Decimals, powers of ten and bounds

A city can have several million inhabitants, while a pollen grain is only a few hundredths of a millimetre long. To write very large or very small numbers without a long row of zeros, we use powers of 10. In this lesson you write a decimal in the form $a\times10^n$ with $n$ an integer, you bound a decimal and you give approximate values.

Powers of 10 with integer exponents

For a whole number $n$: $10^n=1\,0\dots0$ ($n$ zeros) and $10^{-n}=\dfrac{1}{10^n}=0.0\dots01$ (the digit $1$ is in the $n$-th place after the decimal point).

Rules: $10^m\times10^n=10^{m+n}$ and $\dfrac{10^m}{10^n}=10^{m-n}$.

Multiplying by $10^n$ moves the decimal point $n$ places to the right; multiplying by $10^{-n}$ moves it $n$ places to the left.

The form $a\times10^n$ and scientific notation

Every decimal can be written as a whole number times a power of 10: $3.45=345\times10^{-2}$.

Scientific notation is the form $a\times10^n$ where $1\le a<10$ and $n$ is an integer:

• $3\,450=3.45\times10^3$;
• $0.007\,2=7.2\times10^{-3}$.

To find $n$, count how many places the decimal point must move to get a number $a$ between $1$ and $10$. A number smaller than $1$ has a negative exponent.

Bounds and approximate values

To bound $x$ is to find two numbers $a$ and $b$ such that $a<x<b$. The difference $b-a$ is the width of the bound. For $x=7.468$:

• to the unit: $7<x<8$;
• to the tenth: $7.4<x<7.5$;
• to the hundredth: $7.46<x<7.47$.

The left number is the lower approximate value, the right one the upper approximate value.

Rounding means choosing the closer of the two. Look at the next digit: if it is $5$ or more, take the upper value; otherwise, take the lower value. $7.468$ rounded to the hundredth is $7.47$.