Large and small positive numbers

Writing large numbers with powers of 10

Some numbers are so large that writing all the zeros is not practical. Instead, we can write them as a single digit multiplied by a power of 10.

A power of 10 is written as $10^n$, where $n$ tells us how many zeros follow $1$.
For example:
- $1{,}000 = 1 \times 10^3$
- $7{,}500 = 7.5 \times 10^3$

This form is called scientific notation. It makes it easier to estimate, compare, and calculate with large numbers.

Writing small numbers with powers of 10

Small numbers can also be expressed using powers of 10, but with negative exponents. A negative exponent means division by 10.

For example:
- $0.1 = 1 \times 10^{-1}$
- $0.005 = 5 \times 10^{-3}$

Another way to see this is as a unit fraction:

$0.005 = \dfrac{5}{1000} = 5 \times \dfrac{1}{10^3} = 5 \times 10^{-3}$

This helps us write tiny quantities more clearly and compare them more easily.

Comparing quantities

Once numbers are in exponential form, it is easier to see how many times bigger one number is than another.

Example:
- Earth’s population $\approx 8 \times 10^9$
- A city’s population $\approx 2 \times 10^6$
- Earth’s population is about $4{,}000$ times larger:
  $$\dfrac{8 \times 10^9}{2 \times 10^6} = 4 \times 10^3 = 4{,}000$$

This shows how exponential form makes comparisons simpler.

Real-world examples

We often use scientific notation in science, technology, and daily life.

Distance in space
The distance from Earth to the Sun is about $150{,}000{,}000$ km. In exponential form, that is $1.5 \times 10^8$ km.
Tiny objects
The diameter of a human hair is about $0.00008$ m. That can be written as $8 \times 10^{-5}$ m.
Data storage
A smartphone may store about $256{,}000{,}000{,}000$ bytes. In exponential form, this is $2.56 \times 10^{11}$ bytes.

Summing it up