Scientific notation calculator

Scientific notation is a compact way to write very large or very small numbers without a long row of zeros. The number becomes:

a × 10ⁿ, with a between 1 and 10 (the "mantissa") and n the exponent.

The exponent counts how many places the decimal point moved: to the right for small numbers (negative n), to the left for large ones (positive n).

Step by step

• 0.000123 → the point moves 4 places right → 1.23 × 10⁻⁴
• 45,000,000 → the point moves 7 places left → 4.5 × 10⁷

Real-world orders of magnitude

Quantity	Decimal form	Scientific notation
Avogadro constant	602,200,000,000,000,000,000,000	6.022 × 10²³
Earth–Sun distance	149,600,000 km	1.496 × 10⁸ km
Speed of light	299,792,458 m/s	≈ 3 × 10⁸ m/s
World population	8,100,000,000	8.1 × 10⁹
Thickness of a human hair	0.00007 m	7 × 10⁻⁵ m
Hydrogen atom radius	0.000000000053 m	5.3 × 10⁻¹¹ m
Electron mass	0.00000…9109 kg	9.109 × 10⁻³¹ kg

This is where the notation earns its keep: comparing 10⁻⁵ with 10⁻¹¹ is instant — six orders of magnitude, a million times over. Comparing the two decimal forms means counting zeros by eye and getting it wrong.

Arithmetic in scientific notation

Multiply: multiply the mantissas and add the exponents.
(2 × 10³) × (3 × 10⁴) = 6 × 10⁷

Divide: divide the mantissas and subtract the exponents.
(6 × 10⁸) ÷ (2 × 10³) = 3 × 10⁵

Add or subtract: here is the most common mistake. You cannot add directly — the exponents must match first.
(3 × 10⁵) + (4 × 10⁴) → rewrite the second as 0.4 × 10⁵ → 3.4 × 10⁵

If a product overflows the range (say 5 × 10³ times 4 × 10³ = 20 × 10⁶), normalize it: 2 × 10⁷.

Engineering notation

Same idea with one constraint: the exponent is always a multiple of 3. That makes the number line up with SI prefixes — kilo (10³), mega (10⁶), giga (10⁹), milli (10⁻³), micro (10⁻⁶), nano (10⁻⁹).

So 47,000 Ω is 4.7 × 10⁴ in scientific notation, but 47 × 10³ Ω = 47 kΩ in engineering notation — which is how the value is printed on the component.

Significant figures

Scientific notation resolves an ambiguity decimal form has. Writing 4500 does not say whether the measurement has 2, 3 or 4 reliable digits. But:

• 4.5 × 10³ → 2 significant figures
• 4.50 × 10³ → 3 figures
• 4.500 × 10³ → 4 figures

In a lab that distinction is not fussiness: it reports the precision of the instrument that took the measurement.

The calculator "e"

When the screen shows 6.022e23 or 6.022E23, the "e" means exponent — it is 6.022 × 10²³. It is not Euler's number and not exponentiation: 6.022e23 is nothing like 6.022²³. Spreadsheets and programming languages accept the same form as input.