Scientific Notation Calculator

What Is Scientific Notation?

Scientific notation (also called standard form) is a way to write numbers as a × 10ⁿ. The coefficient a has exactly one non-zero digit before the decimal point, so 1 ≤ |a| < 10; the base is always 10; and the exponent n is a whole number that says how many places the decimal point has moved.

In 5.2 × 10⁶, the coefficient is 5.2, the base is 10 and the exponent is 6, so the number is 5.2 with the decimal point moved 6 places to the right: 5,200,000. It is useful because it replaces long strings of zeros with a single exponent, makes the size of a number obvious at a glance, and shows exactly which digits are significant.

How to Convert to Scientific Notation

Step 1: move the decimal point until only one non-zero digit is to its left. Step 2: count how many places it moved. Step 3: use that count as the exponent. Step 4: make the exponent positive if the original number was 10 or more (the point moved left) and negative if it was between 0 and 1 (the point moved right).

Large number: 4,500,000 → move the point 6 places left to get 4.5, so 4.5 × 10⁶. Small number: 0.000072 → move the point 5 places right to get 7.2, so 7.2 × 10⁻⁵. To go back to a decimal, reverse it: 6.3 × 10⁴ moves the point 4 places right to give 63,000, and 8.2 × 10⁻⁶ moves it 6 places left to give 0.0000082.

Scientific Notation Rules

The coefficient must satisfy 1 ≤ |a| < 10. Writing 12 × 10⁵ is not normalized; move the point one place left and add 1 to the exponent to get 1.2 × 10⁶. Likewise 0.5 × 10³ becomes 5 × 10².

A positive exponent means the number is at least 10 (3 × 10⁴ = 30,000). A negative exponent means it is between −1 and 1 (3 × 10⁻⁴ = 0.0003). An exponent of 0 leaves the coefficient unchanged, because 10⁰ = 1 (3 × 10⁰ = 3).

Negative numbers keep their sign on the coefficient: −4500 = −4.5 × 10³ and −0.00082 = −8.2 × 10⁻⁴. The sign of the number and the sign of the exponent are independent. Zero is a special case: no power of 10 turns 0 into a number between 1 and 10, so it is written simply as 0.

Arithmetic With Scientific Notation

To multiply, multiply the coefficients and add the exponents: (3.2 × 10⁵) × (4 × 10²) = 12.8 × 10⁷, which normalizes to 1.28 × 10⁸. To divide, divide the coefficients and subtract the exponents: (8 × 10⁷) ÷ (2 × 10³) = 4 × 10⁴.

To add or subtract, the exponents must match first; you cannot just add the coefficients. For 3 × 10⁵ + 4 × 10³, rewrite 4 × 10³ as 0.04 × 10⁵, then add: 3 + 0.04 = 3.04, giving 3.04 × 10⁵. For subtraction, the result may need normalizing the other way: 1.2 × 10⁵ − 1.1 × 10⁵ = 0.1 × 10⁵ = 1 × 10⁴.

Significant Figures in Scientific Notation

Every digit of the coefficient is significant, which is why scientific notation is the clearest way to show precision. Leading zeros are never significant: 0.0045678 has 5 significant figures (4, 5, 6, 7, 8), and in scientific notation it is 4.5678 × 10⁻³. Trailing zeros after a decimal point are significant, so 2.50 has 3.

Trailing zeros in a whole number are ambiguous: 1200 could have 2, 3 or 4 significant figures. Writing 1.2 × 10³, 1.20 × 10³ or 1.200 × 10³ removes the doubt. To round, keep the required number of digits and look at the next one: 12,345.678 to 3 significant figures is 1.23 × 10⁴, and 0.0045678 is 4.57 × 10⁻³.

Engineering Notation

Engineering notation is like scientific notation, but the exponent is always a multiple of 3 and the coefficient can be from 1 up to 999. So 12,300 is 1.23 × 10⁴ in scientific notation but 12.3 × 10³ in engineering notation, and 0.000000345 is 345 × 10⁻⁹.

The multiples of 3 line up with metric prefixes: 10³ is kilo, 10⁶ mega, 10⁹ giga, 10⁻³ milli, 10⁻⁶ micro and 10⁻⁹ nano. That makes engineering notation convenient in electronics and engineering, where 4.7 × 10³ Ω reads directly as 4.7 kΩ.

Real-World Uses of Scientific Notation

Physics: the speed of light is about 2.998 × 10⁸ m/s, and an electron's charge is about 1.602 × 10⁻¹⁹ coulombs. Astronomy: the Sun is about 1.496 × 10¹¹ m from Earth, and a light-year is about 9.461 × 10¹⁵ m. Chemistry: Avogadro's number is 6.022 × 10²³ particles per mole, and a hydrogen atom is roughly 1 × 10⁻¹⁰ m across.

Biology: a typical bacterium is about 2 × 10⁻⁶ m long, and the human body contains roughly 3 × 10¹³ cells. Computer science: a terabyte is 10¹² bytes, and calculators and programming languages print very large or small results in e-notation, such as 6.022e23. In all of these, scientific notation makes the size of the number obvious and keeps the significant figures honest.