Mathematics Year 7: Scientific notation: very large numbers, written short
Mathematics, Year 7 (Portugal): Write a big number as a number from 1 to under 10 times a power of 10, then compare and calculate with it.
A number from 1 to 10, times a power of 10
A number is in scientific notation when it is written a × 10ⁿ, with a at least 1 and less than 10, and n a positive whole number. To write 45,300,000, move the decimal point left until only one non-zero digit is in front of it: 4.53. You moved 7 places, so 45,300,000 = 4.53 × 10⁷. It also works with decimals: 3,400.5 = 3.4005 × 10³. To compare, look at the exponent first: the greater exponent, the greater number; only if they are equal do you compare the values of a.
Why scientists write numbers this way
In physics and chemistry you meet huge quantities: the Earth is about 150,000,000 km from the Sun, that is, 1.5 × 10⁸ km. Counting zeros invites mistakes and comparing numbers full of zeros is tiring; in scientific notation, the exponent shows at once how big the number is. Simple calculations get easy too: for double, triple, half or a percentage you only change a and leave the power of 10 alone (then tidy a if it leaves the range from 1 to 10).
Writing and calculating
Writing 45,300,000: the decimal point moves 7 places, so 4.53 × 10⁷. Double of 6 × 10⁵: 2 × 6 = 12, giving 12 × 10⁵, but 12 is not between 1 and 10; 12 = 1.2 × 10, so 1.2 × 10 × 10⁵ = 1.2 × 10⁶. Half of 3 × 10⁷: 3 ÷ 2 = 1.5, so 1.5 × 10⁷. 25% of 8 × 10⁶: 25% of 8 is 2, so 2 × 10⁶. Check: 2 × 10⁶ = 2,000,000 and 25% of 8,000,000 = 2,000,000.
A bigger a does not make a bigger number
Someone comparing 8.5 × 10⁶ with 2.1 × 10⁷ looks at 8.5 and 2.1 and decides the first is greater. That is the classic mistake: the exponent weighs more. Written out, 8.5 × 10⁶ = 8,500,000 and 2.1 × 10⁷ = 21,000,000, so the greater is 2.1 × 10⁷. With a between 1 and 10, the number with the greater exponent is always the greater, because it has more digits before the decimal point. Only when the exponents are equal does a decide.
On the calculator screen and in science class
When a result does not fit on the screen, a calculator writes it in scientific notation, for example 4.53E12 (the exact format depends on the model). The E means “times 10 to the power”: 4.53E12 = 4.53 × 10¹², that is, 4,530,000,000,000. It is not the number e (2.718…). In science class, the speed of light, about 3 × 10⁸ m/s, and distances in space are usually written like this.
Test what you learned
Which of these is 45,300,000 written in scientific notation?
- a) 4.53 × 10⁶
- b) 45.3 × 10⁶
- c) 453 × 10⁵
- d) 4.53 × 10⁷
Correct answer: d) 4.53 × 10⁷ — Yes: 4.53 is between 1 and 10 and the point moved 7 places, so 45,300,000 = 4.53 × 10⁷.
What is the triple of 4 × 10⁷, written in scientific notation?
- a) 12 × 10⁷
- b) 1.2 × 10⁸
- c) 1.2 × 10⁷
- d) 4 × 10¹⁰
Correct answer: b) 1.2 × 10⁸ — Yes: 3 × 4 = 12, and 12 × 10⁷ = 1.2 × 10 × 10⁷ = 1.2 × 10⁸, which is 120,000,000.
Which is greater: 3.2 × 10⁵ or 9.8 × 10⁴?
- a) 3.2 × 10⁵
- b) 9.8 × 10⁴
- c) You cannot tell without writing both out in full
Correct answer: a) 3.2 × 10⁵ — Yes: both a are between 1 and 10, so the greater exponent wins (5 > 4). Written out, 320,000 is greater than 98,000.
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