Scientific Notation Calculator

This calculator consists of two parts – scientific notation converter and scientific notation calculator. The first part allows you to convert an input number into the following formats:

• Scientific notation
• Engineering notation
• E-notation
• Real number format

You can enter a number in any of the formats above, and the calculator will convert it to the remaining formats.

The second part performs various mathematical operations with numbers in scientific notation. You can perform the following operations:

• Addition
• Subtraction
• Multiplication
• Division
• Raising to a power
• Finding square root
• Finding the square

Directions for use

Scientific notation converter

To use the scientific notation converter, simply enter the known number and press “Convert.” Input values can be positive or negative integers and decimals, with the exception of 0.

To enter a number in scientific notation, use the following representation: ax10^b, for example, 4x10^-3.

To enter a number in e-notation, use the following representation: aeb, for example, 5.2e12.

To enter a decimal real number, separate the whole number part from the decimal part by a dot, for example, 3.876. You can use spaces or commas to separate orders of magnitude, but it’s not necessary.

Scientific notation calculator

The scientific notation calculator performs operations with two numbers: X and Y. To use the calculator, enter the whole number parts of X and Y, and the corresponding powers of 10. Then enter a positive integer in the precision field. Precision represents the number of digits after the decimal point in the final answer. Finally, choose the necessary operation at the bottom of the calculator. The calculation will start automatically.

Definitions and algorithms

Notations

Scientific notation – is a convenient way to write very big or very small numbers. The numbers are written in the following format: a × 10ᵇ. For example,

9,000 = 9 × 10³

0.000005 = 5 × 10⁻ᵇ

Scientists, mathematicians, and engineers often have to deal with numbers that are very large or very small, therefore, they use this notation very frequently.

To convert a number to scientific notation, follow the algorithm below:

1. Write down the significant digits of the number, putting the decimal point after the first digit. This part of a number is sometimes called the significand.
2. Determine the power of 10 in the final number by counting how many positions should a decimal point be moved to get the original number. If to get the original number the decimal point has to be moved to the right, the power of 10 will be positive. If it has to be moved to the left, the power of 10 will be negative. The power of 10 is called the exponent of the number.

For example, let’s convert 678000 to scientific notation:

1. Writing down the significant digits of the number, putting the decimal point after the first digit, we get: 6.78.
2. We see that in step 1 we have moved the decimal point 5 positions to the left, therefore, to get the original number we would need to move the decimal point 5 positions to the right. Th exponent will be +5.

678,000 = 6.78 × 10⁵

Engineering notation – is almost the same as scientific notation, but the exponents can only be represented by multiples of 3. For example, 4.45 × 10⁶, 1.15 × 10⁻¹². This notation was developed, so that it’s easier to read out the numbers, since the powers of 10 in this notation align with the SI prefixes.

For example, imagine a scientist measured the length of a very short signal, and it turned out to be 0.00000004 seconds. Converting this number to engineering notation, we get:

0.00000004 = 4 × 10⁻⁸ = 40 × 10⁻⁹

If you have to read this number out loud, you would quickly notice that pronouncing 4 × 10⁻⁸ in scientific notation takes a rather long time. In engineering notation, however, 10⁻⁹ corresponds to the SI prefix “nano,” so 40 × 10⁻⁹ seconds can be read as “forty nanoseconds.”

E-notation is the same as scientific notation, but “10 to the power of” is replaced by “e.” For example, 2 × 10⁴ would be 2e⁴, or 2E⁴, in e-notation. This notation is used when the exponents in scientific or engineering notations cannot be conveniently displayed, for example, in some calculators.

Mathematical operations

Addition and subtraction

To add or subtract numbers in scientific notation, follow the steps below:

1. Convert all numbers to numbers with the same power of 10.
2. Perform addition and subtraction for the significant digits of numbers from step 1.
3. If necessary, convert the result to scientific notation.

For example, let’s calculate (5 × 10⁸) + (3.5 × 10¹⁰):

1. (5 × 10⁸) + (3.5 × 10¹⁰) = (5 × 10⁸) + (350 × 10⁸)
2. 5 + 350 = 355
3. (5 × 10⁸) + (3.5 × 10¹⁰) = (5 × 10⁸) + (350 × 10⁸) = 355 × 10⁸ = 3.55 × 10¹⁰

Multiplication and division

To multiply or divide numbers in scientific notation, follow the steps below:

1. Separate significands from the exponents.
2. Multiply or divide the significands, following the rules for real numbers.
3. Add the exponents for multiplication, or subtract the exponents for division.
4. Convert the result to scientific notation, if necessary.

For example, let’s calculate (3.2 × 10⁻⁵) / (1.6 × 10⁻⁷):

1. Significands are 3.2 and 1.6. Exponents are (⁻⁵) and (⁻⁷).
2. Dividing the significands, we get 3.2/1.6 = 2
3. We perform the division operation, therefore, the exponents must be subtracted: (⁻⁵) - (⁻⁷) = 2.
4. (3.2 × 10⁻⁵) / (1.6 × 10⁻⁷) = 2 × 10². This number is already in scientific notation, therefore, further conversion is not necessary.

Finding the square

To find the square of a number in scientific notation, you have to multiply the number by itself, following the multiplication algorithm.

Finding the square root

To find the square root of a number in scientific notation, first identify if the exponent of the number is even or uneven. If the exponent is even, take the following steps:

1. Find the square root of the significand.
2. Divide the exponent by 2.
3. If necessary, convert the result to scientific notation.

If the exponent is uneven, take the following steps:

1. Multiply the significand by 10, and reduce the exponent by 1, to get an equivalent number with an even exponent.
2. Follow the algorithm for finding the square root of a number with an even exponent.

Real life examples

Scientific notation is not only used by scientists. Many of us use it in our daily lives.

For example, the population of Earth is estimated as around 8,000,000,000 people. In scientific or engineering notation, this can be expressed as 8 × 10⁹ people. Or, using a SI prefix, 8 billion people.

Let’s look at a very small number: a computer chip has a linewidth of 0.00000013 meters. This can be written much easier in scientific notation: 0.00000013 = 1.3 × 10⁻⁷ meters. Or, in engineering notation, 130 × 10⁻⁹ = 0.13 × 10⁻⁶ = 130 nanometers or 0.13 micrometers.