how do you use significant digits?

Question

I’m ashamed to be a senior majoring in math and I still haven’t grasped the concept of amusing significant digits! I seriously need it explained to me like I’m a child.

In: Mathematics

Answer 1

Significant digits are a way of writing numbers to indicate their precision. Every digit except zero is always a significant digit. Zero is only significant if writing it wasn’t necessary to show how big or small the number is. So the zeroes in 100, 20, 0.3, 0.04 are not significant; those numbers all have one significant digit each. On the other hand the zeros in 1.0, and 20.03 are significant.

The assumed precision of numbers is plus or minus half a unit of the least significant (right-most) significant digit, e.g.

* 1 means 1±0.5
* 2.0 means 2.0±0.05
* 30 means 30±5
* 31 means 31±0.5
* 400 means 400±50
* 5600 means 5600±50
* 0.007 means 0.007±0.0005

Sometimes a number has more precision than you can show this way. For example, Everest was once measured to be 29 000 feet high, with high accuracy, but just writing 29 000 implies ±500 feet. The surveyor added two feet to write 29 002, to better indicate the precision implied.

The scientific way out of this issue is to use scientific notation which always has one digit followed by a decimal point, and then by as many extra significant digits as required. The size of the number is indicated by multiplying by a power of ten, e.g.:

* 2.9000×10^4 (the measured height of Everest)
* 1.00×10^3 (equivalent to 1000±5)
* 3.00×10^(−2) (equivalent to 0.03±0.000 05)

Usually you should retain the number of significant digits in calculations. You sometimes see journalists convert measurements in a silly way, so that “1000 miles” becomes “1609 km,” even though the second value implies much more precision than does the first. This one is a tricky case, since staying strictly with the one significant figure of the first measurement would require writing “2000 km,” which will often be misleading. In my view, adding one more significant digit in cases with just a single significant digit that’s a 1 or 2, and writing “1600 km,” is acceptable, provided that no more digits are added in subsequent calculations.

Edit: added example.