Scientific Notation Converter

Scientific notation calculator

This scientific notation calculator converts the inserted number to the following notations:

• scientific notation,
• scientific e-notation,
• engineering notation,
• standard form,
• real number form,
• word form.

The calculator also identifies the order of magnitude of a number for scientific notation and standard form.

Directions for use

To use the scientific notation converter, enter a number and press "Calculate." The calculator will return the number in all the forms listed above and the number's order of magnitude.

Note that this notation calculator only takes the following numbers as inputs: integers, decimals, numbers in scientific notation or standard form, numbers in engineering notation, and numbers in scientific e-notation. Fractions and numbers in word form are not accepted.

To enter a number in scientific e-notation, use the following form: aeb, for example, 3e5. To enter a number in scientific notation, use the circumflex (caret) symbol ^ to represent the powers of 10, for example, 3 × 10^5.

Important definitions

Let's define the special notations returned by the calculator.

Scientific notation

Scientific notation is very convenient for writing very big or very small numbers. The general form of a number in scientific notation looks like this:

a×10ᵇ

Where the modulus of a is greater than or equal to 1 and less than 10:

1≤|a|<10

And ᵇ is an integer. Remember that integers are positive AND negative whole numbers. Therefore, the power of 10 can be positive as well as negative. When the power of 10 is positive, the scientific notation represents a number greater than or equal to 10. When the power of 10 is negative, the scientific notation represents a number smaller than 1. When the power of 10 is zero, the scientific notation represents a number greater than or equal to one and less than 10.

For example, 86,000,000 can be written as 8.6×10⁷, 0.00056 can be written as 5.6×10⁻⁴ and 7.8 can be written as 7.8×10⁰.

How to convert a number to scientific notation

To express the number in scientific notation a×10ᵇ, you need to take the following steps:

1. Move the decimal point to such a position that there is only one digit on the left side of the decimal point. For example, you have the number 654.7. You need to move the decimal point to the position between 6 and 5 so that the number looks like 6.547. The resulting number (6.547 in our case) is A.

2. Count the number of spaces that the decimal point moved and identify the direction of its movement. The number of spaces the decimal point moved will be the absolute value of b, the power of 10 of the number. The direction of the movement defines the sign of B. If the decimal point moves left, B will be positive: b>0. If the decimal point moves right, B will be negative: b<0. In our previous example, we had to move the decimal point 2 spaces to the left. Therefore, b=2.

3. Write down the number in scientific notation. In our previous example:

654.7=6.547×10²

4. Check if there are any trailing zeros and whether they were initially before or after the decimal point. If the zeros were before the decimal point (this usually happens when we convert large numbers), we can omit them. If the zeros were after the decimal point, they are considered significant figures; therefore, you must keep them in the final answer. For example:

0.0007800=7.800×10⁻⁴

Here we do not omit the trailing zeros since they were after the decimal point in the original number. But:

38,000=3.8000×10⁴=3.8×10⁴

The trailing zeros can be omitted here since they were initially before the decimal point.

Note that when the trailing zeros were before AND after the decimal point in the original number, all of them must be kept in the final number. For example:

4000.000=4.000000×10³

Scientific e-notation

Scientific e-notation is a different way of writing the standard scientific notation. A number a×10ᵇ in e-notation will look as aeb. To convert the number to scientific e-notation, convert it to the standard scientific notation, and then write it replacing ×10ᵇ with eb. For example:

26,000=2.6000×10⁴=2.6×10⁴=2.6e4

Scientific e-notation is often used when superscripts or circumflexes are unavailable.

Engineering notation

Engineering notation is very similar to scientific notation, with an additional limitation of B represented only by the multiples of 3 (3, 6, 9, etc.). Therefore, in engineering notation the absolute value of A lies in the following range: 1≤|a|<1000.

Engineering notation is very often used in scientific and engineering communication since the powers of 10 match the metric prefixes. For example, 35×10⁻⁹ can be written as 35ns (pronounced 35 nanoseconds). In many cases it is much more convenient than writing the standard form of scientific notation: 3.5×10⁻⁸. It can be pronounced as "3.5 times ten to the power of minus eight seconds".

Standard form

Standard form is just another name for scientific notation. Therefore, a number in standard form looks exactly like a number in scientific notation: a×10ᵇ.

Calculation example

Write the given number in the following notations: scientific notation, scientific e-notation, engineering notation, standard form, real number form, and word form. What is the order of magnitude of the given number?

Given: 654.901

Solution:

To convert this number to scientific notation, let's first identify the value of A:

a=6.54901

To find the value of A, we had to move the decimal point two steps to the left. Therefore, b=2.

Writing the number in scientific notation, we get:

6.54901×10²

In scientific e-notation, this number will look as follows:

6.54901e2

In engineering notation, B is limited to multiples of 3. However, in our case, b<3. Therefore, we will write it with b=0 so that the corresponding physical value doesn't have any prefix. The number in engineering notation will, therefore, look like this:

654.901×10⁰

Standard form is just another way of defining scientific notation. Therefore, the number in standard form looks the same as the number in scientific notation:

6.54901×10²

The real number form looks as follows:

654.901

And in word form, we can describe it as this number:

"six hundred fifty-four and nine hundred one thousandths"

The number's order of magnitude is defined by the power of 10 in its scientific notation. So in our case, the order of magnitude is 2.