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Orders of magnitude: from atoms to Earth

Estimate familiar length scales, compare powers of ten in a common unit, and distinguish an exponent gap from an exact size ratio.

  • SEC G3 Physics 2027

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An atom and the Earth cannot be usefully compared on an ordinary ruler. Powers of ten let you compare their length scales without writing long strings of zeros. First express every length in metres, then read its standard-form exponent.

Give the kind of length as well as the number

A “size” might be a diameter, height or thickness. Name what is being compared, and treat representative values as estimates rather than universal dimensions.

Object and lengthRepresentative value in metresExponent scale
Gold atom: approximate diameter3 × 10⁻¹⁰ m10⁻¹⁰ m
Sheet of paper: approximate thickness1 × 10⁻⁴ m10⁻⁴ m
Person: illustrative height of 1.7 m1.7 × 10⁰ m10⁰ m
Earth: approximate equatorial diameter1.3 × 10⁷ m10⁷ m

The National Nanotechnology Coordination Office gives approximate atomic and paper dimensions; NASA’s Earth facts give Earth’s equatorial diameter, rounded here. The person’s height is a supplied illustrative example. Actual atoms, paper sheets and people vary; you need their rough scales rather than a list of supposed exact measurements.

Length scales from an atom to EarthA logarithmic number line from 10 to the power minus 10 metres to 10 to the power 8 metres. Each small interval multiplies length by ten. Dots show a gold atom's approximate diameter, 3 times 10 to the power minus 10 metres; paper thickness, 10 to the power minus 4 metres; an illustrative person's height, 1.7 metres; and Earth's approximate equatorial diameter, 1.3 times 10 to the power 7 metres.Equal spacing means an equal ratio10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²10⁰10²10⁴10⁶10⁸Gold atom3 × 10⁻¹⁰ mPaper thickness1 × 10⁻⁴ mPerson's height1.7 mEarth's diameter1.3 × 10⁷ mLength / m
Scroll across the figure to read all labels.
Each small interval is a factor of 10; each interval between labelled ticks is a factor of 100. The dots show lengths, not pictures of objects at their relative sizes. Atom, paper and Earth values are approximate; the person's height is an example.

Count changes in exponent scale

One order of magnitude is a factor of ten in scale; two correspond to a factor of about 100. Using the standard-form exponents above, paper thickness is six exponent steps above the gold atom’s diameter:

-4-(-10) = 6

Negative exponents are smaller scales, not negative lengths. A value of 10⁻¹⁰ m is a very small positive length.

Keep coefficients when finding an actual ratio

The exponent gap compares scales. It does not necessarily give the exact ratio of the values. Using the rounded paper and atom values:

(1 × 10⁻⁴ m)/(3 × 10⁻¹⁰ m) = 1/3 × 10⁶ ≈ 3 × 10⁵

The paper thickness is roughly 300,000 times that atom’s diameter in this comparison, rather than exactly one million times. The length units cancel because both measurements are in metres.

When a question asks specifically for the difference between standard-form exponents, subtract the exponents. If it defines a different approximation convention, such as a nearest power of ten, follow that stated convention. For a numerical ratio, divide the complete values including their coefficients.

Use scale as a plausibility check

Convert an estimate to a common unit and compare it with something familiar. A proposed paper thickness of 0.1 m means 10 cm: it is not plausible for an ordinary sheet. The approximate thickness 10⁻⁴ m instead corresponds to 0.1 mm. Such a check can reveal a reversed prefix conversion.

Compare exponent scales in a worked example

Exam-style question 1

Compare orders of magnitude

2 marks

Examination question

Quantities A = 2.0 × 10⁻³ m and B = 5.0 × 10⁴ m are already in standard form. Using their exponents, state how many orders of magnitude separate their scales. [2 marks]

Compare exponents, not coefficients

Show solution step by step

  1. Identify both exponents

    1 mark

    Method

    Read -3 for A and 4 for B.

    Reason

    Both coefficients already satisfy the standard-form range.

    Working

    nA = -3, nB = 4

  2. Find the separation

    1 mark

    Method

    Subtract the smaller exponent from the larger.

    Reason

    Each exponent step represents a factor-of-ten change in scale; the coefficients are needed for an exact ratio.

    Working

    4-(-3) = 7 orders of magnitude

Try without prompts

Check your understanding 1: Compare exponent scales

The radius of Earth is about 6.4 × 10⁶ m and an atom’s diameter is about 1.0 × 10⁻¹⁰ m. Using the exponents in standard form, how many orders of magnitude separate their scales?

Show answer

The exponents are 6 and -10:

6-(-10) = 16

Their exponent scales are separated by 16 orders of magnitude. This does not mean their exact size ratio is precisely 10¹⁶; the coefficients also affect the ratio.

Mind stretcher 1: Does the exponent gap give the exact ratio?Extension

Length P is 4.0 × 10⁻⁶ m and length Q is 2.0 × 10⁻³ m. Compare their standard-form exponent scales, then calculate Q divided by P. Assess the claim that Q must be exactly 1000 times P because the exponents differ by three.

Show answer

The exponent gap is -3-(-6) = 3. The complete ratio is (2.0/4.0) × 10³ = 500, so Q is 500 times P, not exactly 1000 times. The exponent gap gives the power-of-ten scale comparison; the coefficients also affect the ratio.

Compare like units before comparing scales

State the dimension you mean, express lengths in the same unit and use representative powers of ten for rough scale. Retain coefficients when calculating a ratio, and treat approximate object sizes as approximate.

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