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0.8 Scientific notation by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.

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=3.2×10^1`

`320=3.2×10^2`

`3200=3.2×10^3`...

Each number is ten times bigger than the previous one.

Since `10^1` is ten times smaller than `10^2` , it makes sense to use the notation `10^0` to stand for one, the number that is in turn ten times smaller than `10^1`. Continuing on, we can write `10^(?1)` to stand for `0.1`, the number ten times smaller than `10^0` . Negative exponents are used for small numbers:

`3.2=3.2×10^0`

`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:

`3.2×10^6` | (written notation) |

`3.2`E+`6` | (notation on some calculators) |

`3.2^6` | (notation on some other calculators) |

The last example is particularly unfortunate, because `3.2^6` really stands for the number `3.2×3.2×3.2×3.2×3.2×3.2=1074`, a totally different number from `3.2×10^6=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 E**

A student learns that `10^4` bacteria, standing in line to register for classes at Paramecium Community College, would form a queue of this size:

The student concludes that 10^{2} bacteria would form a line of this length.

Why is the student incorrect? (answer in the back of the PDF version of the book)

0.8 Scientific notation by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.