Scientific notation

Physics deals with numbers that no human being can read correctly: the mass of an electron is 0.000 000 000 000 000 000 000 000 000 000 91 kg.

Scientific notation brings everyone into agreement. A number is written in the form:

a x 10^n        avec  1 <= a < 10

The mass of the electron becomes 9,1 x 10^-31 kg. It is the same number, but we can finally read it, compare it and calculate with it.

Some examples

Number	Scientific notation
1,500	1,5 x 10^3
150,000,000,000	1,5 x 10^11
0.00007	7 x 10^-5

The rule of thumb: move the decimal point until there is only one digit to the left of it, and count the places. To the left, the exponent is positive; to the right, it is negative.

Order of magnitude

This is the nearest power of 10. It answers the question ‘how many zeros?’ without getting bogged down in the details.

• height of a human: 10^0 m (approximately 1 m)
• height of the Eiffel Tower: 10^2 m
• radius of the Earth: 10^7 m
• distance between the Earth and the Sun: 10^11 m

Comparing two orders of magnitude is straightforward: between 10^7 and 10^11, there is a factor of 10^4, or ten thousand. The distance from the Earth to the Sun is ten thousand times the Earth’s radius — a calculation done in your head, without writing down a single operation.

What’s the real point?

To spot absurd results. If a calculation gives you a speed of 10^9 m/s, you know straight away that there’s a mistake: that’s faster than the speed of light (3 x 10^8 m/s).

This order-of-magnitude check takes three seconds and catches the vast majority of calculation errors. It’s the most cost-effective reflex in all of physics.