Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information....

# Scientific Notation

Most of the interesting phenomena in our universe are not on the human scale. It would take about 1,000,000,000,000,000,000,000 bacteria to equal the mass of a human body. When the physicist Thomas Young discovered that light was a wave, it was back in the bad old days before scientific notation, and he was obliged to write that the time required for one vibration of the wave was 1/500 of a millionth of a millionth of a second. Scientific notation is a less awkward way to write very large and very small numbers such as these. Here's a quick review.

Scientific notation means writing a number in terms of a product of something from 1 to 10 and something else that is a power of ten. For instance,

32 = 3.2 × 101
320 = 3.2 × 102
3200 = 3.2 × 103 . . .

Each number is ten times bigger than the previous one.

Since 101 is ten times smaller than 102 , it makes sense to use the notation 100 to stand for one, the number that is in turn ten times smaller than 101 . Continuing on, we can write 10-1 to stand for 0.1, the number ten times smaller than 100 . Negative exponents are used for small numbers:

3.2 = 3.2 × 100
0.32 = 3.2 × 10-1
0.032 = 3.2 × 10-2 . . .

A common source of confusion is the notation used on the displays of many calculators. Examples:

3.2 × 106 (written notation)
3.2E+6 (notation on some calculators)
3.26 (notation on some other calculators)

The last example is particularly unfortunate, because 3.26 really stands for the number 3.2 × 3.2 × 3.2 × 3.2 × 3.2 × 3.2 = 1074, a totally different number from 3.2 × 106 = 3200000. The calculator notation should never be used in writing. It's just a way for the manufacturer to save money by making a simpler display.

Self-Check A student learns that 104 bacteria, standing in line to register for classes at Paramecium Community College, would form a queue of this size:

The student concludes that 102 bacteria would form a line of this length:

Why is the student incorrect?

Answer Exponents have to do with multiplication, not addition. The first line should be 100 times longer than the second, not just twice as long.

Last Update: 2009-06-21