Standard form and order of magnitude

Eight real things, laid out by size. The strip is one step per power of ten. Slide the marker and read how many times bigger it is than a grain of sand.

10⁻¹⁰

a hydrogen atomroughly this size

0.0000001×a grain of sand

Three steps along the strip is not three times bigger. It is a thousand times.

The two rules

Standard form is a × 10^k where 1 ≤ a < 10 and k is an integer. Both conditions, every time. If a is 10 or more, or less than 1, it is not in standard form yet however correct the value is.

Written	Standard form?	Correct version
4.5 × 10⁷	yes	,
52 × 10⁶	no, a is 10 or more	5.2 × 10⁷
0.52 × 10⁸	no, a is under 1	5.2 × 10⁷
5.2E7	no, calculator notation	5.2 × 10⁷

Never write 5.2E7 in an answer. It is what the calculator shows, not what mathematics looks like, and it is explicitly not accepted. Copy it off the screen and rewrite it.

Arithmetic

multiply→ (3 × 10⁵)(2 × 10³) = 6 × 10⁸ multiply the fronts, add the powers divide→ (6 × 10⁸) ÷ (3 × 10⁵) = 2 × 10³ add→ 3 × 10⁵ + 2 × 10⁴ = 3.2 × 10⁵

Adding is the one that catches people. You cannot add the powers: 3 × 10⁵ + 2 × 10⁴ is not 5 × 10⁹. Rewrite both with the same power first, so 2 × 10⁴ becomes 0.2 × 10⁵, then add the fronts.

Worked example

A single red blood cell is about 7 × 10⁻⁶ m across. A person has about 2.5 × 10¹³ of them.

Laid in a line, how far would they stretch? Give your answer in standard form, and in kilometres.

Total = (7 × 10⁻⁶) × (2.5 × 10¹³)One cell's width, that many times. Multiplying, not adding.
= (7 × 2.5) × (10⁻⁶ × 10¹³)Group the fronts and the powers. This is the whole method.
= 17.5 × 10⁷Multiply the fronts, add the powers: −6 + 13 = 7.
= 1.75 × 10⁸ m17.5 is not between 1 and 10, so move the point one place and raise the power by one.
1.75 × 10⁵ km, about 175,000 kmA thousand metres to a kilometre, so the power drops by 3. That is nearly half way to the Moon, from one person's blood.

Your turn

1. Write 52 × 10⁶ in standard form. Give the value of a.

2. How many times bigger is 10⁶ than 10³?

3. 3 × 10⁵ + 2 × 10⁴ equals:

5 × 10⁹ 3.2 × 10⁵ 5 × 10⁵ 3.2 × 10⁹

Where the marks go

Giving the final answer in proper standard form, with 1 ≤ a < 10. A correct value written as 52 × 10⁶ is not finished.

Rewriting calculator output. 5.2E7 scores nothing.

Matching the powers before adding or subtracting, which is a method mark in its own right.

Come back to this

These three are deliberately not all about this page. In an examination the hardest step is deciding which method applies, and questions that arrive straight after the method never make you decide. Answer from memory before you reveal anything.

Come back to these in a week, and again in a month.

Which method?

A sequence begins 5, 8, 11, 14. Find the 20th term.

Arithmetic, not standard form. The gaps are equal, so uₙ = u₁ + (n−1)d gives 5 + 19(3) = 62. Equal gaps means add; equal ratios would mean multiply.
Which method?

Light travels 9.46 × 10¹⁵ m in a year. How many times further is that than the Earth's diameter, 1.27 × 10⁷ m?

Standard form division: divide the fronts, subtract the powers. 9.46/1.27 ≈ 7.45 and 15 − 7 = 8, so about 7.45 × 10⁸ times.
Which method?

1000 is invested at 5% compounded yearly. What is it worth after 10 years?

Compound interest, a geometric process. 1000 × 1.05¹⁰ = 1628.89. The clue is a percentage applied repeatedly.

The same idea elsewhere

4.9, the normal distribution: Standard form appears whenever a probability is tiny, such as e⁻⁶ = 2.48 × 10⁻³.
1.6, approximation and bounds: Significant figures and standard form are the same idea: which digits actually carry information.

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