Standard Form Calculator
Convert a number to standard form by entering the decimal value below. You can also convert a number in standard form to decimal.
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How to Write a Number in Standard Form
Standard form, also referred to as scientific notation, is a way of representing very large numbers or very small numbers in a concise format that is easier to read.
Numbers in standard form are written as an expression of a coefficient multiplied by 10 to the nth power.
For instance, the number 25,970,000 can be expressed using standard form as 2.597 × 107.
The number of digits in the expression’s coefficient will vary depending on the number of significant figures in the number. You can use a sig fig calculator to find the significant figures.
You can convert a number from decimal to standard form in a few simple steps.
Step One: Create the Equation
First, create an expression with the initial number in decimal form as the coefficient multiplied by 10 to the power of 0 (recall 100 = 1).
For example, let’s create the initial standard form equation for the decimal 25,970,000.
25,970,000 = 25,970,000 × 100
Step Two: Move the Decimal Point
The next step is to move the decimal point in the coefficient until there is one significant digit to the left of the decimal point. For small numbers less than 1, you would move the decimal point to the right.
For each space the decimal point is moved to the left, increase the 10’s exponent by 1. For numbers less than 1, reduce the exponent by 1 for each space the decimal point is moved to the right.
Continuing the example above, let’s finish converting 25,970,000 to standard form.
25,970,000 = 25,970,000.0 × 100
25,970,000 = 2,597,000.0 × 101
25,970,000 = 259,700.0 × 102
25,970,000 = 25,970.0 × 103
25,970,000 = 2,597.0 × 104
25,970,000 = 259.7 × 105
25,970,000 = 25.97 × 106
25,970,000 = 2.597 × 107
You can use our exponent calculator to learn more about working with exponents in a standard form equation.
How to Express a Small Number in Standard Form
You can also represent small numbers, much less than one, in standard form as well. The main difference is instead of moving the decimal point to the left, you move it to the right, and instead of increasing the 10’s exponent, you decrease it.
For example, let’s express the number 0.00000659 in standard form:
0.00000659 = 0.00000659 × 100
0.00000659 = 0.0000659 × 10-1
0.00000659 = 0.000659 × 10-2
0.00000659 = 0.00659 × 10-3
0.00000659 = 0.0659 × 10-4
0.00000659 = 0.659 × 10-5
0.00000659 = 6.59 × 10-6
How to Write Precise Decimals in Standard Form
What if you have a long, precise decimal? How do you write that in standard form? The standard practice is to write the standard form coefficient with the same number of significant figures as the decimal.
For example, the number 72,001,000 has five significant figures and can be written in standard form as 7.2001 × 107.
Another example of a very precise decimal is 8,756,992.34 (with nine sig figs) and can be written in standard form as 8.75699234 × 106. You may notice that in the latter example, it does not seem much simpler to write the decimal in standard form.
In practice, if one is dealing with very precise numbers like these, it is best to either leave them in decimal form or round them to a few significant figures, purely for the purpose of simpler viewing. For example, you may round 8.75699234 × 107 to 8.76 × 107.
However, it is very important to remember that when actually performing calculations, you must use the full decimal or standard form, not the rounded value.
How to Use Standard Form in Equations
When dealing with very large or very small numbers in equations, it is often beneficial to rewrite these values using standard form. For example, you can write an equation of a line such as y = 5600000x + 820000000, but this is not ideal to read.
Instead, you can rewrite these values into standard form:
y = 5.6 × 106x + 8.2 × 108
Although this is much cleaner, do you notice the problem? By writing the numbers in standard form, we have introduced multiplication signs that can be confused for the variable x.
Therefore, when using standard form in equations, we can use a different practice commonly used in mathematics. That is, replace the multiplication sign and the 10 with the letter e.
As an example, 1.12 × 109 can be rewritten as 1.12e9.
To help you remember this simpler form, think of e as standing for “exponent”.
Continuing our example from above, we can rewrite y = 5.6 × 106x + 8.2 × 108 as y = 5.6e6x + 8.2e8, and the confusion between the multiplication sign and the variable x disappears!
Frequently Asked Questions
Why do we use standard form?
We use standard form because equations and calculations with very large or very small values become unwieldy and cumbersome. Also, by removing the leading or trailing zeros, all the numbers left in the coefficient are significant.
Are standard form and scientific notation the same thing?
Yes, standard form and scientific notation are different names for the same format.
Can negative numbers be written in standard form?
Absolutely! Negative numbers can be written in standard form in the exact same way as positive numbers. For example, -0.0045 can be written as -4.5 × 10-3 or -4.5e-3.