A cientific notation is a representation of numbers using powers of base 10. This type of representation is essential for writing numbers with many digits in a simpler and more objective way. Remember that in our decimal system, digits are the symbols from 0 to 9: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
Read too: Potentiation — how to deal with numbers that have powers?
Summary about scientific notation
- Scientific notation is the writing of a number using powers of base 10.
- A number represented in scientific notation has the following format, where 1 ≤ to <10 It is n is integer:
\(a\times{10}^n\)
- The properties of potentiation are fundamental for writing a number in scientific notation.
Video lesson on scientific notation
What is scientific notation?
Scientific notation is the representation of a number in the following format:
\(a\times{10}^n\)
On what:
- The is a rational number (in decimal representation) greater than or equal to 1 and less than 10, that is, 1 ≤ to <10 ;
- It is n is an integer.
Examples:
Decimal representation |
Representation in scientific notation |
0,35 |
3,5×10-1 |
407 |
4,07×102 |
120.000 |
1,2×105 |
What is scientific notation for?
Scientific notation is used to represent numbers with many digits. This is the case with very large numbers (such as the distance between celestial bodies) and very small numbers (such as the size of molecules).
Examples of numbers with many digits:
- The approximate distance between the Sun and Earth is 149,600,000,000 meters.
- The diameter of a carbon atom is approximately 0.000000015 centimeters.
Let's look at how to write each of these numbers in scientific notation.
How to transform a number into scientific notation?
To transform a number into scientific notation, we need to write it in the form:
\(a\times{10}^n\)
With 1 ≤ to <10 It is n whole.
For that, It is essential to know the properties of potentiation, mainly in relation to the comma shift when we multiply a number by a power of base 10 and in relation to the sign of the respective exponent.
Example: Represent each number below in scientific notation.
- 3.700.000
This number can be written as 3,700,000.0. Note that in this case, The should be equal to 3.7. Therefore, it is necessary to move the decimal point six places to the left.
Soon,\( 3.7\times{10}^6\) is the representation in scientific notation of 3,700,000, that is:
\(3,700,000=3.7\times{10}^6\)
Observation: To check if the representation is correct, just solve the multiplication \(3.7\times{10}^6\) and observe that the result is equal to 3,700,000.
- 149.600.000.000
This number can be written as 149,600,000,000.0. Note that in this case, The should be equal to 1.496. Therefore, it is necessary to shift the decimal point 11 places to the left.
Soon,\( 1,496\times{10}^{11}\) is the representation in scientific notation of 149,600,000,000, that is:
\(149,600,000,000=1,496\times{10}^{11}\)
Observation: To check whether the representation is correct, simply solve the multiplication \(1,496\times{10}^{11}\) and observe that the result is equal to 149,600,000,000.
- 0,002
Note that for this number, The must be equal to 2. Therefore, it is necessary to move the decimal point three decimal places to the right.
Soon,\(2.0\times{10}^{-3}\) is the representation in scientific notation of 0.002, that is:
\(0.002=2.0\times{10}^{-3}\)
Observation: To check whether the representation is correct, simply solve the multiplication \(2.0\times{10}^{-3}\) and observe that the result is equal to 0.002.
- 0,000000015
Note that for this number, The should be equal to 1.5. Therefore, it is necessary to shift the decimal point eight decimal places to the right.
Soon, \(1.5\times{10}^{-8}\) is the representation in scientific notation of 0.000000015, that is:
\(0.000000015=1.5\times{10}^{-8}\)
Observation: To check whether the representation is correct, simply solve the multiplication 1,5×10-8 and observe that the result is equal to 0.000000015.
Operations with scientific notation
Addition and subtraction in scientific notation
In the case of addition and subtraction operations with numbers in scientific notation, we must ensure that the respective powers of 10 in each number have the same exponent and highlight them.
Example 1: Calculate \(1.4\times{10}^7+3.1\times{10}^8\).
The first step is to write both numbers with the same power of 10. Let's, for example, rewrite the number \(1.4\times{10}^7\). Note that:
\(1.4\times{10}^7=0.14\times{10}^8\)
Therefore:
\(\color{red}{\mathbf{1},\mathbf{4}\times{\mathbf{10}}^\mathbf{7}}+3,1\times{10}^8=\color{ red}{\ \mathbf{0},\mathbf{14}\times{\mathbf{10}}^\mathbf{8}}+3,1\times{10}^8\)
Putting the power \({10}^8\) In evidence, we have that:
\(0.14\times{10}^8+3.1\times{10}^8=\left (0.14+3.1\right)\times{10}^8\)
\(=3.24\times{10}^8\)
Example 2: Calculate \(9.2\times{10}^{15}-6.0\times{10}^{14}\).
