Standard Form Calculator

This calculator converts input numbers to standard form or standard notation. The calculator takes positive or negative decimals and integers as inputs.

Directions for use

To use this standard form converter, enter the given number into the input field and press “Calculate.”

Limitations on the input values

• Input values greater than or equal to 1 cannot start with a zero. For example, to convert 6 to standard form, you should enter 6, not 0006.
• Input values can be given in number form (integer or decimal), e-notation, or scientific notation. See more details on scientific notation below. Fractions are not accepted.
• You can use commas to separate different orders of magnitude, but it’s unnecessary. For example, both 32,000,000,000 and 32000000000 are valid inputs.

Standard form definition

In simple words, a number is in standard form if it consists of a decimal number greater than zero and less than ten and (though not always) 10 raised to some power. This notation is often used to describe very big or small numbers.

For example, the mass of the Earth is currently estimated to be 5,972,200,000,000,000,000,000,000 kg. Saying or even writing down this number is cumbersome, but in standard form, it can be written as 5.9722 × 10²⁴ kg! Note that this number consists of two parts – a decimal 0 < 5.9722 < 10 and 10 to the power of 24.

For an example of a very small number let’s look at the mass of a grain of sand. The average grain of sand is estimated to weigh about 0.0000128 kg. This number can be written as 1.28 × 10⁻⁵ kg in standard form. It consists of two parts – a decimal 0 < 1.28 < 10 and 10 to the power of -5.

Standard form vs. scientific notation

The terms “standard form” and “scientific notation” describe the same thing. The term “standard form” is mainly used in the US and other countries following the US conventions. “Scientific notation” is largely used in the United Kingdom and other countries following the UK conventions. Therefore, while this standard notation calculator accepts “scientific notation” as an input, converting scientific notation to standard form will not change how the number is written.

How to convert a number to standard form

Let’s look at the conversion algorithm in several examples. For an example of a very big number, let’s convert 34,000,000 to standard form. We will take the following steps:

1. Write down the first significant digit of the number followed by the decimal point: 3.
2. Write all remaining significant digits after the decimal point: 3.4
3. Count how many digits are there after the first digit. In our case, the first digit is 3, with 7 digits after it. 7 will be the power of 10 in the final number.
4. The final number is 3.4 × 10⁷.

For an example of a very small number, let’s convert 0.00065 to standard form. We will take the following steps:

1. Just like during the conversion of a large number, write down the first significant digit of the number, followed by the decimal point. In our example, the first significant digit is 6, so we write 6.
2. The second step is similar to the large number conversion process: write all remaining significant digits after the decimal point. In our example, we will write: 6.5
3. Count how many digits in the original number stand before the first significant digit (including the first zero). The negative of this number will be the power of 10 in the standard form. In our example, there are 4 digits before the 6. Therefore, the standard form will have 10⁻⁴.
4. The final answer will be 6.5 × 10⁻⁴.

Alternatively, the conversion process can be described as follows:

1. Move the decimal point to the position right after the first significant digit of the number.
2. Count the number of steps the decimal point has moved. This will be the power of 10 in the standard form. If the decimal point was moved to the right, the power of 10 would be negative. If it was moved to the left, the power of 10 would be positive.

Let’s convert 456,000 to scientific notation following the alternative algorithm:

1. Moving the decimal point, we get 4.56
2. The given number is whole. Therefore, the decimal point would be at the end of the original number: 456,000 = 456,000.00. To get 4.56 we have moved it 5 steps to the left. This means that the final number will be multiplied by 10⁵.
3. Finally, 456,000 = 4.56 × 10⁵.

0 in a standard form

Since 0 multiplied by any number is still 0, it is also 0 when multiplied by 10 to any power. This means that 0 can be written in standard form in an infinite number of ways: 0 = 0 × 10⁰ = 0 × 10¹ = 0 × 10² = 0 × 10³ = …

Real-life examples

Standard form, or scientific notation, is widely used by scientists, engineers, and even in everyday life to describe very small or very large numbers. Below are some examples of values that are often described in standard form:

• The speed of light is estimated to be approximately 300,000,000 m/s. Let’s convert this number to standard form following the alternative algorithm. Moving the decimal point, we get 3. We had to move the decimal point 8 positions to the left. Therefore, the final number will be multiplied by 10⁸. 300,000,000 = 3 × 10⁸ m/s.
• The diameter of the SARS-CoV-2 (COVID-19) virus is estimated at approximately 0.0000001 m. Moving the decimal point, we get 1. The decimal point has moved 7 steps to the right. Therefore, the final number will be multiplied by 10⁻⁷. Finally, 0.0000001 = 1 × 10⁻⁷. Note that the size of the COVID-19 coronavirus is also often described in nanometers. 1 nanometer amounts to 10⁻⁹ meters. 0.0000001 m = 1 × 10⁻⁷ m = 100 × 10⁻⁹ m = 100 nm.