Table of Contents
Welcome to scientific notation, our website about expressing decimal numbers in standard index form.
In contrast to decimal form (notation without exponents), the base ten notation is a very convenient way to express, both, large and small numbers.
It is frequently used in engineering, math and science.
Here you can learn everything about scientific notation representations, including the definition and rules for the mathematical operations.
Keep reading to understand all about it, and make sure to check out our converter further below in this article.
Result and Calculator
What is Scientific Notation?
Every decimal number can be transformed into the form m × 10n.
The coefficient m is called significand (the significant digits); 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 and introduce the normalized notation.
Definition
The strict scientific notation figure definition goes as follows: It 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 definition.
In normalized aka standard scientific notation form, m is chosen such that 1 ≤ |m| < 10; the level of precision is not affected.
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.
Example: 500 in decimal notation, the number is written 5 × 102.
Engineering notation, 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 move the decimal separator point of the original number n places to put the it’s coefficient within a desired range, between 1 and 10 for normalized notation.
The coefficient m represents the significant figures which add to the precision, and is kept short by eliminating any potential trailing zeros to the right of the non-zero digits after the decimal separator.
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.
Negative exponents like 10-n stand for the division 1 / n.
You can have it way easier by using our scientific notation calculator below the intro .
Our tool converts your number back and forth, on the fly.
Scientific Notation Converter
Note that the numbers are shown in ^-form which means that ^ replaces superscript; the rest is self-explanatory.
More about e-notation can be found further down on this page.
Observe that no button must be pressed unless you want to change the type of notation.
Frequently converted numbers on our site, among many others, include:
- 149.6 Billion in Standard Index Form
- 149.6 Million in Standard Index Form
- 704 in Standard Index Form
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 × 10n × b × 10m = (a × b) × 10n+m
-1.5 103 × 9.2 × 10-2 = (-1.5 × 9.2) × 103-2 = -13.8 × 101 = -1.38 × 102
Dividing Scientific Notation
For the division, build the quotient of the significands, then subtract the exponents.
For the purpose of normalization, you may have to move the decimal point such that 1 ≤ coefficient < 10: a × 10n / b × 10m = a/b × 10n-m
12.5 × 104 / 2.5 × 102 / = 12.5 / 2.5 × 104-2 = 5 × 102
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: 1.5 × 103 + 2.1 × 102 = 1.5 × 103 + 0.21 × 103 = 1.71 × 103
Subtracting Scientific Notation
For the subtraction all numbers must be converted to the same base-10 exponent before the significands are subtracted.
For the sake of the normalized form, the decimal point may needs to be shifted for 1 ≤ significand < 10: 1.5 × 103 – 2.1 × 102 = 1.5 × 103 – 0.21 × 103 = 1.29 × 103
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 in the opposite direction: 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.
E-notation
E-notation is the way many calculators display their output when you calculate large or very small figures.
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[.
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.
Scientific Notation Examples
For your convenience, we have also stated examples of 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
Take the opportunity to check out our student exercises (in place of a scientific notation worksheet):
The table below also includes the names of the decimals, make sure to check it out!
