Scientific Notation / Standard Form
Scientific Notation (also called Standard Form) is where a number
is written in two parts:
- Just the digits (with the decimal point placed after the first digit), followed by
- ×10 to a power that would put the decimal point back where it should be (ie it shows how many places to move the decimal point).
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| In this example, 5326.6 is written as 5.3266 × 103, because 5326.6 = 5.3266 × 1000 = 5326.6 × 103 |
To figure out the power of 10, think "how many places do I move the decimal point?"
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If the number is 10 or greater, the decimal place has to move to the left, and the power of 10 will be positive. |
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If the number is smaller than 1, then decimal place has to move to the right, so the power of 10 will be negative:
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Example: 0.0055 would be written as 5.5 × 10-3, because 0.0055 = 5.5 × 0.001 = 5.5 × 10-3
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Check
After putting the number in Scientific Notation, just check that:
- The "digits" part is between 1 and 10 (it can be 1, but never 10)
- The "power" part shows exactly how many places you moved the decimal point
Why Use It?
Because it makes it easier when you are dealing with very big or very small numbers, which are common in Scientific and Engineering work.
For example it is easier to write (and read) 1.3 × 10-9 than 0.0000000013
And it can make calculations easier, as in this example:
Example: a tiny space inside a computer chip has been measured to be
0.00000256m wide, 0.00000014m long and 0.000275m high.
What is its volume?
Let's first convert the three lengths into scientific notation:
- width:
0.000 002 56m = 2.56×10-6
- length: 0.000 000 14m = 1.4×10-7
- height: 0.000 275m = 2.75×10-4
Then multiply the digits together (ignoring the ×10s):
2.56 × 1.4 × 2.75 = 9.856
Last, multiply the ×10s:
10-6
× 10-7 × 10-4 = 10-17 (this was easy: I just added -6, -4 and -7 together)
The result is 9.856×10-17 m3
Engineering Notation
Engineering Notation is like Scientific Notation, except that you only use powers of ten that are multiples of 3 (such as 103, 10-3, 1012 etc).
Example: 19,300 would be written as 19.3 × 103
Example: 0.00012 would be written as 120 × 10-6
Notice that the "digits" part can now be between 1 and 1,000 (it can be 1, but never 1,000).
The advantage is that you can replace the ×10s with Metric Numbers. So you can use standard words (such as thousand or million) prefixes (such as kilo, mega) or the symbol (k, M, etc)
Example: 19,300 meters would be written as 19.3 × 103 m, or 19.3 km
Example: 0.00012 seconds would be written as 120 × 10-6 s, or 120 microseconds
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