How To Write a Number in Scientific Notation (With Examples)
Updated February 3, 2023
When working with large and complex equations, scientists, mathematicians and statisticians may try to simplify numbers to easily calculate them. Using scientific notation can make a complex number easier for people to read and understand. You can write a number in scientific notation by inputting your numbers into a simple equation.
In this article, we explain what scientific notation is, describe how to write a number in scientific notation and highlight industries that use scientific notation regularly.
What is scientific notation?
Scientific notation is used to present very large or small numbers in a simpler way. Many scientists or engineers use scientific notation to write numbers that are often too long to write or understand. When a number converts into scientific notation, it's a decimal rather than an extended amount of numbers that may be unnecessary to the overall equation. The new decimal value makes the number simpler to understand and input into an equation.
For example, here are standard numbers in scientific notation:
500 = 5 x 10²
5,500,000,000 = 5.5 x 10⁹
0.000000055 = 5.5 x 10⁻⁹
Elements of scientific notation
There are certain factors of scientific notation to include for presenting an accurate solution. All numbers in scientific notation follow the form "m x 10ⁿ. The following elements are important parts for calculating the scientific notation:
Decimal: To find the scientific notation, you move the decimal a certain number of times to the right or left of the coefficient until it becomes a number equal to or larger than one and less than 10.
Coefficient: This number entails the decimal point moving a specific number of times to determine the coefficient. A coefficient is a number that's equal to or greater than one and less than 10.
Base: The base is always the number 10. When you're multiplying to find the final solution, the exponent is how many times you multiply 10 by itself.
Exponent: In scientific notation, the exponent can be a positive or negative number that represents the number of times you move a decimal to form the final coefficient.
Power of 10: The exponent and base together are the power of 10. These numbers give you the final total of the scientific notation.
5,500,000 = 5.5 (coefficient) x 10 (base) ⁶ (exponent)
0.00555 = 5.55 (coefficient) x 10 (base) ⁻³ (exponent)
For example, 1,000 has three zeros, so there are three exponents in this number. In scientific notation, 1,000 would be 10³. If you want to write in scientific notation for numbers smaller than one, you use negative numbers for the exponent, such as 0.01 would be 10⁻².
How to write a number in scientific notation
You can find scientific notation by inputting your numbers into a simple equation. Follow the steps below to learn how to write a number in scientific notation:
1. Find the original location of the decimal
Before you can find the scientific notation, you should evaluate your number to find the current location of the decimal. If the number doesn't have a visible placement in the equation, it may rest on the far right or left side. Take the number 5,200 as an example. Since this is a positive number, the decimal is at the far right end of the value.
Example: The value is 5,200.0.
2. Determine if the number is positive or negative
As you decide where the decimal is located, you can determine if your exponent is a positive or negative number. If the decimal moves to the left to make the integer a whole number, your exponent and final integer are positive numbers. If the decimal moves to the right, the exponent is a negative number.
Example: 5,200.0 is a positive number because the decimal was moved to the left.
3. Move the decimal point until it creates a number between one and 10
You can now take your number and calculate its scientific notation. Find the decimal point you just located and move it to the right or left until the number becomes a positive or negative value greater or equal to one and less than 10.
Example: Take the number 5,200 and move the decimal to the left until it shows as 5.200.
4. Count how many times you moved the decimal
Now we can determine the exponent by counting how many times you moved the decimal to the left.
Example: With 5.200, you moved the decimal three times, giving you an exponent of three, which is how many times you multiply 10 by itself.
5. Input your numbers into the equation and present your final scientific notation
You can now input your numbers to determine the final scientific notation. You can put this number into other equations and calculations to help you easily compare other numbers or simplify other calculations.
Example: With the number 5.200, the scientific notation is 5.2 x 10³.
6. Be consistent how you write notation
Most commonly, you can write a number in scientific notation by using the formula "m x 10ⁿ." You can also indicate the exponent by writing a caret symbol (^) after the 10 and before the exponent. Still, you may write a number in scientific notation by using the letter E. This use is common on scientific calculators. Choose which notation style fits your purpose and then use it consistently throughout the report or project.
Example: 5.2 x 10³ is the same as 5.2 x 10^3 or 5.2e + 3.
