Scientific Notation Calculator
Convert to and from scientific, E and engineering notation
๐ฌ Enter a value
โจ Scientific notation
๐ Every form
๐ช Step by step
- Start with the value 0.00042.
- Move the decimal point so exactly one non-zero digit sits to its left, giving the coefficient 4.2.
- Count the places moved. You moved it 4 places right, so the exponent is -4.
- Write it as 4.2 ร 10-4.
Results are rounded to your chosen number of significant figures. Conversions use standard IEEE-754 double precision, so values beyond ~15-16 significant digits may show tiny rounding artifacts.
Last updated June 2026
Method: Numbers are normalized to the standard form a × 10n with 1 ≤ |a| < 10, then re-expressed as E-notation and as engineering notation (exponent a multiple of 3). Conversions use IEEE-754 double-precision arithmetic.
Included: Decimal → scientific and scientific → decimal conversion, E-notation, engineering notation, the expanded form, a step-by-step walkthrough, a significant-figures control and a significant-figures count.
Not included: Arithmetic on two numbers in scientific notation (use a scientific calculator), symbolic algebra, and exact representation beyond ~15-16 significant digits.
Scientific notation calculator: the complete guide
Some numbers are too big or too small to write comfortably. The speed of light is 299,792,458 meters per second; the mass of a single proton is 0.00000000000000000000000000167 kilograms. Writing those out is error-prone - it is far too easy to add or drop a zero. Scientific notation fixes this by expressing every number as a coefficient times a power of ten, so the speed of light becomes a tidy 2.9979 × 108 and the proton mass becomes 1.67 × 10-27. This scientific notation calculator converts numbers to and from that form, and also shows E-notation and engineering notation, with the working laid out step by step.
The definition and formula
A number is in scientific notation (also called standard form in many countries) when it is written as a coefficient multiplied by a power of ten:
value = a × 10n, where 1 ≤ |a| < 10 and n is an integer Here a is the coefficient (sometimes called the mantissa or significand) and n is the exponent. The rule 1 ≤ |a| < 10 means the coefficient always has exactly one non-zero digit before the decimal point. That single constraint is what makes the form normalized - every number has one and only one correct scientific-notation representation.
A first worked example
Take 91,200. Slide the decimal point (which currently sits after the final zero) to the left until just one non-zero digit remains in front of it: 9.1200, i.e. 9.12. You moved it 4 places, and because you moved left the exponent is positive: 9.12 × 104. To check, 104 = 10,000, and 9.12 × 10,000 = 91,200. Now a small number: 0.00042. Slide the decimal point right until one non-zero digit is in front of it - past the three zeros and onto the 4 - giving 4.2 after moving 4 places right, so the exponent is negative: 4.2 × 10-4.
How to use this calculator
The tool works in both directions. Pick a mode at the top, then read the results below.
- Choose a direction. "Decimal → scientific" turns an ordinary number into scientific notation; "Scientific → decimal" turns a coefficient and exponent back into a plain number.
- Type the number. In decimal mode you can paste figures with commas (91,200) and even E-notation (6.022e23). In scientific mode, enter the coefficient and the whole-number exponent separately.
- Set the significant figures. The slider controls how many significant figures the coefficient keeps, which is handy for tidying up long decimals.
- Read every form. The big result shows the normalized scientific notation; the cards below give E-notation, engineering notation and the expanded form, plus a step-by-step breakdown.
Who this calculator is for
- Students in middle-school, high-school or college math, physics and chemistry classes learning standard form and significant figures.
- Science and engineering professionals who routinely handle quantities spanning many orders of magnitude.
- Programmers and data analysts who meet E-notation in spreadsheets, logs and floating-point output and need to read it as a normal number.
- Anyone double-checking a homework answer, a lab measurement, or a calculator display they do not quite trust.
Scientific, E and engineering notation compared
These three forms describe the same value with the same digits; only the presentation differs.
- Scientific notation: the printed form a × 10n with one digit before the point, e.g. 9.12 × 104.
- E-notation: a keyboard-friendly version where "× 10^" is replaced by the letter E, e.g. 9.12E+4. You will see this on calculators and in code.
- Engineering notation: the exponent is forced to a multiple of 3 to match metric prefixes, e.g. 91.2 × 103 (that is 91.2 kilo-somethings).
