Scientific Notation

Welcome to scientific notation, our website about expressing decimal numbers in standard index form.

In contrast to decimal notation, the base ten notation is a very convenient way to express, both, large and small numbers, and frequently used in engineering, math and science.

Here you can learn everything about writing numbers in scientific, normalized and e-notation, including the definition and rules for mathematical operations.

Keep reading to understand all about it, and make sure to check out our converter further below in this article.

What is Scientific Notation?

Every decimal number can be transformed into the form m × 10n, m \thinspace \in \thinspace \mathbb{R} and n \thinspace \in \thinspace \mathbb{Z}.

The coefficient m is called significand; m can be any real number, and n is an integer.

The term m × 10n is pronounced m times ten to the power of n, an exponentiation with the base 10 and the exponent n, as detailed in the second reference at the bottom of this page.

For example, 500 can be written as 5 × 102 and 0.25 as 2.5 × 10-1.

In the next section we look at the definition of scientific notation and introduce the normalized notation.

Definition

The strict scientific notation definition goes as follows:

Scientific notation is a method of expressing real numbers in terms of a significand, multiplied by a power of 10.

For 500, not only 5 × 102, but also, for example, 0.5 × 103 or 50 × 101 comply with the above defintion.

In normalized scientific notation, m is chosen such that 1 ≤ |m| < 10.

Normalized scientific notation is a method of expressing real numbers in terms of a significand m, multiplied by a power of 10, such that 1 ≤ |m| < 10.

The term scientific notation, also know as exponential notation and base 10 notation, usually means normalized scientific notation.

On this site we discuss the normalized representation, unless stated differently.

Next, we are going to show you how to convert decimal numbers.

Convert to Scientific Notation

To convert a number to scientific notation move the decimal separator point n places, to put the number’s coefficient within a desired range, between 1 and 10 for normalized notation.

If the decimal was moved to the left, append × 10n; else append × 10-n. For example, for the number -79896725978 we move the decimal point 10 places to the left: -7.9896725978.

We then append × 1010 and get -7.9896725978 × 1010.

As a second example, for a small number like 0.86651112 we first move the decimal point 1 place to the right: 8.6651112.

We then append × 10-1: 0.86651112 = 8.6651112 × 10-1.

You can have it way easier by using our converter below. Our tool converts your number back and forth, on the fly.

Scientific Notation Converter

Note that the numbers are shown in ^-notation which means that ^ replaces superscript; the rest is self-explanatory.

More about e-notation can be found further down on this page.

Change decimal to scientific


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Observe that no button must be pressed unless you want to change from a decimal to standard index form transformation to a scientific to decimal conversion.

Frequently converted numbers on our site, among many others, include:

In the next part of our article we are going to discuss the rules for the arithmetic operations multiplication, division, addition as well as subtraction.

Scientific Notation Rules

Once the coefficients have been multiplied or divided, the multiplications and divisions of the indices are conducted according to rules detailed in exponentiation.

With respect to the sum and difference, follow the instructions and check out the example conversions, once in abstract form and once with a real example.

Also bear in mind that you could always convert the numbers to decimal format, do the math, and then convert the result back to scientific or exponential form.

Multiplying Scientific Notation

For the multiplication, build the product of the significands, then add the exponents.

To achieve normalization, you may have to move the decimal point such that 1 ≤ significand < 10:

a\times 10^{n} \hspace{5px}\times\hspace{5px} b \times 10^{m}
= (a\times b) \times 10^{n+m} = ab10^{n+m}

-1.5\times 10^{3} \hspace{5px}\times\hspace{5px} 9.2 \times 10^{-2}
= (-1.5\times 9.2 ) \times 10^{3-2}
= -13.8\times10^{1}
= -1.38\times10^{2}

Dividing Scientific Notation

For the division, build the quotient of the significands, then add the exponents.

For the purpose of normalization, you may have to move the decimal point such that 1 ≤ coefficient < 10:

\frac{a\times 10^{n}}{b\times 10^{m}} = \frac{a}{b} \times 10^{n-m} \hspace{25px} b\neq 0

\frac{12.5\times 10^{4}}{2.5\times 10^{2}}
= \frac{12.5}{2.5} \times 10^{4-2}
= 5 \times 10^{2}

Adding Scientific Notation

For the addition all numbers must be converted to the same power of 10 before the coefficients are added.

