Scientific notation

The sun is about 150,000,000 kilometres away, a virus measures around 0.0000001 metres. Writing such numbers by hand is tedious and error prone. Scientific notation makes them short, clear and comparable.

The idea: a times a power of ten

You write every number as $a \cdot 10^n$. Here $a$ is at least 1 and smaller than 10, and $n$ is a whole number. The factor $a$ names the meaningful digits, the power $10^n$ just shifts the decimal point. That separates the meaning from the sheer size. The rule for $a$ is strict: $40 \cdot 10^2$ does give 4000, yet it is not scientific notation, because 40 is not smaller than 10. Written correctly the number is $4 \cdot 10^3$.

Writing large numbers

For large numbers the exponent tells you how often you move the decimal point to the right. $6 \cdot 10^4$ means: take the 6 and move the point four places to the right, giving 60,000. The larger $n$, the larger the number, because each step in $n$ is a factor of 10.

Small numbers and negative exponents

For very small numbers the exponent turns negative: the point moves to the left. $1 \cdot 10^{-9}$ is one nanometre, that is 0.000000001 metres. A negative $n$ does not mean a negative number, it means a tiny fraction of 1.

Calculating with powers of ten

You multiply two numbers in this notation in two separate steps: multiply the front factors, meaning the numbers in front of the power of ten, and add the exponents. So $(2 \cdot 10^3) \cdot (3 \cdot 10^2)$ becomes $6 \cdot 10^5$. Adding works because every step in the exponent stands for one more factor of 10: multiplying by 10 three times and then twice more means multiplying by 10 five times in total.