Reference table: common values in scientific notation
A few familiar quantities, and how they look in each form:
| Quantity | Expanded | Scientific | Engineering |
|---|---|---|---|
| Thousand | 1,000 | 1 × 103 | 1 × 103 |
| Million | 1,000,000 | 1 × 106 | 1 × 106 |
| Speed of light (m/s) | 299,792,458 | 2.998 × 108 | 299.8 × 106 |
| Avogadro's number | 602,200,000,000,000,000,000,000 | 6.022 × 1023 | 602.2 × 1021 |
| One thousandth | 0.001 | 1 × 10-3 | 1 × 10-3 |
| Example small value | 0.00042 | 4.2 × 10-4 | 420 × 10-6 |
| Proton mass (kg) | 0.00000000000000000000000000167 | 1.67 × 10-27 | 1.67 × 10-27 |
Reading the exponent at a glance
The exponent is really just a place-value counter. For a positive exponent n, the number is a 1 followed by n zeros (times the coefficient). For a negative exponent, count zeros after the decimal point.
| Power of 10 | Value | Metric prefix |
|---|---|---|
| 109 | 1,000,000,000 | giga (G) |
| 106 | 1,000,000 | mega (M) |
| 103 | 1,000 | kilo (k) |
| 100 | 1 | — |
| 10-3 | 0.001 | milli (m) |
| 10-6 | 0.000001 | micro (µ) |
| 10-9 | 0.000000001 | nano (n) |
Worked example 2: converting back to a plain number
Suppose a problem gives you 6.022 × 1023 (Avogadro's number) and asks for the expanded form. A positive exponent of 23 means you shift the decimal point 23 places to the right. Starting from 6.022, the first three shifts use the existing digits 0, 2, 2, and the remaining 20 shifts append zeros. The result is 602,200,000,000,000,000,000,000. In practice nobody writes that out - which is exactly why scientific notation exists - but the mechanic is simply "exponent = how far the decimal point moves."
Worked example 3: a negative number
Negative numbers keep their sign on the coefficient. Convert -0.0000005: ignore the sign for a moment, move the decimal point right past six zeros and onto the 5 to get the coefficient 5, having moved 7 places, so the exponent is -7. Reattach the sign and the answer is -5 × 10-7. The normalized rule 1 ≤ |a| < 10 is about the magnitude of the coefficient, so a negative coefficient is perfectly valid.
Significant figures and why notation helps
Significant figures are the digits in a measurement that actually carry meaning. Ordinary decimals are ambiguous: does "4,200" have two significant figures or four? You cannot tell. Scientific notation removes the doubt because every digit in the coefficient is significant. Writing 4.2 × 103 declares two significant figures, while 4.200 × 103 declares four. The table below summarizes the counting rules.
| Rule | Example | Sig figs |
|---|---|---|
| All non-zero digits count | 3.14 | 3 |
| Zeros between non-zeros count | 5.007 | 4 |
| Leading zeros do not count | 0.0034 | 2 |
| Trailing zeros after a decimal count | 2.300 | 4 |
| In scientific notation, every coefficient digit counts | 4.20 × 103 | 3 |
Tips for working with scientific notation
- Sanity-check the exponent's sign. Big numbers get positive exponents, numbers below 1 get negative exponents. If your sign feels wrong, you probably moved the decimal the wrong way.
- Count places, not zeros. The exponent is how many places the decimal moved, which is not always the same as the number of zeros (think 91,200).
- Use engineering notation for units. If you want to attach a metric prefix, snap the exponent to a multiple of 3 first.
- Keep significant figures consistent. When you round, do it in the coefficient; the exponent never affects how many significant figures you have.
Common pitfalls explained
Most mistakes come from the decimal-point dance rather than the concept itself. Getting the direction of the shift backwards flips the sign of the exponent; forgetting that the coefficient must stay below 10 leaves you with a non-normalized answer like 91.2 × 103, which is engineering notation, not standard scientific notation. Another frequent slip is treating the exponent as "number of zeros" - true for round numbers like 1,000,000 but false the moment any non-zero digits follow the leading one.
Related concepts
- Orders of magnitude: the exponent itself is the "order of magnitude," so 108 is three orders of magnitude larger than 105.
- Logarithms: the exponent in scientific notation is essentially the integer part of the base-10 logarithm of the number.
- Floating-point numbers: computers store decimals internally in a binary version of scientific notation, which is why E-notation appears in code and spreadsheets.
- Metric prefixes: kilo, mega, milli and micro are just engineering-notation exponents given names.
โ ๏ธ Common mistakes & edge cases
Getting the sign of the exponent backwards
Big numbers take a positive exponent; numbers smaller than 1 take a negative one. Writing 0.00042 as 4.2 × 104 instead of 4.2 × 10-4 is the single most common error - it makes the number a billion times too large.
Leaving the coefficient out of range
The coefficient must satisfy 1 ≤ |a| < 10. Answers like 91.2 × 103 or 0.5 × 10-2 are not normalized scientific notation - normalize them to 9.12 × 104 and 5 × 10-3.
Counting zeros instead of decimal places
The exponent is how many places the decimal point moves, not the number of zeros. For 91,200 the decimal moves 4 places (exponent 4) even though only two trailing zeros are present.
Confusing engineering notation with scientific notation
Engineering notation deliberately uses exponents that are multiples of 3 and lets the coefficient run up to 1000. That is a feature, not a mistake - but do not submit 420 × 10-6 when a question asks for strict scientific notation.
❓ Frequently asked questions
What is scientific notation?