For the sake of the normalized form, the decimal separator may needs to be shifted for 1 ≤ coefficient < 10:

Addition

1.5 \times 10^{3} + 2.1 \times 10^{2}
= 1.5 \times 10^{3} + 0.21 \times 10^{3}
= 1.71 \times 10^{3}

Subtracting Scientific Notation

For the subtraction all numbers must be converted to the same exponent of 10 before the significands are subtracted. For the sake of the normalized form, the decimal point may needs to be shifted for 1 ≤ significand < 10:

Subtraction

1.5 \times 10^{3} - 2.1 \times 10^{2}
= 1.5 \times 10^{3} - 0.21 \times 10^{3}
= 1.29 \times 10^{3}

Convert Scientific Notation to Decimal

To convert a number from scientific to decimal notation, first take notice of the value of the exponent n, then remove the appendix.

Next move the decimal separator n places to the left for numbers with an absolute value less than one (|n| ≤ 1); else move the decimal point n places to the right:

8.6651112 × 10-1: The exponent -1 demands moving the point one digit to the left → 0.86651112
-7.9896725978 × 1010: The index 10 demands moving the point to the right → -79896725978

As you can see, the procedure is the same for both, negative as well as positive numbers. What matters is the absolute value of the index.

Ahead is our explanation of the e-notation.

E-notation

E-notation is the way many calculators display the normalized scientific notation.

Instead of m × 10n, they display a number as me±n, thereby avoiding the use of superscript or the caret symbol.

In contrast to exponential notation, in e-notation the absolute value of the significand m is always in the interval [1;10[.

In the context of our article, the frequently asked question include, for example:

  • What is Scientific Notation
  • How to do scientific notation
  • How to write in scientific notation

Taking into account all of the above, you already know the answers. However, if something remains unclear just fill in the comment form or send us your issue or question by email.

Next you can find more examples of standard index notations, which can also be used as practice, followed by a table and the summary of this page.

Examples of Scientific Notation

For your convenience, we have also stated the example numbers in e-notation.

  • 2390 → 2.39e+3 = 2.39 × 103
  • 1128368 → 1.128368e+6 = 1.128368 × 106
  • 636487 → 6.36487e+5 = 6.36487 × 105
  • 76.58953 → 7.658953e+1 = 7.658953 × 101
  • -6291773 → -6.291773e+6 = -6.291773 × 106
  • 0.1272324 → 1.272324e-1 = 1.272324 × 10-1
  • 100000 → 1e+5 = 1 × 105

Table

Last, but not least, here’s our table containing common values:

NameDecimal ValueScientific Notation
one trillionth0.0000000000011 × 10-12
one hundred-billionth0.000000000011 × 10-11
one ten-billionth0.00000000011 × 10-10
one billionth0.0000000011 × 10-9
one hundred-millionth0.000000011 x 10-8
one ten-millionth0.00000011 x 10-7
one millionth0.0000011 x 10-6
one hundred-thousandth0.000011 x 10-5
one ten-thousandth0.00011 x 10-4
one thousandth0.0011 x 10-3
one hundredth0.011 x 10-2
one tenth0.11 x 10-1
one11 x 100
ten101 x 101
one hundred1001 x 102
one thousand1,0001 x 103
ten thousand10,0001 x 104
one hundred thousand100,0001 x 105
one million1,000,0001 x 106
ten million10,000,0001 x 107
one hundred million100,000,0001 x 108
one billion1,000,000,0001 x 109
ten billion10,000,000,0001 x 1010
one hundred billion100,000,000,0001 x 1011
one trillion1,000,000,000,0001 x 1012

Ahead is the summary of our content.

Summary

You have reached the end of our article which we sum up using the following image:

Scientific Notation

Writing numbers this way, using a coefficient multiplied by a power of 10, is very convenient to express very small numbers and very large numbers.

Sometimes the ^ symbol is used: m × 10^n.

Note that instead of our converter, you may also use the search form in the sidebar, or at the bottom, depending on your device, because we have already converted many numbers for you.

If something remains unclear, then you can check out our FAQs.

If you are happy with our content, then hit the sharing buttons to let all your friends know about our site, and don’t forget to bookmark us.

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2 comments on “Scientific Notation

  1. pete says:

    8.6651112 × 10-1: The exponent -1 demands moving the point one digit to the left →8.6651112 ?

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