Exponents: Scientific Notation

Why does scientific notation exist?

Scientists are often working with very large and very small numbers. Rather than writing out the entire number, scientific notation allows for shortening the expressions. For instance, instead of saying that a water molecule is about 0.00000000000275 meters across, we can say that its width is about 2.75×10⁻¹² meters.

In day-to-day life, most people don't encounter gigantic or tiny numbers. Scientific notation was invented to make life easier for scientists, who often deal with astronomical numbers (in the literal sense of measuring distances between stars) and microscopic numbers (in the literal sense of needing some sort of microscope to see what they're measuring). Scientific notation makes things easier to handle.

How does scientific notation work?

The format for writing a number in scientific notation is fairly simple: (first [non-zero] digit of the number) followed by (the decimal point) and then (all the rest of the digits of the number), times (10, raised to an appropriate power).

To convert a big number to scientific notation, follow these steps:

1. Take the original value, and move the decimal point from after the last digit to after the first digit. For instance, take 7,375,498,555 and move the (understood) decimal point from after the final 5 to after the 7, to get 7.375498555.
2. Count the number of spots that the decimal point has been moved. In this example, the point moved 9 places.
3. Multiply the converted number by 10, raised to the power of however many places the point has been moved. In this example, the scientific notation is 7.375498555×10⁹.

The process for converting a small number to scientific notation is similar:

1. Take the original value, and move the decimal point from its original location to the spot right after the first non-zero digit. For example, take 0.00000000000275 and move the decimal point from after the first zero to right after the 2, to get 2.75.
2. Count the number of spots that the decimal point has been moved. In this example, the point moved 12 places.
3. Multiply the converted number by 10, raised to the power of the negative of however many places the point has been moved. In this example, the scientific notation is 2.75×10⁻¹².

Whether you're converting large or small numbers to scientific notation, the part that comes before the "times 10 to some power" is called the coefficient; the 10 is the base; and the power is, well, the power.

How do you know what sign to use for the power on 10?

A large number is going to involve 10 to some positive value, such as 10⁷, which is 1,000,000. A small number is going to involve 10 to some negative value, such as 10⁻⁷, which is 0.0000001. So expect positive powers when you're converting big numbers to scientific notation, and negative powers when you're converting small numbers.

In practice, the conversion process is fairly simple.

• Write 124 in scientific notation.

In the example above, I used mathematical reasoning to explain why the power on 10 needed to be 2. But converting between "regular" notation and scientific notation is even simpler than I just showed, because all you really need to do is count decimal places. To do the conversion for the previous example, I'd count the number of decimal places I'd moved the decimal point. Since I'd moved it two places, then I'd be dealing with a power of 2 on 10. But should it be a positive or a negative power of 2? Since the original number (124) was bigger than the converted form (1.24), then the power should be positive.

• Write in decimal notation: 3.6 × 10¹²

...or 3.6 trillion.

Idiomatic note: "Trillion" means a thousand billion — that is, a thousand thousand million — in American parlance; the British-English term for the American "billion" would be "a milliard", so the American "trillion" (above) would be a British "thousand milliard".

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• Write 0.000 000 000 043 6 in scientific notation.

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• Convert 4.2 × 10⁻⁷ to decimal notation.

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• Convert 0.000 000 005 78 to scientific notation.

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• Convert 93,000,000 to scientific notation.

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Warning: A negative on an exponent and a negative on a number mean two very different things! For instance:

Don't confuse these!

You might be asked to multiply and divide numbers in scientific notation. I've never really seen the point of this, since, in "real life", you'd be dealing with these messy numbers by using a calculator, but here's the process, if you have to "show your work":

• Simplify and express in scientific notation: (2.6 × 10⁵) (9.2 × 10⁻¹³)

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Dividing numbers in scientific notation works about the same way.

• Simplify and express in scientific notation: (1.247 × 10⁻³) ÷ (2.9 × 10⁻²)

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If you are required to do problems like these, remember that you can always check your answers in your calculator. For instance, entering "1.247 EE −3 ÷ 2.9 EE −2" on my calculator returns "0.043", which equals 4.3 × 10⁻² in scientific notation.

If you have to do a lot of these problems, you may find it useful to set your calculator to display all values in scientific notation. Check your owner's manual for instructions.