KS3 · Maths

Ordering and comparing numbers in standard form

Earth's radius is about 6 × 10⁶ m, Neptune's about 2.5 × 10⁷ m. Which is bigger? Your eyes say Earth. Your eyes are wrong.

Standard form · comparing and ordering

Four planets, one line

Each radius below is written in standard form, A × 10ⁿ. The line is measured in millions of metres, so 6 on the line means 6 × 10⁶ m. Slide each planet to where you think its radius sits, then check. Make your guesses first: the line will show you what the powers of 10 are really doing.

Tidying a number that is not quite in standard form

Problem

Write each of these in standard form: 2,390 × 10⁴, 0.87 × 10⁵ and 9,800 × 10⁻⁸.

Standard form · ordering

Put them in order

Order these five numbers from the smallest to the largest. Two of them are not quite in standard form, so convert those first.

1 · Smallest5 · Largest

  1. 4.2 × 10⁻³

  2. 1.5 × 10²

  3. 9.1 × 10⁻⁴

  4. 0.6 × 10⁻²

  5. 35 × 10⁻⁴

Standard form · very big numbers

How long is a billion seconds?

A billion seconds is 1 × 10⁹ seconds. How many years is that? Slide to your guess, then lock it in.

Your estimate

50 years

0 years100 years

Standard form · prefixes

Giga, tera and friends

A drive holds 30 terabytes. How many bytes is that, written in standard form?

  1. A prefix is a power of 10 in disguise. Tera means 10¹², so 30 terabytes = 30 × 10¹² bytes.
  2. missing step
Which line is step 2?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Why the exponent shouts first and A only whispers

What you need to know

  • Standard form is A × 10ⁿ, where A is at least 1 and less than 10, and n is an integer (positive, negative or zero). There is always a × in it.
  • A number like 67 × 10¹¹ or 0.76 × 10ⁿ is not quite in standard form, because A is 10 or more, or less than 1. Fix A, then change the exponent to pay for it: add to it if A was too big, subtract from it if A was too small.
  • Once every number is in standard form, the greater exponent means the greater number. If the exponents are the same, the greater A means the greater number.
  • A positive exponent gives a large number and a negative exponent gives a small one. Each extra power of 10 makes a number 10 times as big.
  • Prefixes are powers of 10: kilo = 10³, mega = 10⁶, giga = 10⁹, tera = 10¹², micro = 10⁻⁶.

The big picture

Standard form lets you compare huge and tiny numbers quickly. Put every number into standard form first, compare the exponents, and look at A only when the exponents match. Each extra power of 10 makes a number ten times as big.

Key points

1Standard form: A × 10ⁿ with 1 ≤ A < 10 and n an integer (positive, negative or zero).
2Not quite standard form? Fix A, then adjust the exponent. A too big: add. A too small: subtract.
3Compare in standard form: exponent first, then A only if the exponents are equal.
4One more power of 10 means ten times as big: 6 × 10⁷ is 10 times 6 × 10⁶.
5Symbols for comparing: = equal, ≠ not equal, < less than, > greater than, ≤ less than or equal, ≥ greater than or equal.
6Standard form is how people write very large and very small numbers in science, such as a billion seconds, which is about 31.7 years.

Worked example

Problem

Which is bigger: 7.2 × 10⁵ or 41 × 10⁴?

⚠ Watch out

Judging by A first. 9.9 × 10³ has the bigger A, but 1.1 × 10⁵ is the bigger number, because the exponent decides before A gets a say. And do not compare numbers until they are all properly in standard form.

🧠

Memory hook

The exponent is the size of the box; A is where you stand inside it. Different boxes: the bigger box wins. Same box: whoever stands further along wins. (Each box is ten times the last, not just a bit bigger.)

✓

Check yourself

Is 85 × 10⁻⁴ bigger or smaller than 6 × 10⁻³? Convert, then exponent, then A. Answer: 85 × 10⁻⁴ = 8.5 × 10⁻³, and 8.5 > 6, so it is bigger.

Flashcards

(14)
What is standard form?
A number written as A × 10ⁿ, where A is at least 1 and less than 10, and n is a whole number (positive, negative or zero). There is always a × between A and the power of 10.
Why is 67 × 10¹¹ not quite in standard form?
A = 67 is too big. A has to be less than 10.
Why is 0.76 × 10ⁿ not quite in standard form?
A = 0.76 is too small. A has to be at least 1.
A is too big. What do you do to the exponent?
Write A as a number between 1 and 10 times a power of 10, then add to the exponent. For example 960 × 10⁹ = 9.6 × 10² × 10⁹ = 9.6 × 10¹¹.
A is too small. What do you do to the exponent?
Each 1/10 you take out of A means one fewer 10 multiplied, so you subtract from the exponent. For example 0.0034 × 10⁷ = 3.4 × 10⁴.
Two numbers are both in standard form. What do you compare first?
The exponents. The greater the exponent, the greater the number.
The exponents are the same. How do you decide which is bigger?
Compare the values of A. For example 6.4 × 10⁵ is greater than 6.1 × 10⁵.
What does the exponent tell you about the size of a number?
Roughly how large or how small it is: a positive exponent gives a large number and a negative exponent gives a small one.
How many times bigger is one number than another when they have the same A and the exponent is one more?
10 times bigger. Each extra power of 10 multiplies by another 10, so 2 × 10⁸ is 10 times 2 × 10⁷.
Why convert to standard form before comparing?
Because the rule 'exponent first, then A' only works when every number is in standard form. A number like 35 × 10⁻⁴ hides its true exponent.
What do the prefixes kilo, mega, giga, tera and micro mean?
kilo = 10³, mega = 10⁶ (a million), giga = 10⁹ (a billion), tera = 10¹², micro = 10⁻⁶.
Convert 128 gigabytes to bytes in standard form.
128 × 10⁹ = 1.28 × 10² × 10⁹ = 1.28 × 10¹¹ bytes.
What do the symbols =, ≠, <, >, ≤ and ≥ mean?
= equal to; ≠ not equal to; < less than; > greater than; ≤ less than or equal to; ≥ greater than or equal to.
About how many years is a billion seconds?
About 31.7 years, using 365 days in a year. A billion is 1 × 10⁹, which is 1,000 times a million.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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