In each line, I moved the decimal point one position to the left, which divides the decimal number by 10. But I compensated for that by increasing the exponent of 10 by one, which multiplies that part by 10. The result is that all five of the lines represent an equivalent number.

No matter what the number, you can always pick a representation so that the decimal part is a number that is at least 1 and less than 10. This is called scientific notation in "standard form".

The reason it is called "scientific" notation is that scientists use very large numbers and very small numbers. Consider what would happen if a scientist needed to multiply 11,000,000,000,000 by 0.00000000003.

It's hard to figure this out without scientific notation. But with scientific notation, the first number is 1.1 x 10¹³ and the second is 3 x 10^-11. To find the product, just multiply the decimal parts and add the exponents: 3.3 x 10².

Here's what you need to know:

1 is 10⁰
10 is 10¹
100 is 10²
... etc.

.000987 is 9.87 divided by 10,000. That's 9.87 times 10^-4. Standard scientific notation has a single non-zero digit in front of the decimal point, and this decimal number, called the mantissa, is multiplied by a power of 10, which is called the exponent.

The translation of your example to scientific notation is

9.87 x 10^-4

Sometimes, this is written

9.87e-4.

The exponent is the number of places the decimal point has been shifted. Negative exponents represent very small numbers -- less than one. Positive exponents represent large numbers -- 10 or more. A zero exponent means the mantissa has not had any shifting of the decimal point.

If you need to multiply two numbers in scientific notation, you multiply their mantissas, and add their exponents. Then you may need to renormalize by shifting the decimal point one place to the left, and adding one to the exponent. The following example illustrates all these points:

Example: Multiply 3 x 10² by 4 x 10^-1.

3 x 4 is 12
2 + (-1) is 1
So the answer is 12 x 10¹

Renormalizing this, we get 1.2 x 10².

When you divide, you divide the mantissas and subtract the exponents. You may need to renormalize by moving the decimal point one position to the right, and subtracting one from the exponent.

Example: divide 1.2 x 10² by 3 x 10².

To do this we divide 1.2 by 3, which gives us 0.4, and subtract 2 from 2, giving zero. The result is 0.4 x 10⁰. Normalizing, we get 4 x 10^-1.

Related pages on this website

Help with Unit Conversions: Arithmetic using Physics Units

The webmaster and author of the Math Help site is Graeme McRae.
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