Exponential Notation Calculator

What is an exponential notation calculator?

An exponential notation calculator converts numbers between ordinary decimal form and scientific notation. Scientific notation, also called exponential notation, writes a number as a mantissa multiplied by a power of ten:

$$m \times 10^{e}$$

where the mantissa $m$ satisfies $1 \le |m| < 10$ and the exponent $e$ is an integer. This compact form makes very large and very small numbers easy to read and compare, which is why it is the standard way of expressing quantities in science and engineering.

This calculator works in both directions. Choose Scientific from number to enter an ordinary number and read off its mantissa and exponent, or choose Number from scientific to enter a mantissa and exponent and recover the standard decimal value.

Key concepts

• Mantissa (m) — the significant part of the number, always written with a single nonzero digit before the decimal point so that $1 \le |m| < 10$.
• Exponent (e) — the integer power of ten that scales the mantissa. A positive exponent shifts the decimal point to the right (large numbers); a negative exponent shifts it to the left (small numbers).
• Base — in scientific notation the base is always 10.

How does the calculator work?

To convert a number to scientific notation, the calculator finds how many places the decimal point must move so that exactly one nonzero digit remains in front of it. That number of places is the exponent, and the shifted value is the mantissa.

Formulas

The exponent is the floor of the base-10 logarithm of the absolute value of the number:

$$e = \lfloor \log_{10} |x| \rfloor$$

The mantissa is then the number divided by that power of ten:

$$m = \frac{x}{10^{e}}$$

To go the other way, multiply the mantissa by the power of ten:

$$x = m \times 10^{e}$$

Worked examples

Example 1: a large number

Convert $12345$ to scientific notation. The decimal point moves four places to the left:

$$12345 = 1.2345 \times 10^{4}$$

Example 2: a small number

Convert $0.00067$ to scientific notation. The decimal point moves four places to the right, giving a negative exponent:

$$0.00067 = 6.7 \times 10^{-4}$$

Example 3: back to standard form

Given a mantissa of $3.2$ and an exponent of $5$, the standard number is:

$$3.2 \times 10^{5} = 320000$$

Practical uses

• Science — expressing physical constants such as Avogadro’s number or the charge of an electron without writing long strings of zeros.
• Engineering — recording measurements that span many orders of magnitude in a uniform, comparable format.
• Computing — floating-point numbers are stored internally in a form closely related to scientific notation.
• Education — practising the relationship between place value, powers of ten, and the powers of two used in binary representations.

Notes

• The mantissa always carries the sign of the original number, so $-4500 = -4.5 \times 10^{3}$.
• Zero has no unique scientific notation; by convention this calculator reports it as $0 \times 10^{0}$.
• The exponent is always a whole number, while the mantissa may have a decimal part.