# Scientific Notation

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.

## Result and Calculator

Scientific Notation:
×10
E Notation:
Decimal Notation:

BTW: Scientific calculators extend the basic operations of calculators with functions such as square roots, just to name a few, and shall not be confused with scientific notation calculators like the one above.

## 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.

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:

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

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:

NameDecimal FormScientific 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
two22 x 100
three33 x 100
four44 x 100
five55 x 100
six66 x 100
seven77 x 100
eight88 x 100
nine99 x 100
ten101 x 101
twenty202 x 101
thirty303 x 101
forty404 x 101
fifty505 x 101
sixty606 x 101
seventy707 x 101
eighty808 x 101
ninety909 x 101
one hundred1001 x 102
two hundred2002 x 102
three hundred3003 x 102
four hundred4004 x 102
five hundred5005 x 102
six hundred6006 x 102
seven hundred7007 x 102
eight hundred8008 x 102
nine hundred9009 x 102
one thousand1,0001 x 103
two thousand2,0002 x 103
three thousand3,0003 x 103
four thousand4,0004 x 103
five thousand5,0005 x 103
six thousand6,0006 x 103
seven thousand7,0007 x 103
eight thousand8,0008 x 103
nine thousand9,0009 x 103
ten thousand10,0001 x 104
twenty thousand20,0002 x 104
thirty thousand30,0003 x 104
forty thousand40,0004 x 104
fifty thousand50,0005 x 104
sixty thousand60,0006 x 104
seventy thousand70,0007 x 104
eighty thousand80,0008 x 104
ninety thousand90,0009 x 104
one hundred thousand100,0001 x 105
two hundred thousand200,0002 x 105
three hundred thousand300,0003 x 105
four hundred thousand400,0004 x 105
five hundred thousand500,0005 x 105
six hundred thousand600,0006 x 105
seven hundred thousand700,0007 x 105
eight hundred thousand800,0008 x 105
nine hundred thousand900,0009 x 105
one million1,000,0001 x 106
two million2,000,0002 x 106
three million3,000,0003 x 106
four million4,000,0004 x 106
five million5,000,0005 x 106
six million6,000,0006 x 106
seven million7,000,0007 x 106
eight million8,000,0008 x 106
nine million9,000,0009 x 106
ten million10,000,0001 x 107
twenty million20,000,0002 x 107
thirty million30,000,0003 x 107
forty million40,000,0004 x 107
fifty million50,000,0005 x 107
sixty million60,000,0006 x 107
seventy million70,000,0007 x 107
eight million80,000,0008 x 107
ninety million90,000,0009 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:

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.

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