# Small Numbers

## Scientific Notation of Numbers

Scientific notation is the way of writing numbers that are very large or very small in an abbreviated form, using exponentiation.
According to scientific notation, the number will be represented as the product of a certain number between $1$ and $10$ multiplied by $10$ raised to some power.
That is:

$m \times 10^e$

$m$ will be a number between $0$ and $1$
If $e$ is a positive integer, the entire expression will be some number greater than $1$
If $e$ is a negative integer, the entire expression will be some number less than $1$

### Small Numbers

Steps for the notation of very small numbers according to scientific notation:

1. We will move the decimal point to the right until it is placed after a number that is less than $10$.
2. We will count how many steps we have moved the decimal point to the right, and the number of steps will be the exponent of $10$, only this time, in negative.
3. We will multiply the $10$ raised to the power we found (in negative) by our number $m$ and thus arrive at the scientific notation.

## Scientific Notation of Numbers

### What does it mean?

In certain scientific subjects such as, for example, biology and chemistry, there are extremely large or infinitesimally small numbers.
For example:
The mass of planet Earth is $6,000,000,000,000,000,000,000,000$ kg.
or
The radius of a carbon atom is $0.000,000,000,07$ m.
To express these numbers in a simple and practical way, without having to write a lot of digits, we can use powers.

Scientific notation is a system for expressing very large or very small numbers in a practical way.
According to scientific notation, the number will be represented as the product of a certain number between $1$ and $10$ multiplied by $10$ raised to some power.
That is:

$m \times 10^e$

$m$ will be a number between $0$ and $1$
If $e$ is a positive integer, the whole expression will be some number greater than $1$
If $e$ is a negative integer, the whole expression will be some number less than $1$

Let's remember that, when we have a decimal number like, for example:
$5.32$
and we move the decimal point one step to the right, we are actually multiplying the number by $10$.
That is, if we multiply
$5.32$
by $10$
We will get:
$53.2$
Similarly, if we move the decimal point one step to the left, we are actually dividing the number by $10$.

To write large and small numbers in a practical way we will use the powers of $10$.
You'll catch on right away.
Let's take an example of a number that is not too large: $183$
If we move our imaginary decimal point one step to the left, we are actually dividing the number by $10$.
Therefore, to not alter the numerical value we must immediately multiply it by $10$.
That is:

$183=18.3 \times 10$

If we move the point another step back, we must multiply by $100$.
That is:
$183=18.3 \times 10=1.83 \times 100$
We know that, $100$ can be written as $10^2$
Therefore, we can express $183$ as: $1.83 \times 10^2$
Similarly, if we take a larger number, for example: $5,000,000$
We can say it is equivalent to:
$5 \times 1,000,000$
We know that, $1,000,000$ is equivalent to $10^6$
Therefore:
$5,000,000=5 \times 1,000,000=5 \times 10^6$
So how can we write numbers using scientific notation without getting confused?

### Small Numbers

Steps for the notation of very large numbers according to scientific notation:

1. We will move the decimal point to the right until it is placed after a number that is less than $10$.
2. We will count how many steps we have moved the decimal point to the right. The number of steps taken will be the exponent of $10$, only this time, in negative.
3. We will multiply the $10$ raised to the power we found (in negative) by our number $-m$ and that will give us the scientific notation.

Let's take the following number as an example:
$0.00654$

We moved the decimal point to the right until it was behind a number greater than $0$.
We counted the number of steps taken and got $3$. Since we moved the decimal point to the right, the exponent will be the number of steps taken in negative, that is $-3$.

Therefore, we will obtain:
$6.54 \times 10^{-3}$

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