Scientific notation writes a number as a x 10^n, where the coefficient a has exactly one non-zero digit before the decimal point (so 1 <= |a| < 10) and n is a whole-number power of 10. For example, 91,200 becomes 9.12 x 10^4 and 0.00042 becomes 4.2 x 10^-4. It is the standard way to write very large or very small numbers compactly.
How do I convert a number to scientific notation?
Move the decimal point until exactly one non-zero digit remains to its left - that gives the coefficient. Count how many places you moved it: that count is the exponent. Moving the decimal to the left (for big numbers) makes the exponent positive; moving it to the right (for small numbers) makes the exponent negative. Example: 4,500 -> move the point 3 places left -> 4.5 x 10^3.
What is the difference between scientific and E-notation?
They mean the same thing, just written differently. Scientific notation uses the printed form a x 10^n (for example 6.022 x 10^23). E-notation replaces 'x 10^' with the letter E for calculators and computers, so the same value is 6.022E+23 or 6.022e23. The coefficient and exponent are identical.
What is engineering notation?
Engineering notation is a variant of scientific notation in which the exponent is always a multiple of 3 (..., -6, -3, 0, 3, 6, ...) so it lines up with metric prefixes like kilo (10^3), mega (10^6), milli (10^-3) and micro (10^-6). The coefficient is allowed to be between 1 and 1000. For example, 91,200 is 9.12 x 10^4 in scientific notation but 91.2 x 10^3 in engineering notation.
How do I write a small number like 0.00042 in scientific notation?
For numbers smaller than 1 the exponent is negative. Move the decimal point to the right until one non-zero digit is in front of it: 0.00042 becomes 4.2, and you moved the point 4 places right, so the exponent is -4. The result is 4.2 x 10^-4.
What is the standard form of a number?
'Standard form' is the British and international term for scientific notation - the same a x 10^n format taught in schools. In the United States, 'standard form' more often means the ordinary expanded way of writing a number (for example 5,300 instead of 5.3 x 10^3). This calculator shows both the scientific form and the expanded form, so it covers either meaning.
How do I convert scientific notation back to a regular number?
Take the coefficient and shift its decimal point by the exponent: a positive exponent moves the point right (making the number bigger), a negative exponent moves it left (making it smaller), padding with zeros as needed. For 4.2 x 10^-4 you move the point 4 places left to get 0.00042; for 9.12 x 10^4 you move it 4 places right to get 91,200. Switch this tool to 'Scientific -> decimal' mode to do it automatically.
How are significant figures related to scientific notation?
One of the big advantages of scientific notation is that every digit shown in the coefficient is significant - there is no ambiguity from leading or trailing zeros. The coefficient of 4.20 x 10^3 clearly has three significant figures, whereas writing 4,200 leaves it unclear whether the trailing zeros count. This calculator lets you choose how many significant figures to keep in the result.
Can scientific notation have a negative coefficient?
Yes. The sign of the whole number stays with the coefficient, so a negative number has a negative coefficient. For example, -0.0000005 is written as -5 x 10^-7. The rule that 1 <= |a| < 10 uses the absolute value, so the magnitude of the coefficient is still between 1 and 10.
What is the exponent of a number that is already between 1 and 10?
Numbers from 1 up to (but not including) 10 are already in the coefficient range, so the exponent is 0 - and 10^0 equals 1. For example, 7.5 in scientific notation is simply 7.5 x 10^0. The number 1 itself is 1 x 10^0.
How do you handle zero in scientific notation?
Zero is a special case: it has no non-zero digit to place in front of the decimal, so there is no unique exponent. By convention zero is written as 0 (or 0 x 10^0). This calculator treats 0 as 0 rather than forcing an artificial exponent.
Why does my result show a tiny rounding difference?
Computers store decimals in binary (IEEE-754 double precision), which cannot represent every decimal exactly. For numbers with more than about 15-16 significant digits, or extreme exponents, you may see a trailing digit drift slightly. Lowering the significant-figures setting cleans this up for everyday use.
๐ก Good to know
"Standard form" can mean two things
In the UK and much of the world, "standard form" is scientific notation (a × 10n). In US schools it often means the ordinary expanded number. This tool shows both, so you are covered either way.
E-notation is just shorthand
When a calculator or spreadsheet shows 6.022E+23, the E means "times ten to the power of." It is identical to 6.022 × 1023 - the same digits, the same value, only easier to type.
Every coefficient digit is significant
Scientific notation removes the ambiguity of trailing zeros. 4.20 × 103 clearly states three significant figures, which is impossible to convey unambiguously by writing 4,200.
Related Calculators
Percentage Calculator
Solve any percentage problem - what is X% of Y and more
Fraction Calculator
Add, subtract, multiply and divide fractions
Scientific Calculator
A full scientific calculator with trig, logs and more
Average Calculator
Calculate the mean, median and mode of numbers
GPA Calculator
Calculate your grade point average (weighted or unweighted)
Ratio Calculator
Simplify, scale and solve ratios and proportions