, so 0.000 32 = 3.2 × 10−4.
Back to an ordinary number. 6.07 × 105 = 607,000 (point 5 places right). 2.5 × 10−3 = 0.0025 (point 3 places left).
A quick check: numbers bigger than 10 have a positive power of 10; numbers between 0 and 1 have a negative power.
Multiplying. Multiply the numbers, then add the indices: (3 × 104) × (5 × 106) = 15 × 1010. Since 15 is not less than 10, adjust: 15 × 1010 = 1.5 × 1011.
Dividing. Divide the numbers, then subtract the indices: (8.4 × 106) ÷ (2.1 × 10−2) = 4 × 106 − ( − 2) = 4 × 108.
Adding and subtracting. First give both numbers the same power of 10: 3.2 × 105 + 4.5 × 104 = 3.2 × 105 + 0.45 × 105 = 3.65 × 105.
Use 13,246,394 people and 26,338 km2 to estimate the number of people per km2.
Step 1 Round to 3 significant figures and write in standard form: 13,246,394 ≈ 1.32 × 107 and 26,338 ≈ 2.63 × 104.
Step 2 Divide: 1.32 ÷ 2.63 ≈ 0.502, and 107 ÷ 104 = 103. So the density is about 0.502 × 103 = 5.02 × 102.
Answer: about 500 people per km2. (The exact division 13,246,394 ÷ 26,338 gives 502.9, so the estimate is sound.) This makes Rwanda one of the most densely populated countries in Africa.
Tip
Exam technique. After every calculation, check that the front number A obeys 1 ≤ A < 10. If it does not, adjust it and change the index the opposite way: 24 × 103 = 2.4 × 104 (A got 10 times smaller, so the power got 1 bigger). Examiners often give no mark for a correct value that is not in standard form.
Write 0.000 604 in standard form.
Solution: The point moves 4 places right to give 6.04, so 6.04 × 10−4.
Write 7.2 × 106 as an ordinary number.
Solution: Move the point 6 places right: 7,200,000.
Calculate (6 × 105) × (4 × 10−2), giving the answer in standard form.
Solution: 6 × 4 = 24 and 105 + ( − 2) = 103. 24 × 103 = 2.4 × 104.
Calculate (9.6 × 108) ÷ (3.2 × 103).
Solution: 9.6 ÷ 3.2 = 3 and 108 − 3 = 105. Answer: 3 × 105.
Calculate 4.3 × 106 − 8 × 105.
Solution: 8 × 105 = 0.8 × 106. Then 4.3 − 0.8 = 3.5, so the answer is 3.5 × 106.
Remember
Sources and credits
The Rwandan worked examples, the practice items and the exercises were written by Ganzaa.
