Scientific notation: formula, steps, and examples

Scientific notation is a compact way to write very large or very small numbers without a long trail of zeros. Instead of writing 0.0000000047 or 5,300,000,000, you write the same value as a mantissa (also called the coefficient) multiplied by a power of ten:

Scientific notation form: a × 10ⁿ

• a is the mantissa (usually 1 ≤ |a| < 10 in “normalized” scientific notation).
• n is the exponent (an integer telling you how many places the decimal moved).

This page gives you a fast, shareable Scientific Notation Calculator for homework, labs, coding, finance spreadsheets, and anytime you want fewer zeros and fewer mistakes. It converts:

• Standard → Scientific: 5300000000 → 5.3 × 10⁹
• Scientific → Standard: 4.7 × 10^-9 → 0.0000000047
• “e” notation: 1.2e6 ↔ 1.2 × 10⁶

Why scientific notation is viral (and actually useful)

It’s one of those “tiny math tools” that shows up everywhere: science class, engineering, statistics, finance growth models, and even social media (“how big is 10⁹ seconds?”). People share these conversions because they make numbers feel real and readable. Converting huge values into a quick “a × 10ⁿ” format is also a cheat code for mental math: you can compare magnitudes instantly.

The core idea (formula breakdown)

To convert any non‑zero number x into normalized scientific notation, you pick an integer exponent n so that dividing by 10ⁿ leaves a mantissa a between 1 and 10 (in absolute value):

x = a × 10ⁿ with 1 ≤ |a| < 10

A fast way to find the exponent is:

• n = floor(log₁₀(|x|)) (for x ≠ 0)
• a = x / 10ⁿ

Example: x = 5300000000. |x| = 5.3 × 10⁹. log₁₀(|x|) is a bit more than 9, so n = 9. Then a = 5300000000 / 10⁹ = 5.3. Done.

How rounding / significant figures work

Most real‑world uses of scientific notation include rounding to a chosen number of significant figures (sig figs). Significant figures are the digits that carry meaning based on measurement precision. For example:

• 3.14159 has 6 significant figures
• 5.30 × 10² has 3 significant figures (the trailing zero matters)
• 0.00470 has 3 significant figures (leading zeros don’t count)

In this calculator, you can choose a “Sig figs” setting. We round the mantissa to that many significant digits. If rounding pushes the mantissa to 10.0 (for example 9.99 → 10.0), we shift it back into the normalized range by dividing by 10 and increasing the exponent by 1.

Step-by-step: Standard → Scientific

1. Start with your number (example: 0.000072).
2. Move the decimal until you have one non‑zero digit to the left (7.2).
3. Count moves: you moved 5 places to the right, so the exponent is negative (−5).
4. Write it: 7.2 × 10⁻⁵.

Why negative? Because the original number is smaller than 1. Moving the decimal to the right made it bigger, so you “undo” that by multiplying by 10 raised to a negative power.

Step-by-step: Scientific → Standard

If you already have a × 10ⁿ, convert back by shifting the decimal in a:

• If n is positive, move the decimal right.
• If n is negative, move the decimal left.

Example: 4.7 × 10⁻⁹. Move the decimal 9 places left: 0.0000000047.

Examples (copy/paste friendly)

• Large number: 987000000000 → 9.87 × 10¹¹
• Small number: 0.0000012 → 1.2 × 10⁻⁶
• Negative value: −45000 → −4.5 × 10⁴
• E notation: 3.5e8 = 3.5 × 10⁸

How this calculator works (under the hood)

When you click “Convert”, the calculator reads your input in one of three forms:

• Standard decimal: 1200, 0.0047, −98.6
• Scientific: 1.2 × 10^6, 1.2*10^6, 1.2 × 10⁶
• E notation: 1.2e6, 1.2E6

It then parses that into a single numeric value and produces:

• Standard output (plain decimal, with an optional comma toggle)
• Scientific output (normalized mantissa + exponent, rounded to your chosen sig figs)

For extremely large exponents, plain decimals can get very long. To keep the page fast and readable, we format standard output in a “safe” way (not scientific) when it’s reasonable, and otherwise we show a clean string representation that still matches the value.

FAQ

• What’s the difference between scientific notation and engineering notation?

  Scientific notation uses any integer exponent and keeps the mantissa between 1 and 10. Engineering notation forces the exponent to be a multiple of 3 (… −6, −3, 0, 3, 6 …) so it pairs nicely with metric prefixes (micro, milli, kilo, mega, etc.).

• Why does the exponent become negative for small numbers?

  Because you move the decimal to the right to get a mantissa between 1 and 10. Moving right makes the number bigger, so you compensate by multiplying by 10 raised to a negative power.

• Does 5.0 × 10³ mean something different than 5 × 10³?

  In many lab and measurement contexts, yes: 5.0 has two significant figures while 5 has one. That “extra” zero can communicate precision. In pure math, both represent 5000 exactly.

• My calculator shows “E” instead of “× 10^”. Is that the same?

  Yes. 1.23E4 means 1.23 × 10⁴. It’s just a compact computer-friendly format.

• Can scientific notation help with multiplying and dividing big numbers?

  Absolutely. Multiply the mantissas and add the exponents: (a × 10^m)(b × 10ⁿ) = (ab) × 10^m+n. Divide the mantissas and subtract the exponents: (a × 10^m)/(b × 10ⁿ) = (a/b) × 10^m−n. Then re-normalize if needed.

• What if my number is zero?

  Zero is a special case. It doesn’t have a meaningful exponent because 0 divided by any power of ten is still 0. We simply show “0” and “0 × 10⁰”.

• Why do I get slightly different decimals sometimes?

  Computers store many decimals using binary floating‑point, which can create tiny rounding artifacts (like 0.1 + 0.2 = 0.30000000000000004). This calculator formats outputs to be clean and human-friendly while keeping the underlying value consistent.

Tip: If you’re doing homework, also try the Significant Figures Calculator and the Rounding Numbers Calculator to keep your final answers in the right format.

Sanity-check your result

Use these fast rules to catch mistakes in seconds:

• Mantissa range: In normalized scientific notation, the mantissa should be between 1 and 10 (in absolute value).
• Exponent sign: If the original number is less than 1 (but not 0), the exponent should be negative.
• Magnitude: Bigger numbers have bigger exponents. 10⁶ is a million; 10⁹ is a billion.
• Move check: If you move the decimal left to form the mantissa, the exponent is positive; if you move it right, the exponent is negative.
• Reasonable rounding: If you set 3 sig figs, the mantissa should have ~3 meaningful digits.

If you’re working with measurements, your final step is often rounding to the correct significant figures.

Common powers of ten (fast reference)

• 10³ = 1,000 (thousand)
• 10⁶ = 1,000,000 (million)
• 10⁹ = 1,000,000,000 (billion)
• 10¹² = 1,000,000,000,000 (trillion)
• 10⁻³ = 0.001 (milli)
• 10⁻⁶ = 0.000001 (micro)
• 10⁻⁹ = 0.000000001 (nano)

Engineering notation uses exponents in multiples of 3 to match these prefixes.