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.

You may also be interested:

  • List of scientific notation exercises
  • Monomials – What are they? What are worth for? How to do operations between monomials?
  • Rule of Three – See the Types and Learn How to Calculate

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