The first step is to write both numbers with the same power of 10. Let's, for example, rewrite the number \(6.0\times{10}^{14}\). Note that:
\(6.0\times{10}^{14}=0.6\times{10}^{15}\)
Therefore:
\(9.2\times{10}^{15}-\color{red}{\mathbf{6},\mathbf{0}\times{\mathbf{10}}^{\mathbf{14}}} =9.2\times{10}^{15}-\color{red}{\mathbf{0},\mathbf{6}\times{\mathbf{10}}^{\mathbf{15}}}\ )
Putting the power 1015 In evidence, we have that:
\(9.2\times{10}^{15}-0.6\times{10}^{15}=\left (9.2-0.6\right)\times{10}^{15} \)
\(=8.6\times{10}^{15}\)
Multiplication and division in scientific notation
To multiply and divide two numbers written in scientific notation, we must operate the numbers that follow the powers of 10 with each other and operate the powers of 10 with each other.
Two essential potentiation properties in these operations are:
\(x^m\cdot x^n=x^{m+n}\)
\(x^m\div x^n=x^{m-n}\)
Example 1: Calculate \(\left (2.0\times{10}^9\right)\cdot\left (4.3\times{10}^7\right)\).
\(\left (2,0\times{10}^9\right)\cdot\left (4,3\times{10}^7\right)=\left (2,0\cdot4,3\right) \times\left({10}^9\cdot{10}^7\right)\)
\(=8.6\times{10}^{9+7}\)
\(=8.6\times{10}^{16}\)
Example 2: Calculate \(\left (5.1\times{10}^{13}\right)\div\left (3.0\times{10}^4\right)\).
\(\left (5,1\times{10}^{13}\right)\div\left (3,0\times{10}^4\right)=\left (5,1\div3,0\ right)\times\left({10}^{13}\div{10}^4\right)\)
\(=1.7\times{10}^{13-4}\)
\(=1.7\times{10}^9\)
Read too: Decimal numbers — review how to do operations with these numbers
Exercises on scientific notation
Question 1
(Enem) Influenza is a short-term acute respiratory infection caused by the influenza virus. When this virus enters our body through the nose, it multiplies, spreading to the throat and other parts of the respiratory tract, including the lungs.
The influenza virus is a spherical particle that has an internal diameter of 0.00011 mm.
Available at: www.gripenet.pt. Accessed on: 2 Nov. 2013 (adapted).
In scientific notation, the internal diameter of the influenza virus, in mm, is
a) 1.1×10-1.
b) 1.1×10-2.
c) 1.1×10-3.
d) 1.1×10-4.
e) 1.1×10-5.
Resolution
In scientific notation, the The for the number 0.00011 it is 1.1. Thus, the decimal point must be moved four decimal places to the left, that is:
\(0.00011=1.1\times{10}^{-4}\)
Alternative D
Question 2
(Enem) Researchers at the Vienna University of Technology, Austria, produced miniature objects using high-precision 3D printers. When activated, these printers launch laser beams onto a type of resin, sculpting the desired object. The final print product is a three-dimensional microscopic sculpture, as seen in the enlarged image.
The sculpture presented is a miniature of a Formula 1 car, 100 micrometers long. A micrometer is one millionth of a meter.
Using scientific notation, what is the representation of the length of this miniature, in meters?
a) 1.0×10-1
b) 1.0×10-3
c) 1.0×10-4
d) 1.0×10-6
e) 1.0×10-7
Resolution
According to the text, 1 micrometer is \(\frac{1}{1000000}=0.000001\) subway. Thus, 100 micrometers are \(100\cdot0.000001=0.0001\) meters.
Writing in scientific notation, we have:
\(0.0001=1.0\times{10}^{-4}\)
Alternative C
Sources:
ANASTACIO, M. A. S.; VOELZKE, M. A. Astronomy Topics as Prior Organizers in the Study of Scientific Notation and Units of Measurement. Abakós, v. 10, no. 2, p. 130-142, 29 nov. 2022. Available in https://periodicos.pucminas.br/index.php/abakos/article/view/27417 .
NAISSINGER, M. A. Cientific notation: a contextualized approach. Monograph (Specialization in Mathematics, Digital Media and Didactics) — Federal University of Rio Grande do Sul, Porto Alegre, 2010. Available in http://hdl.handle.net/10183/31581.
