If you work through this section you should be able to:
If you are unsure of the basic rules of working with powers, or wish to refresh your memory, see the pages on powers in the Mathematical operations section before looking at standard form.
Any number can be expressed as a value between 1 and 10, multiplied by a power of 10
4786 = 4.786 × 103
A number expressed in this way is said to be written in standard form. The term scientific notation may also be encountered.
This way of writing numbers is very useful, particularly if the numbers are very large or very small.
Because our number system is based on the number 10, it is particularly easy to multiply and divide by 10 or any power of 10, such as 100 (102), 1000 (103), etc.
To convert a large number to standard form, move the decimal point to the left until there is only one digit to the left of the point, so you know you have a number greater than 1 and less than, or equal to, 10. The number of places you have moved the point is the required power of 10 by which you will need to multiply this new number.
Move the decimal point n places to the left, until there is only one digit to the left of the decimal point, then multiply this new number by 10n.
245 = 2.45 × 100 = 2.45 × 102
49,827 = 4.9827 × 10,000 = 4.9827 × 104
53,000,000 = 5.3 × 10,000,000 = 5.3 × 107
It is common to round the base number at this stage.
So, 123456789 = 1.23456789 × 108, which is probably more precise than is required so this can be rounded to, say, 1.23 × 108.
Special case
The base number can actually be 1, so:
1,000,000 = 1 × 106
We can also express numbers that are less than 1 in standard form.
0.0034 = 3.4 ÷ 103
However, instead of using the division sign we use the notation of a negative power to indicate dividing by that power of 10.
i.e. 10−n = 1 over 10 to the power of n
So, to convert a small number to standard form, move the decimal point to the right until there is one non-zero digit to the left of the decimal point. Leading zeroes are then not shown. The number of places you have moved the point is the required negative power of 10 by which you will need to multiply this new number.
Move the decimal point n places to the right until there is one non-zero digit to the left of the decimal point.
Then multiply this new number by 10−n.
0.012 = 1.2 ÷ 102 = 1.2 × 10−2 (had to move 2 places so 10−2)
0.0047 = 4.7 ÷ 103 = 4.7 × 10−3 (had to move 3 places so 10−3)
0.000002 = 1 ÷ 106 = 2 × 10−6 (had to move 6 places so 10−6)
You can download a version of this Standard form activity in Word format:
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