Scientific Notation Calculator

Scientific notation is a method of writing numbers that are too large or too small to be conveniently written in standard decimal form. Widely used in chemistry, physics, astronomy, and engineering, it simplifies calculations and helps clarify the number of significant digits. In scientific notation, numbers are written in the format:

b × 10^n

Where b is the significand (or base), which must be a real number between 1 and 10 (exclusive of 10, meaning 1 ≤ |b| < 10), and n is the order of magnitude (an integer exponent).

Our free Scientific Notation Calculator and Converter converts numbers between decimal form, standard scientific notation, engineering notation, and E-notation. You can also perform math operations (addition, subtraction, multiplication, division, power, and roots) on variables written in scientific notation with high-precision outputs.

Arithmetic Operations in Scientific Notation

Performing calculations with scientific notation involves applying exponent laws. Below are the step-by-step methods for hand calculations:

Addition and Subtraction

To add or subtract numbers, you must first convert all terms to have the exact same power of 10. Once the exponents match, add or subtract the significands and keep the common power of 10:

1.432 × 10^2 + 800 × 10^-1 - 0.001 × 10^5

Convert all terms to 10^2:

– 1.432 × 10^2

– 800 × 10^-1 = 0.8 × 10^2

– 0.001 × 10^5 = 1 × 10^2

Add/Subtract:

(1.432 + 0.8 - 1) × 10^2 = 1.232 × 10^2.

Multiplication

To multiply, separate the significands and exponents. Multiply the significands normally, and add the exponents together:

(1.432 × 10^2) × (800 × 10^-1) × (0.001 × 10^5)

Multiply significands: 1.432 × 800 × 0.001 = 1.1456

Sum exponents: 2 + (-1) + 5 = 6

Result: 1.1456 × 10^6.

Division

To divide, separate the significands and exponents. Divide the significands normally, and subtract the exponents of the divisor from the dividend:

(1.432 × 10^2) ÷ (800 × 10^-1) ÷ (0.001 × 10^5)

Divide significands: 1.432 ÷ 800 ÷ 0.001 = 1.79

Subtract exponents: 2 - (-1) - 5 = -2

Result: 1.79 × 10^-2.

Note: If the significand is not between 1 and 10, shift the decimal and adjust the exponent accordingly (e.g., 0.179 × 10^-2 = 1.79 × 10^-3).

Engineering Notation vs. E-Notation

Our converter displays two alternative formats used in computing and manufacturing:

Engineering Notation

Engineering notation is identical to scientific notation, with one exception: the exponent n must be a multiple of 3 (e.g., 0, 3, 6, 9, -3, -6). This allows numbers to match standard SI metric prefixes:

• 10^3 = kilo (k)
• 10^6 = mega (M)
• 10^-6 = micro (μ)

Example: The scientific notation value 1.234 × 10^8 is rewritten in engineering notation as 123.4 × 10^6 (123.4 Mega).

E-Notation

E-notation (or exponential notation) is used by calculators, spreadsheets, and programming languages when superscript exponents cannot be easily displayed. The × 10^n is replaced by the letter E or e:

Solve exponential equations using our Exponent Calculator or solve roots with the Root Calculator.

Frequently Asked Questions (FAQ)

What is a significand?

The significand (also called the mantissa or base) is the portion of a number in scientific notation that contains its significant digits. By standard convention, it must be a number whose absolute value is at least 1 and strictly less than 10 (e.g., in 3.45 × 10^6, the significand is 3.45).

How do you convert a decimal to scientific notation?

Move the decimal point until you have a single non-zero digit to the left of the decimal. Count the number of places you moved it:

• If you moved the decimal to the left, the exponent is positive (e.g., 1568938 ⇒ 1.568938 × 10^6).
• If you moved the decimal to the right, the exponent is negative (e.g., 0.0004 ⇒ 4.0 × 10^-4).

What is the difference between scientific and engineering notation?

In scientific notation, the significand is always between 1 and 10, and the exponent can be any integer. In engineering notation, the significand can range from 1 to 1000, and the exponent must be a multiple of 3 (e.g., 0, 3, 6, 9, -3, -6), making it easier to read with SI prefixes.

Does e-notation mean Euler’s number?

No. In E-notation (such as 2.3e11), the “e” or “E” stands for “exponent” or “times ten raised to the power of.” It is completely different from the mathematical constant e ≈ 2.71828 used as the base of natural logarithms.