# Scientific Notation

Some people have a strong resistance to using scientific notation^{*}. Almost every time I teach an introductory science class, I have one or two students with good math skills who insist on doing all of their calculations in standard notation. Doing this invariably results in mistakes that lead to lost points on exams and homework.

Making sense of scientific notation does take a bit of effort, but once learned, it becomes an effective way to save time and avoid mistakes when working with very large and very small numbers. This is especially useful in applied work where calculators and computers are used. It is much easier to avoid mistakes when writing 4.6x10^{-14} as compared to writing 0.000000000000046.

Converting a number from standard to scientific notation involves moving the decimal place until there is one digit to the left of the decimal, then multiplying it by 10 raised to the number of places moved. 300 written in scientific notation is 3x10^{2}. Three times ten to the two. Ten raised to the power of two means that to get back to standard notation, the decimal needs to be moved two places. The sign of the exponent^{*} indicates which way to move the decimal.

A positive exponent means the number is greater than one and the the decimal will be moved to the right to convert from scientific to standard notation. 3.0 x10^{2} becomes 300 when the decimal is moved.

A negative exponent means the number is less than one and the decimal will be moved to the left to convertfrom scientific to standard notation. 3.0 x10^{-2} becomes 0.03 when the decimal is moved.

The mathematical explanation of the link between the number of places the decimal is moved and the power 10 is raised to stems from the fact that moving the decimal one place is the same as dividing or multiplying the number by 10. To change 300 to 3.0 x 10^{2}, 300 is divided by 10 two times:

- First division by 10: 300 ÷ 10 = 30 x 10
^{1} - Second division by 10: 30 x 10
^{1}÷ 10 = 3 x 10^{2}

Therefore, 3.0 x 10^{2}, 3.0 x 10 x 10 and 300 all represent the same value.

The same pattern works regardless of the size of the nubmer:

2.3 x x 10^{12} = 2.3 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 2,300,000,000,000

Very small numbers work the same way except that converting from standard to scientific notation involves multiplying by 10, not dividing. Multiplying by 10 moves the decimal to the right. To write the number 0.0003 in scientific notation, the number is repeatedly multiplied by 10 until the first non-zero number (in this case a 3) is just to the left of the decimal place:

- First multiplication by 10: 0.0003 x 10 = 0.003 x 10
^{-1} - Second multiplication by 10: 0.003 x 10
^{-1}x 10 = 0.03 x 10^{-2} - Third multiplication by 10: 0.03 x 10
^{-2}x 10 = 0.3 x 10^{-3} - Fourth multiplication by 10: 0.3 x 10
^{-3}x 10 = 3.0 x 10^{-4}

The exponent is negative because the the original number is less than one.

Two different formats can be used for scientific notation. The more traditional “x10” can be replaced with the letter E so that 3200 can be written as 3200, 3.2E3 or 3.2 x10^{3}. The use of E has become more common with the increased use of computers.

The following illustration explores the relationship between numbers written in standard and scientific notation. A video demonstration can be found at the bottom of the post. Be sure to also take a look at the scientific notation practice problems to test your understanding of the concepts covered by this illustration.

Watch the demonstration video or just click 'Set/Reset' to begin.

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