Standard form - Higher

When writing and working with very large or very small numbers, we use standard form. It makes the numbers easier to write down.

Standard form shows the size of numbers as powers of ten.

Standard form numbers are written as:

A × 10n

A is a number greater than one but less than 10

n is the index or power. It tells you how many lots of 10 to multiply or divide A by. A positive index means multiply and a negative index means divide.

The following are examples of ways in which standard form may be used.

UseExample
Large numbersA population of 120,000,000 microorganisms could be written as 1.2 × 108. We can write the number above as 120,000,000.0. If we move the decimal place eight spaces to the left we get 1.2. This number is between one and 10. Now we put x 108 after 1.2 to show that we moved the decimal place 8 times. Because the original number is greater than one metre we don't put a minus sign before the 8. We have to multiply 1.2 by 10 a total of 8 times to get the 'original' number. It makes a very large number easier to write down.
Small numbersA red blood cell's diameter of 7 μm or 0.000007 m could be written as 7 × 10-6 m. The small number above is written as 0.000007. If we move the decimal place six spaces to the right we get 7.0. This number is between one and 10. We put x 10-6 after 7 to show that we moved the decimal place 6 times. Because the original number is less than one metre we put a minus sign before the 6. We have to divide 7.0 by 10 a total of 6 times to get the 'real' number. It makes a very small number easier to write down.
Question

A cell is viewed with a light microscope with a magnification of ×1,000.

The diameter of the image is 0.02 m.

Use the formula: real size of object = size of image/magnification to calculate the actual diameter of the cell.

Give your answer in metres, using standard form.

0.02 ÷ 1,000

= 0.00002 m

= 2 × 10-5 m

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