Notation with caret
Notation with E
4.32987 x 105
4.32987 x 10^5
7.654 x 102
7.654 x 10^2
3.400 x 10-3
3.400 x 10^-3
Powers of 10 scientific notation
When using scientific notation, the powers of 10 represent the number of times 10 is multiplied by itself. It can help you quickly calculate scientific notation. Common positive powers of 10 you might regularly find in a mathematical equation include:
10⁰ = 1
10¹ = 10
10² = 100
10³ = 1,000
10⁴ = 10,000
10⁵ = 100,000
10¹⁰ = 10,000,000,000
Basically, a positive power of 10, listed as 10x, equals 1 followed by x number of zeros.
Depending on the equation you're solving, you may use negative powers of 10 as well. Here are common negative powers of 10:
10⁻³ = 0.001
10⁻⁶ = 0.000001
10⁻⁹ = 0.000000001
10⁻¹² = 0.000000000001
Basically, a negative power of 10, listed as 10⁻ᵡ, equals 0. followed by x (minus 1) number of zeros.
Example of scientific notation
You can use scientific notation to help you find comparisons, track analytical data and determine certain distances.
For example, if you aim to find the scientific notation of the distance between New York and Australia, you can compare this amount to distances between other cities as well.
For our example, the distance between Australia and New York is 10,512 miles.
You can take the decimal at the end of the number and move it until it creates a number equal to or greater than one and less than 10. With this example, our integer would be 1. The decimal moves four places to the left to give you 1.051. Since it moved four places to the left, your exponent is 4. This makes your scientific notation 1.051 x 10⁴.
Addition and subtraction
When using scientific notation for subtraction or addition, it's important to make sure all of the exponents in the equation are the same. For example, (4 x 10³) + (5 x 10³) is the same as 9 x 10³ because 4 + 5 equals 9, and the exponent is the same.
If your bases have different exponents, you can adjust one of the numbers so that both components have the same exponent. For example, (6 x 10⁴) + (8 x 10⁵). You can rewrite it as (6 x 10⁴) + (80 x 10⁴). The result is 86 x 10⁴, rewritten to 8.6 x 10⁵ in scientific notation.
When using scientific notation in multiplication, the exponents don't need to be the same as they do for addition and subtraction. Instead, you can simply add the exponents to get the correct answer. For example, 10³ x 10² = 10⁵ because 2 + 3 equals 5. Another example is (4 x 10²) x (2 x 10³), rewritten as 8 x 10⁵.
When using scientific notation for division, subtract the exponents from each other for each representative number. For example, 10⁸ / 10⁵ = 10³. Another example is (4 x 10⁵) / (2 x 10³).This equals 2 x 10² because 4 divided by 2 equals 2 and 5 minus 3 equals 2.
Example of a calculation using scientific notation
A space is 0.00000256m wide, 0.00000014m long and 0.000275m high. What is its volume?
First, convert the three lengths into scientific notation:
Width: 0.00000256m = 2.56×10⁻⁶
Length: 0.00000014m = 1.4×10⁻⁷
Height: 0.000275m = 2.75×10⁻⁴
Second, multiply the digits together:
2.56 × 1.4 × 2.75 = 9.856
Last, multiply the ×10s:
10⁻⁶ × 10⁻⁷ × 10⁻⁴ = 10⁻¹⁷ by adding exponents
The result is 9.856×10⁻¹⁷ m3
Who uses scientific notation?
Employees who regularly use equations and formulas to find solutions to complex scientific or mathematical challenges practice scientific notation. Below are employees who may regularly use scientific notation while on the job:
Scientists, typically astronomers, may use scientific notation to simplify numbers that are originally too large or small to use in an equation. For example, an astronomer may try to calculate and compare distances between planets, which often involve measurements in millions, billions and trillions. Instead of calculating this distance using a significant amount of zeros, scientists can instead use scientific notation to make the number simpler and easy to compare.
Mathematicians may regularly use scientific notation to calculate basic equations. They may calculate distances, speed and the mass of various objects. Many mathematicians work in academia where they regularly use scientific notation to find solutions for various concepts, analyze data and design mathematical models.
A statistician develops or applies statistical methods, mathematical theories and models to solve real-world problems in business, science or other fields. They use computers with specialized software to analyze data and look for relationships and trends for decision-making. They often work closely with others to identify solutions to problems. For example, they may work closely with industrial designers to evaluate and improve a car's aerodynamics.
Related: Learn About Being a Statistician
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