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Scientific Notation Calculator

Result

1.28 × 10³

Scientific notation

E notation
1.28e3
Ordinary notation
1280

Scientific notation writes a number as a coefficient between 1 and 10 times a power of ten, so 45000 becomes 4.5 × 10⁴ and 0.00045 becomes 4.5 × 10⁻⁴. Its value is that the size of the number moves into the exponent: multiplying and dividing stop being about counting zeros and become about adding and subtracting small integers. This calculator takes two numbers and one operation and prints the result in three forms at once — scientific notation, e notation, and the ordinary decimal. The exponent rules are the whole of the arithmetic. To multiply, multiply the coefficients and add the exponents: (3.2 × 10⁵) × (4 × 10⁻³) = 12.8 × 10², which is then normalised to 1.28 × 10³. To divide, divide the coefficients and subtract the exponents. Adding and subtracting are the awkward ones, because the exponents have to line up first: 1.5 × 10³ plus 2.5 × 10² is 1500 plus 250, and the answer is 1.75 × 10³. Nothing about that is different from ordinary decimal arithmetic — it is the same sum — but the notation makes the alignment step easy to forget. Inputs are accepted in either form. The e notation that programmers type, 3.2e5 and 4e-3, is read directly, and so is the ordinary decimal 45000; the two are the same number and the page treats them identically. A written form such as 4.5 × 10⁴ is not accepted, and the reason is worth knowing rather than working around: recognising it means accepting five spellings of the same idea — a multiplication sign, an x, an asterisk, a superscript exponent, a caret — and reading one of them wrong means computing with the wrong number. The e notation line of the result is there to be pasted back into the inputs. Both numbers may carry up to fifteen significant digits, and their exponents may run from −99 to 99; the answer may go past that, since 1e99 × 1e99 is 1 × 10¹⁹⁸. One caution applies to the arithmetic in a way it does not to the other pages here. Addition and multiplication of decimals are done in double precision, which cannot represent every decimal exactly, so 0.1 + 0.2 is 0.30000000000000004 before it is anything else. The result is rounded to fifteen significant digits and its trailing zeros are dropped, which prints 3 × 10⁻¹ and 0.3 — the tail is at the sixteenth and seventeenth digit. That fifteen is a display width, not a claim: an answer computed from two numbers does not know how many of its digits were measured, and no number of digits printed here would tell you. Deciding how many significant figures an answer deserves is a separate question, and it belongs to the significant figures page rather than to this one. Two conventions meet on this page and are worth separating once. In American usage this notation is scientific notation; in British usage, and in the GCSE syllabus, the same thing is usually called standard form. The two names describe the same coefficient-and-exponent form, and the standard form page handles the conversion half of the job — one number in, the same number written the other way — while this page does the arithmetic half.

Powers of ten, written as scientific notation, as e notation and as a decimal

powerordinarye notation
1 × 10⁻⁹0.0000000011e-9
1 × 10⁻⁶0.0000011e-6
1 × 10⁻³0.0011e-3
1 × 10⁻²0.011e-2
1 × 10⁻¹0.11e-1
1 × 10⁰11e0
1 × 10¹101e1
1 × 10²1001e2
1 × 10³10001e3
1 × 10⁴100001e4
1 × 10⁵1000001e5
1 × 10⁶10000001e6
1 × 10⁹10000000001e9

Thirteen powers of ten, from 10⁻⁹ to 10⁹, with the same number written all three ways. Read the first column as the exponent and the third as the form to type: 10⁻³ is 0.001, which is 1e-3. The rows in the middle are the ones that cause trouble in practice — 10⁰ is 1 and 10⁻¹ is 0.1, so a missing negative sign moves a value by a factor of ten, and 10⁻² and 10⁻³ are the two that turn up most often in measurement work. Note where the decimal form stops being comfortable: below 10⁻⁶ the zeros on the right start to be hard to count, which is the usual argument for writing small values in this notation at all. Above 10⁶ the ordinary form is unreadable without separators, and this page never prints separators, because a comma means a decimal point in several of the languages this site serves. Every cell is computed from the same kernel the panel uses, so a row can never disagree with an answer above it.

Formula

(a × 10ᵐ) × (b × 10ⁿ) = (a × b) × 10ᵐ⁺ⁿ; (a × 10ᵐ) ÷ (b × 10ⁿ) = (a ÷ b) × 10ᵐ⁻ⁿ; addition and subtraction align first: 1.5 × 10³ + 2.5 × 10² = 1500 + 250 = 1.75 × 10³; 3.2e5 × 4e-3 = 1.28 × 10³ = 1.28e3 = 1280

