Scientific Notation Converter

Scientific notation calculator

This versatile scientific notation calculator instantly converts any given number into the following formats:

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

Additionally, the tool automatically determines the order of magnitude for both scientific notation and standard form, making complex math much easier to digest.

Directions for use

To use this scientific notation converter, simply enter your number into the input field and click "Calculate." The tool will rapidly process your input and display the number in all the formats listed above, along with its precise order of magnitude.

Please note that this scientific notation calculator accepts a specific range of numerical inputs: integers, decimals, and numbers already formatted in scientific, standard, engineering, or e-notation. Fractions and numbers written purely in word form are not supported.

To input a value in scientific e-notation, use the aeb format, for example, 3e5. For standard scientific notation, use the circumflex (caret) symbol ^ to indicate powers of 10, for example, 3 × 10^5.

Important definitions

Let's explore the specific mathematical notations generated by this calculator.

Scientific notation

Scientific notation is a highly convenient method for writing excessively large or infinitesimally small numbers. The general form of a number expressed in scientific notation looks like this:

a×10ᵇ

Where the absolute value (modulus) of a is greater than or equal to 1 and strictly less than 10:

1≤|a|<10

And ᵇ represents an integer. Keep in mind that integers include both positive AND negative whole numbers, meaning the power of 10 can be positive or negative. When the exponent is positive, the scientific notation represents a number greater than or equal to 10. When the exponent is negative, it represents a decimal number smaller than 1. If the power of 10 is zero, the notation represents a number greater than or equal to 1 and strictly less than 10.

For example, 86,000,000 translates to 8.6×10⁷, 0.00056 becomes 5.6×10⁻⁴, and 7.8 is written as 7.8×10⁰.

How to convert a number to scientific notation

To manually convert a number into the scientific notation format a×10ᵇ, follow these steps:

1. Move the decimal point so that only one non-zero digit remains to its left. For instance, if you start with the number 654.7, move the decimal point between the 6 and the 5 to get 6.547. This resulting figure is your A value.

2. Count the number of spaces the decimal point shifted and note the direction. The number of spaces moved determines the absolute value of b (the power of 10). The direction of the shift dictates the sign of B. If the decimal point moves to the left, B is positive: b>0. If it moves to the right, B is negative: b<0. In our example, the decimal shifted 2 spaces to the left, meaning b=2.

3. Write the number in its final scientific notation form. Continuing our example:

654.7=6.547×10²

4. Check for trailing zeros and determine their original position relative to the decimal point. If the trailing zeros appeared before the decimal point (which is common when converting large integers), they can be omitted. However, if the zeros appeared after the decimal point, they are considered significant figures and must be retained in your final answer. For example:

0.0007800=7.800×10⁻⁴

Here, we do not drop the trailing zeros because they were located after the decimal point in the original number. Conversely:

38,000=3.8000×10⁴=3.8×10⁴

In this case, the trailing zeros can be safely omitted since they originally appeared before the decimal point.

Please note: If a number contains trailing zeros both before AND after the decimal point, all of them must be preserved as significant figures in the final scientific notation. For example:

4000.000=4.000000×10³

Scientific e-notation

Scientific e-notation is a practical alternative for writing standard scientific notation, commonly used in programming and digital calculators. A number normally written as a×10ᵇ appears as aeb in e-notation. To convert a value into scientific e-notation, first find its standard scientific form, then rewrite it by replacing the ×10ᵇ with eb. For example:

26,000=2.6000×10⁴=2.6×10⁴=2.6e4

This format is especially useful when typing superscripts or circumflex symbols is not an option.

Engineering notation

Engineering notation closely resembles scientific notation, with one key restriction: the exponent B must be a multiple of 3 (e.g., 3, 6, 9, -3, -6). Because of this rule, the absolute value of A falls within a different range: 1≤|a|<1000.

This notation is widely adopted in scientific and engineering fields because its powers of 10 directly correspond to standard metric prefixes. For example, 35×10⁻⁹ can be written as 35ns (read as 35 nanoseconds). This is often far more intuitive than writing the standard scientific form: 3.5×10⁻⁸, which reads as "3.5 times ten to the power of negative eight seconds".

Standard form

Standard form is simply an alternative term for scientific notation used in certain regions (such as the UK). Therefore, a number written in standard form looks exactly the same as one in scientific notation: a×10ᵇ.

Calculation example

Let's apply these concepts in practice. We will convert a given number into scientific notation, scientific e-notation, engineering notation, standard form, real number form, and word form. We will also determine its order of magnitude.

Given: 654.901

Solution:

To translate this number into scientific notation, we first identify the value of A:

a=6.54901

To achieve this A value, we moved the decimal point two spaces to the left. Therefore, b=2.

Writing the final number in scientific notation, we get:

6.54901×10²

In scientific e-notation, the same number is expressed as:

6.54901e2

For engineering notation, the exponent B must be a multiple of 3. Since our current value is b<3, we adjust the format to use b=0 so the corresponding physical value lacks a prefix. Consequently, the engineering notation is:

654.901×10⁰

Since standard form is synonymous with scientific notation, the value remains identical:

6.54901×10²

The real number form reverts to our original figure:

654.901

In word form, the number is articulated as:

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

Finally, the order of magnitude is determined by the power of 10 in its scientific notation. In this case, the order of magnitude is 2.