How to write a number in scientific notation?

What is scientific notation? Acientific notationis a simpler way of writing numbers that are either very small or very large. With it, numbers like 0.000001 and 3,000,000,000 can be written in a shortened form.

One number written in scientific notation has the following form: $\dpi{120} \mathbf{{{\color{Red} a} \cdot 10^ {\color{Blue}b}}}$ , on what:

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$\dpi{120} \mathbf{{\color{Red} a}}$ is a real number greater than or equal to 1 and less than 10;
$\dpi{120} \mathbf{ {\color{Blue} b}}$ is an integer that will be: $\dpi{120} \bg_white \left\{\begin{matrix} \mathbf{ \negative,\ for \\acute{u}very \ small\ numbers;}\\ \mathbf{positive,\ for \n\ acute{u}numbers\ very \ large \ \ .} \end{matrix}\right.$

see some examplesnumbers written in scientific notation:

Number	Number in scientific notation
0,000001	$\bg_white 1 \cdot 10^{-6}$
0,0000000000815	$\bg_white \bg_white 8.15 \cdot 10^{-11}$
3.000.000.000	$\bg_white \bg_white 3 \cdot 10^{9}$
250.000.000.000.000.000	$\bg_white \bg_white 2.5 \cdot 10^{17}$

But how do you convert a number to scientific notation? Learn this in the topic below.

Writing a number in scientific notation

Case 1. very small numbers

1st step) Let's move the comma to the right until it has a first and only non-zero digit before the decimal point. From this, we get the value of $\dpi{120} \bg_white {\color{Red} \mathbf{a}}$ ;

2nd step) The number of places we move the decimal point will be the

exponent in scientific notation, it will have a minus sign; this will be the value of $\dpi{120} \bg_white \mathbf{{\color{Blue} b}}$ .

Example 1: Let's write the number 0,00052 in scientific notation:

Shifting the decimal point to the right, until it has a first and only non-zero digit before the decimal point, we get the number 00005,2 It is like 00005,2 $\dpi{120} \bg_white$ 5,2, then, $\dpi{120} \mathbf{\color{Red} to \color{Black}{\color{Red} 5.2}}$ .
We shifted the decimal 4 places (we went from 0.00052 to 00005.2), so our exponent is the number 4 with a negative sign, that is, $\dpi{120} \mathbf{\color{Blue} b \color{Black}{\color{Blue} -4}}$ .

So, we have to $\dpi{120} \mathbf{0.00052{\color{Red} 5.2} \cdot 10^{{\color{Blue} -4}}}$ .

Example 2: Let's write the number 0,0000008 in scientific notation:

Shifting the decimal point to the right, until it has a first and only non-zero digit before the decimal point, we get: 00000008,0 It is like 00000008,0 $\dpi{120} \bg_white$ 8,0. Then, $\dpi{120} \mathbf{\color{Red} to \color{Black}{\color{Red} 8.0}}$ .
We shift the decimal 7 places, so our exponent is the number 7 with a negative sign, that is, $\dpi{120} \mathbf{\color{Blue} b \color{Black}{\color{Blue} -7}}$ .

Therefore, $\dpi{120} \mathbf{0.0000008 {\color{Red} 8.0} \cdot 10^{{\color{Blue} -7}}}$ .

Case 2. very large numbers

1st step) Let's move the comma to the left until you have only a digit before the decimal point. Hence, we get the value of $\dpi{120} \bg_white {\color{Red} \mathbf{a}}$ ;

2nd step) The number of places we move the decimal point will be the exponent in scientific notation, it will have a plus sign; this will be the value of $\dpi{120} \bg_white \mathbf{{\color{Blue} b}}$ .

Example 1: Let's write the number 340.000 in scientific notation:

All integers have an implicit comma (2 $\dpi{120} \bg_white$ 2,0 / 11 $\dpi{120} \bg_white$ 11,0 / 200 $\dpi{120} \bg_white$ 200.0 and so on). So, we have to 340.000 $\dpi{120} \bg_white$ 340.000,0.
Then, shifting the decimal point to the left, until you have only a digit before the decimal point, we get: 3,400000 It is like 3,400000 $\dpi{120} \bg_white$ 3,4, then, $\dpi{120} \mathbf{\color{Red} to \color{Black}{\color{Red} 3.4}}$ .
We shift the decimal 5 places, so our exponent is the number 5 with a positive sign, that is, $\dpi{120} \mathbf{\color{Blue} b \color{Black}{\color{Blue} 5}}$ .

With that, we have to $\dpi{120} \mathbf{340,000{\color{Red} 3.4} \cdot 10^{{\color{Blue} 5}}}$ .

Example 2: Let's write the number 90.000.000 in scientific notation:

We have to 90.000.000 $\dpi{120} \bg_white$ 90.000.000,0. Then, shifting the decimal point to the left, until you have only a number before the comma, we get: 9,00000000 It is like 9,00000000 $\dpi{120} \bg_white$ 9, then, $\dpi{120} \mathbf{\color{Red} a \color{Black}{\color{Red} 9}}$ .
We shift the decimal 7 places, so our exponent is the number 7 with a positive sign, that is, $\dpi{120} \mathbf{\color{Blue} b \color{Black}{\color{Blue} 7}}$ .

In this way, we have to $\dpi{120} \mathbf{90,000,000{\color{Red} 9} \cdot 10^{{\color{Blue} 7}}}$ .

more examples

$\dpi{120} {\color{DarkGreen} \mathbf{0.000323.2\cdot 10^{-4}}}$

1st step) We get 00003.2 which is equal to 3.2

2nd step) we get the exponent $\dpi{120} \bg_white -$ 4 as we move 4 houses to the right.

$\dpi{120} {\color{DarkGreen} \mathbf{-0.00007 -7.0\cdot 10^{-5}}}$

1st step) we get $\dpi{120} \bg_white -$ 000007.0 which is equal to $\dpi{120} \bg_white -$ 7,0

2nd step) we get the exponent $\dpi{120} \bg_white -$ 5 as we move 5 houses to the right.

$\dpi{120} {\color{DarkGreen} \mathbf{35.801 3.5801 \cdot 10^{4}}}$

1st step) As $\dpi{120} \bg_white 35,801 35,801.0$ we get $\dpi{120} \bg_white 3.58010$ which is equal to 3.5801

2nd step) We get the exponent 4 since we moved 4 places to the left.

$\dpi{120} {\color{DarkGreen} \mathbf{ 1,000,000 1 \cdot 10^{6}}}$

1st step) As $\dpi{120} \bg_white 1,000,0001,000,000.0$ , we get $\dpi{120} \bg_white 1,0000000 1$

2nd step) We get the exponent 6 by moving 6 places to the left.

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on Jul 22, 2021