# Large 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 is represented as the product of another number that is between $1$ and $10$ multiplied by $10$ and 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 number, the entire expression will be some number greater than $1$
If $e$ is a negative integer number, the entire expression will be some number less than $1$

### Large Numbers

Ways to notate very large numbers using scientific notation:

1. We will move the imaginary decimal point, which is located at the end of the number, until we get a certain number between $1$ and $10$.
2. We will count how many steps we have moved the point to the left. The number of steps taken will be the exponent of $10$.
3. We will multiply the $10$ raised to the power we found 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 such numbers in a simple and practical way, without having to write so many figures, we can use powers.

Scientific notation is a way of writing numbers that are either very large or very small in a convenient form.
According to scientific notation, the number is represented as the product of another number that is between $1$ and $10$ multiplied by $10$ and raised to some power.
That is:

$m\times 10^e$

$m$ will be a number between
if $e$ is a positive integer, the entire expression will be a certain number greater than $1$
if $e$ is a negative integer, the entire expression will be a certain 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 obtain:
$53.2$
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 will see it shortly.
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, in fact, we divide 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 that 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?
We will start with the scientific notation of large numbers.

### Large Numbers

The ways to notate very large numbers using scientific notation:

1. We will move the imaginary decimal point, which is at the end of the number, until we get a certain number between $1$ and $10$.
2. We will count how many steps we have moved the point to the left. The number of steps taken will be the exponent of $10$.
3. We will multiply the $10$ raised to the power we found by our number $m$ and will obtain the scientific notation.

Let's see it in an example:
Let's take the following number:
$180,000,000$
We will note an imaginary decimal point at the end of the number and move it to the left until we get a certain number between $1$ and $10$:

We have moved the decimal point eight times. Consequently, $8$ will be the exponent of $10$.
We will obtain:
$1.8 \times 10^8$

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