What’s the biggest number you can think of? Depending on a kid’s age, they may answer 100 or a million or infinity (infinity is a concept, not a number). What’s the smallest number you can think of? Sometimes kids answer this with 1 or zero or a negative number. With elementary aged kids, I often have to remind them that there are many numbers possible between 1 and 0 (the fractions or decimals).

The problems with very big and very small numbers are* (1) How do you express them?* and *(2) How do you conceptualize or understand them?*

1. To deal with the first problem, we have scientific notation. Scientific notation is a way to write numbers that are too big or two small to write in the conventional decimal notation. (If you are doing a calculation using very big or very small numbers, it can be inconvenient to write all those zeros.) As an example, 3,000,000 written in scientific notation is 3 x 10^{6}, because this is the same as 3 x 10 x 10 x 10 x 10 x 10 x 10. Similarly, a small number such as 0.000003 would be written as 3 x 10^{-6}, because it is the same as 3 / (10 x 10 x 10 x 10 x 10 x 10).

The table below shows some numbers in decimal notation, their American names (Did you know that a billion does not mean the same thing in all countries?), and the scientific notation. Do you have a large number that you don’t know how to name? Try this interactive site.

2. To address the second problem with very large and very small numbers – how to conceptualize them – I like to have kids play with this fun interactive website: Scale of the Universe 2. This site shows the relative sizes of things from theoretical strings and quantum foam (on the order of 10^{-35} meters) to the observable universe (on the order of 10^{27} meters). Seeing the differences in scale is pretty mind-blowing, and students enjoy finding out that the site was developed by a 14-year-old boy with help from his twin brother.

Screenshot from Scale of the Universe 2