Table
Last before the wrap-up, but not least, here’s our table containing common values:
| Name | Decimal Form | Scientific Notation |
|---|---|---|
| one trillionth | 0.000000000001 | 1 × 10-12 |
| one hundred-billionth | 0.00000000001 | 1 × 10-11 |
| one ten-billionth | 0.0000000001 | 1 × 10-10 |
| one billionth | 0.000000001 | 1 × 10-9 |
| one hundred-millionth | 0.00000001 | 1 x 10-8 |
| one ten-millionth | 0.0000001 | 1 x 10-7 |
| one millionth | 0.000001 | 1 x 10-6 |
| one hundred-thousandth | 0.00001 | 1 x 10-5 |
| one ten-thousandth | 0.0001 | 1 x 10-4 |
| one thousandth | 0.001 | 1 x 10-3 |
| one hundredth | 0.01 | 1 x 10-2 |
| one tenth | 0.1 | 1 x 10-1 |
| one | 1 | 1 x 100 |
| two | 2 | 2 x 100 |
| three | 3 | 3 x 100 |
| four | 4 | 4 x 100 |
| five | 5 | 5 x 100 |
| six | 6 | 6 x 100 |
| seven | 7 | 7 x 100 |
| eight | 8 | 8 x 100 |
| nine | 9 | 9 x 100 |
| ten | 10 | 1 x 101 |
| twenty | 20 | 2 x 101 |
| thirty | 30 | 3 x 101 |
| forty | 40 | 4 x 101 |
| fifty | 50 | 5 x 101 |
| sixty | 60 | 6 x 101 |
| seventy | 70 | 7 x 101 |
| eighty | 80 | 8 x 101 |
| ninety | 90 | 9 x 101 |
| one hundred | 100 | 1 x 102 |
| two hundred | 200 | 2 x 102 |
| three hundred | 300 | 3 x 102 |
| four hundred | 400 | 4 x 102 |
| five hundred | 500 | 5 x 102 |
| six hundred | 600 | 6 x 102 |
| seven hundred | 700 | 7 x 102 |
| eight hundred | 800 | 8 x 102 |
| nine hundred | 900 | 9 x 102 |
| one thousand | 1,000 | 1 x 103 |
| two thousand | 2,000 | 2 x 103 |
| three thousand | 3,000 | 3 x 103 |
| four thousand | 4,000 | 4 x 103 |
| five thousand | 5,000 | 5 x 103 |
| six thousand | 6,000 | 6 x 103 |
| seven thousand | 7,000 | 7 x 103 |
| eight thousand | 8,000 | 8 x 103 |
| nine thousand | 9,000 | 9 x 103 |
| ten thousand | 10,000 | 1 x 104 |
| twenty thousand | 20,000 | 2 x 104 |
| thirty thousand | 30,000 | 3 x 104 |
| forty thousand | 40,000 | 4 x 104 |
| fifty thousand | 50,000 | 5 x 104 |
| sixty thousand | 60,000 | 6 x 104 |
| seventy thousand | 70,000 | 7 x 104 |
| eighty thousand | 80,000 | 8 x 104 |
| ninety thousand | 90,000 | 9 x 104 |
| one hundred thousand | 100,000 | 1 x 105 |
| two hundred thousand | 200,000 | 2 x 105 |
| three hundred thousand | 300,000 | 3 x 105 |
| four hundred thousand | 400,000 | 4 x 105 |
| five hundred thousand | 500,000 | 5 x 105 |
| six hundred thousand | 600,000 | 6 x 105 |
| seven hundred thousand | 700,000 | 7 x 105 |
| eight hundred thousand | 800,000 | 8 x 105 |
| nine hundred thousand | 900,000 | 9 x 105 |
| one million | 1,000,000 | 1 x 106 |
| two million | 2,000,000 | 2 x 106 |
| three million | 3,000,000 | 3 x 106 |
| four million | 4,000,000 | 4 x 106 |
| five million | 5,000,000 | 5 x 106 |
| six million | 6,000,000 | 6 x 106 |
| seven million | 7,000,000 | 7 x 106 |
| eight million | 8,000,000 | 8 x 106 |
| nine million | 9,000,000 | 9 x 106 |
| ten million | 10,000,000 | 1 x 107 |
| twenty million | 20,000,000 | 2 x 107 |
| thirty million | 30,000,000 | 3 x 107 |
| forty million | 40,000,000 | 4 x 107 |
| fifty million | 50,000,000 | 5 x 107 |
| sixty million | 60,000,000 | 6 x 107 |
| seventy million | 70,000,000 | 7 x 107 |
| eight million | 80,000,000 | 8 x 107 |
| ninety million | 90,000,000 | 9 x 107 |
| one hundred million | 100,000,000 | 1 x 108 |
| one billion | 1,000,000,000 | 1 x 109 |
| ten billion | 10,000,000,000 | 1 x 1010 |
| one hundred billion | 100,000,000,000 | 1 x 1011 |
| one trillion | 1,000,000,000,000 | 1 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:
The key idea is that 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, not only for mathematicians.
Examples include the time of radioactive decay and astronomical distances, just to name a few.
Sometimes the ^ symbol is used instead of superscripted exponents: m × 10^n.
Put simply: the engineering notation uses less zeros in general, and extra zeros in the significant are truncated.
Observe that a negative power 10^-n means 1 / 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 or get in touch.
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– Article written by Mark, last updated on February 1st, 2024