Scientific Notation

Write numbers as a × 10^n and compute with them.

The idea

Scientific notation is the standard compact form for very large and very small numbers, whose ordinary form is mostly placeholder zeros.

Each power of ten shifts a decimal point by one place, so we can write any nonzero number as a small leading number times a power of ten: $a \times 10^{n}, \qquad 1 \le |a| < 10, \quad n \text{ an integer}.$ The restriction on $a$ puts exactly one nonzero digit in front of the decimal point, so each number has exactly one such form and we can compare two numbers at a glance. To find $n$, move the decimal point until one nonzero digit sits in front of it, and count the places it moved: $n$ is positive for a large number and negative for one smaller than $1$. Zero has no such form and is written $0$.

Since the two parts are multiplied together, arithmetic splits in two as well: handle the leading numbers on their own, and combine the powers of ten with the usual exponent rules.

Ways to work on it

• Walkthrough. Convert large and small numbers to and from a × 10^n form.
• Practice. Convert a standard-form number to scientific notation.
• Hardest. Multiply in scientific notation and renormalize the result.

Not sure where to start? Take the ten-question placement test.