# Scientific Notation

In general, we need scientific notation, if we want to write very big or very small number more compactly.

For example, it is known that mass of the Earth is 5973600000000000000000000 kg. Look how long it is!

To make it shorter, we will use powers of 10. We can count number of zeros and write that $$${5973600000000000000000000}={59736}\cdot{100000000000000000000}={59736}\times{{10}}^{{{20}}}$$$.

But this is not scientific notation, in powers of 10 note, we saw that we can represent number using different powers of ten.

So, how do we define normalized scientific notation?

**Definition.** Scientific notation is when there is only one digit to the left of decimal point, greater than 0.

Let's see on example, what does this mean.

Number 450 can be written in many ways: $$${450}={45}\times{{10}}^{{1}}={4.5}\times{{10}}^{{2}}={0.45}\times{{10}}^{{3}}$$$.

First representation is not applicable, since there are two digits, third representation has one digit to the left of decimal point, but this digit is less than 1. Correct representation is second.

So, $$$450=4.5\times 10^2$$$.

Similarly, $$${0.54}={54}\times{{10}}^{{-{2}}}={5.4}\times{{10}}^{{-{1}}}={0.054}\times{{10}}^{{1}}$$$.

So, $$$0.54=5.4\times 10^{-1}$$$.

In example with the Earth's mass, correct representation is $$${5.9736}\times{{10}}^{{{24}}}$$$.

Note, that if an exponent is $$$0$$$, we just don't write it $$$\left({5}={5}\times{{10}}^{{0}}={5}\right)$$$.

Finally, let's do some exercises.

**Exercise 1.** Write 3562000 in scientific notation.

**Answer**: $$${3562000}={3.562}\times{{10}}^{{{6}}}$$$.

Next exercise.

**Exercise 2.** Write -5678.94 in scientific notation.

**Answer**: $$$-{5678.94}=-{5.67894}\times{{10}}^{{3}}$$$.

Next exercise.

**Exercise 3.** Write $$${0.035}$$$ in scientific notation.

**Answer**: $$${0.035}={3.5}\times{{10}}^{{-{2}}}$$$.

Now, try to convert number in scientific notation back to the number itself.

**Exercise 4.** Write $$${1.35}\times{{10}}^{{3}}$$$ as normal number.

**Answer**: $$${1.35}\times{{10}}^{{3}}={1350}$$$.

Last one.

**Exercise 5.** Write $$$-{3.5}\times{{10}}^{{-{2}}}$$$ as normal number.

**Answer**: $$$-{3.5}\times{{10}}^{{-{2}}}=-{0.035}$$$.