KS3 · Maths

Standard form for large numbers

845 × 10, 84.5 × 10² and 0.845 × 10⁴ all equal 8,450, so which one is standard form? Only one passes the rule.

Maths · Number

Seven true lines. Only one is standard form.

All seven lines equal 8,450, so being 'correct' is not the test. Which one line is standard form, and what is wrong with each of the others?

Still to sort

Standard form (0)

A × 10ⁿ, with A from 1 up to (but not including) 10

Where the line is: A has to be at least 1 and less than 10, and the power of 10 has to be multiplied on.

A is 10 or more (0)

The front number is too big

Where the line is: Standard form stops short of 10. Even A = 10 itself is too big.

A is less than 1 (0)

The front number is too small

Where the line is: Standard form starts at 1. Anything below 1 has not gone far enough.

Divided, not multiplied (0)

It divides by a power of 10

Where the line is: Standard form is a multiplication, so there is always × 10ⁿ and never ÷.

7 of 7 still to sort.

Each line below is a correct way to write the number 8,450. Pick a line, then pick the box it belongs in.

Worked example: the place value chart

Problem

Write 98,400 in standard form, using a place value chart whose headings are powers of 10.

Predict, then check: going back out

Standard form is just a short way of writing an ordinary number. Can you unpack one?

What ordinary number is 5.63 × 10⁷?

Maths · No chart this time

Fill in the missing steps

Write 905,000,000,000 in standard form, without a place value chart.

  1. The digits that matter are 9, 0 and 5. Every one of them stays, the zero too.
  2. missing step
Which line is step 2?

Maths · Calculator check

Which keystroke goes wrong?

Sam wants to enter 3.2 × 10⁵ on a calculator (it should give 320,000). Find the line where it goes wrong.

Sam's keystrokes — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

What you need to know

  • Standard form is A × 10ⁿ, where A is at least 1 but less than 10, and for large numbers n is a positive integer.
  • There are endless ways to write a number as a calculation, but only one is standard form. It is always a multiplication by a power of 10, never a division.
  • Find the first significant figure and count the multiplications of 10 that carry it from the ones column to its real column. That count is n, and the digits with the point after the first one make A.
  • Writing a very large number in standard form makes it shorter, cuts copying errors, and loses no accuracy.
  • On a calculator the power of 10 button already includes the × 10, so you must not type it again.

The big picture

Standard form writes a number as A × 10ⁿ with A from 1 up to (not including) 10. For large numbers n is a positive integer, found by counting the multiplications of 10 that carry the first significant figure from the ones column to where it really sits.

Key points

1Standard form: A × 10ⁿ with 1 ≤ A < 10. A can be 1, but never 10.
289.2 × 10ⁿ is not standard form because 89.2 is 10 or more, and 0.016 × 10ⁿ is not because 0.016 is less than 1.
3Standard form is always multiplied by the power of 10, never divided.
4Place value chart: the first significant figure sits under the heading that gives the power of 10, so that heading is n.
5No chart: make A with the first significant figure in the ones column, keeping every digit (zeros included), then count the multiplications of 10 to get n.
6To go back, multiply: 10³ = 1,000, so 2.3 × 10³ = 2,300. Converting your answer back to standard form checks it.

Worked example

Problem

A concert crowd of 3,400,000 people watched online. Write 3,400,000 in standard form.

⚠ Watch out

Dropping or changing digits when you make A. For 905,000,000,000, A is 9.05, not 905 (a whole number) and not 9.5 (the zero has gone). Check that A is between 1 and 10 and that every digit is still there.

🧠

Memory hook

Front digit, point after it, then count the tens that carry it home. The count is n.

✓

Check yourself

62.5 × 10³ equals 62,500. Is it standard form? Fix it if not. (Answer: no, A = 62.5 is 10 or more. It is 6.25 × 10⁴.)

Flashcards

(13)
What is the rule for standard form for a large number?
A × 10ⁿ, where 1 ≤ A < 10 and n is a positive integer. Always a multiplication by a power of 10.
Is 89.2 × 10⁵ in standard form?
No. A = 89.2 is 10 or more, so it is too big. A has to be less than 10.
Is 0.016 × 10⁵ in standard form?
No. A = 0.016 is less than 1, so it is too small. A has to be at least 1.
Can a division by a power of 10 be standard form?
No. Standard form shows a multiplicative relationship, so there is always a multiplication by the power of 10, never a division.
How many ways can you write 530 as a calculation, and how many are standard form?
Endless ways, but only one is standard form.
Give two reasons to write a very large number in standard form.
It is shorter and more concise, and it reduces errors in copying all the digits. It also loses no accuracy and is easier to use in calculations.
What does 10³ mean, and what is it equal to?
10 × 10 × 10 is the same as 10³, which is 1,000. A power tells you how many 10s are multiplied together.
How does a place value chart give you n?
Head the columns with powers of 10, place the digits, then find the column under the first significant figure. That heading's exponent is n.
Without a chart, how do you find n?
Make A with the first significant figure in the ones column, then count the multiplications of 10 needed to move it to its correct column. That count is n.
Write 2.3 × 10³ as an ordinary number.
2.3 × 1,000 = 2,300.
How do you check a conversion from standard form to an ordinary number?
Convert your ordinary number back into standard form and see if you get the number you started with.
Why must you not type an extra × 10 before the power of 10 button?
The power of 10 button already includes the multiplication by 10. An extra × 10 makes the answer ten times too big.
What does a calculator do when an answer is too big to display in full?
It converts the answer to standard form automatically, for example 82 million × 25,000 is displayed as 2.05 × 10¹².

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