Scientific Notation Converter

About Scientific Notation

Scientific notation expresses numbers as a × 10ⁿ, where 1 ≤ |a| < 10 and n is an integer.

Examples

When to Use

• Very large numbers (astronomy, physics)
• Very small numbers (chemistry, biology)
• Scientific calculations
• Engineering applications

What Is Scientific Notation?

Scientific notation writes very large or very small numbers in the compact form a × 10ⁿ, where the coefficient a sits between 1 and 10 and n is an integer exponent. Instead of writing 6,022,000,000,000,000,000,000,000, chemists write 6.022 × 10²³ — the number of atoms in a mole. Instead of 0.000000001, physicists write 1 × 10⁻⁹ seconds for a nanosecond. The notation keeps the significant digits visible while the exponent carries the scale.

It is the standard language of science and engineering because it makes magnitude obvious at a glance and turns messy multiplication into simple exponent arithmetic. The calculator on this page converts between standard and scientific notation and handles the arithmetic for you.

Rules for Scientific Notation

Three rules keep the notation consistent. First, the coefficient must satisfy 1 ≤ |a| < 10 — exactly one non-zero digit before the decimal point. Second, moving the decimal point right makes the exponent negative (for small numbers), and moving it left makes the exponent positive (for large numbers). Third, in E notation used by calculators and programming, 4.5 × 10⁻⁴ is written as 4.5e-4.

Step-by-Step Conversion

• Step 1: Start with 0.00045. Find the first non-zero digit, which is 4.
• Step 2: Move the decimal point right until exactly one non-zero digit stands before it: 4.5.
• Step 3: Count the moves: the point traveled 4 places.
• Step 4: Because the original number was smaller than 1, the exponent is negative: 4.5 × 10⁻⁴.
• Step 5: Sanity check: 10⁻⁴ = 0.0001, and 4.5 × 0.0001 = 0.00045. Correct.

Worked Examples

Example 1: A tiny length

Convert 0.00000053 meters. Moving the decimal 7 places right gives 5.3, so the result is 5.3 × 10⁻⁷ m — about half a micrometer.

Example 2: Multiplying

Multiply (3 × 10⁴) by (2 × 10⁵). Multiply the coefficients (3 × 2 = 6) and add the exponents (4 + 5 = 9): 6 × 10⁹.

Example 3: Adding

Add 4.2 × 10³ and 1.5 × 10². Convert to standard form: 4200 + 150 = 4350, then re-normalize: 4.35 × 10³. Addition requires matching exponents first.

Example 4: Avogadro's number in practice

Two moles of a substance contain 2 × 6.022 × 10²³ = 12.044 × 10²³ atoms. Re-normalize the coefficient into the 1–10 range: 1.2044 × 10²⁴.

Common Mistakes

• Writing a coefficient of 10 or more, like 12.5 × 10³ — it must be re-normalized to 1.25 × 10⁴.
• Using the wrong exponent sign: small numbers need negative exponents, large numbers positive.
• Adding or subtracting without first converting to the same exponent.
• Confusing 10⁻⁴ (a tiny positive number, 0.0001) with −10⁴ (negative ten thousand).

Frequently Asked Questions

Why must the coefficient be between 1 and 10?

The rule guarantees one unique representation per number, so 4.5 × 10⁻⁴ can never also be written as 45 × 10⁻⁵ in proper scientific notation. Uniqueness makes comparison and arithmetic reliable.

What does the "e" mean in 4.5e-4?

It is shorthand for "times ten to the power of," borrowed from calculator displays and programming languages. 4.5e-4 means exactly 4.5 × 10⁻⁴.

How do I multiply numbers in scientific notation?

Multiply the coefficients and add the exponents, then re-normalize if the coefficient leaves the 1–10 range. Division works the same way, except you subtract the exponents.

Is engineering notation the same thing?

Nearly — engineering notation restricts exponents to multiples of 3 (thousands, millions, billions), matching metric prefixes like kilo, mega, and giga. The coefficient may then range from 1 to 1000.

E Notation: How Calculators Write It

On calculators, in spreadsheets, and in programming languages, scientific notation is written on one line as E notation: the letter e (or E) stands for "times ten to the power of".

Spreadsheets write it slightly differently, as 6.022E+23. The meaning is identical — only the punctuation differs.

Engineering Notation and SI Prefixes

Engineering notation is scientific notation with one extra rule: the exponent must be a multiple of 3 (..., 10^-6, 10^-3, 10^3, 10^6, ...). The coefficient may then range from 1 to 1000. Engineers prefer it because those powers match the metric (SI) prefixes used on every instrument and component.

Worked Example: Scientific to Engineering

Convert 0.00042 to engineering notation. Scientific form is 4.2 x 10^-4. The nearest lower multiple of 3 is -6, so shift the coefficient two places: 420 x 10^-6 — read directly as 420 micrometers when the unit is meters.

Convert 37,000. Scientific form is 3.7 x 10^4. Shift to the multiple of 3 below: 37 x 10^3 — read as 37 thousand of whatever unit applies.

The Full Arithmetic Rules

The page above shows multiplication and addition. The complete set:

Worked Example: Subtraction

7.5 x 10^4 - 2.5 x 10^3. Align exponents: 7.5 x 10^4 - 0.25 x 10^4 = 7.25 x 10^4. Check: 75,000 - 2,500 = 72,500.

Worked Example: Division

(9 x 10^3) / (3 x 10^-2). Divide coefficients (9 / 3 = 3) and subtract exponents (3 - (-2) = 5): 3 x 10^5. Check: 9,000 / 0.03 = 300,000.

Worked Example: Powers

(2 x 10^3)^2 = 2^2 x 10^6 = 4 x 10^6. And (5 x 10^-2)^3 = 125 x 10^-6, re-normalized to 1.25 x 10^-4.

Worked Example: Square Roots

Take the square root of the coefficient and halve the exponent: sqrt(4 x 10^6) = 2 x 10^3 = 2,000. If the exponent is odd, rewrite first: sqrt(9 x 10^-3) = sqrt(90 x 10^-4) = sqrt(90) x 10^-2, about 9.49 x 10^-2.

Powers of Ten Reference

Significant Figures

Scientific notation is the cleanest way to show how many digits of a measurement are trustworthy — its significant figures. The digits in the coefficient are the significant figures; the exponent only sets the scale.

• 0.00520 has 3 significant figures (the leading zeros are placeholders; the trailing zero counts).
• 37,000 is ambiguous — is it 2, 3, 4, or 5 significant figures? Writing 3.7 x 10^4 (2 sig figs) or 3.700 x 10^4 (4 sig figs) removes all doubt.
• When multiplying or dividing, the answer keeps the fewest significant figures of the inputs: 5.67 x 2.34 = 13.2678, reported as 13.3 (3 sig figs) in a lab setting.

Zero and Negative Numbers

Zero has no meaningful scientific notation — it is simply 0, since no coefficient between 1 and 10 times a power of ten equals zero. Negative numbers keep their minus sign in front: -5.2 x 10^-3 = -0.0052. A negative exponent does not make the number negative; it only makes the magnitude smaller than 1.

Orders of Magnitude in the Real World

The exponent is called the order of magnitude: two quantities that differ by 10^3 differ by three orders of magnitude. Comparing exponents is the fastest way to compare scales.