3.2e5 and 4e-3
The two numbers being combined, each written either in e notation (3.2e5) or as an ordinary decimal (320000), and each carrying up to fifteen significant digits. They are the inputs the page reads; the answer is a new number computed from them
× ÷ + −
The operation, chosen from four options. Multiplication is first because it is what scientific notation is for: the exponents add, so the size of the answer is settled before any coefficient is multiplied
10ᵐ × 10ⁿ = 10ᵐ⁺ⁿ
The exponent rule underneath multiplying and dividing. The exponents are added for a product and subtracted for a quotient, which is why 10⁵ × 10⁻³ gives 10² without any counting of zeros
1.5 × 10³ + 2.5 × 10²
Why addition and subtraction are different. The exponents here do not match, so the two numbers have to be written at a common exponent before they can be added — 1500 + 250 — and the result is 1.75 × 10³. Skipping that step is the usual mistake
0.1 + 0.2 = 3 × 10⁻¹
Why the answer is printed to fifteen significant digits. 0.1 and 0.2 cannot both be held exactly in double precision, and their sum is 0.30000000000000004; rounding to fifteen digits, and dropping the trailing zeros that remain, prints 3 × 10⁻¹ and 0.3. Fifteen is a display width rather than a claim about precision
1.28 × 10³, 1.28e3, 1280
The three forms of one answer. The first is what a textbook prints, the second is what a program or a spreadsheet will accept if it is pasted back into an input, and the third is the number written out. They are never three different answers

Scientific notation earns its place wherever the numbers are far from 1. In physics it is the default for anything measured in atoms, metres, or light years — the mass of an electron, the distance to a star, the wavelength of a colour — and the reason is the same at every scale: the exponent carries the size, so a comparison between two quantities becomes a comparison between two exponents. In chemistry the notation pairs with counting: 6.022 × 10²³ is a count of particles, and multiplying it by a measured amount is arithmetic in this form. Engineering leans on it for tolerances at both ends of a drawing, and computing leans on it for the reason e notation exists at all: floating-point numbers are stored as a coefficient and an exponent, and the e notation this page prints is the shortest honest way to write one. Textbooks and exam papers use it as a checkable form. Working a problem in scientific notation and giving the answer as 4.5 × 10⁴ makes the size of the answer visible in a way that a run of zeros does not, which is why a misplaced exponent is caught by eye in this form and missed in the other. What this page is for inside that work is the arithmetic step: it does the exponent bookkeeping and prints the answer in whichever of the three forms the next step needs, and it is most useful when the numbers are being combined and the answer has to be written down rather than estimated. Two related jobs sit either side of it. Writing the same number the other way, with no arithmetic involved, is the standard form page. Deciding how many of the digits in an answer were actually measured — which is what significant figures are about — is the significant figures page, and it is the question this page deliberately does not answer.

Worked examples

  1. Multiplying: 3.2 × 10⁵ by 4 × 10⁻³

    1. Multiply the coefficients: 3.2 × 4 = 12.8
    2. Add the exponents: 10⁵ × 10⁻³ = 10²
    3. The product is 12.8 × 10², which is not scientific notation yet, because the coefficient has to be below 10
    4. Move one factor of ten from the coefficient into the exponent: 12.8 × 10² = 1.28 × 10³
    5. Written out, that is 1280 — and the e notation 1.28e3 can be pasted back into either input

    The default, and the case that shows why the notation is worth using. Both numbers are short, and the arithmetic is two multiplications and an addition; without the notation the same work would be 320000 × 0.004, where the zeros are doing all the work and the answer is easy to misplace by a factor of ten. The normalisation step is the part to watch, and it is the step the panel does silently: 12.8 × 10² and 1.28 × 10³ are the same number, and only the second is in scientific notation.

  2. Dividing: 9.6 × 10⁷ by 3.2 × 10³

    1. Divide the coefficients: 9.6 ÷ 3.2 = 3
    2. Subtract the exponents: 10⁷ ÷ 10³ = 10⁴
    3. The coefficient is already between 1 and 10, so no normalisation is needed
    4. The answer is 3 × 10⁴, which is 3e4 and 30000

    A division where the coefficient comes out clean, so nothing hides the exponent rule. Note that the coefficient has no trailing zeros: the result is 3 and not 3.00, because the page prints fifteen significant digits and then drops the zeros that carry no information. Whether the answer deserves to be quoted as 3 or as 3.00000 is a significant figures question, and the inputs here do not settle it.

  3. Adding 0.1 and 0.2

    1. Exponents do not appear in either input, so both numbers are written out as they are: 0.1 and 0.2
    2. In double precision the stored values are not exactly one tenth and two tenths, and their sum is 0.30000000000000004
    3. The result is rounded to fifteen significant digits, which leaves 0.300000000000000
    4. Trailing zeros are dropped, because the fifteen digits are a display width and not a measurement
    5. What is left is 3 × 10⁻¹, or 3e-1, or 0.3

    The example that justifies the display width. Printing seven significant digits would also hide the floating-point tail, but it would break a different case: 123456789012345 + 0 would print as 123456800000000, which is simply the wrong number. Fifteen digits is the widest setting that is still exact for every value the inputs accept, so it hides the tail and keeps the number. If you have met this sum as a programming puzzle, the puzzle is real — it is just not visible at this width.

Limitations

Each input accepts up to fifteen significant digits, and an exponent from −99 to 99. The answer is not bounded by those limits: 1e99 × 1e99 is 1 × 10¹⁹⁸, and its ordinary form is a line of 199 characters. Written forms are rejected. 4.5 × 10⁴ and 4.5×10^4 do not parse, and neither do thousands separators, so 1,500 is refused rather than read as 1500 — a comma is the decimal mark in several of the languages this site serves, and guessing which one is meant would put a wrong number into the arithmetic. Use 4.5e4, or type the digits out. Division by zero throws rather than returning an infinity, because an infinity is not a result that can be written down. Zero prints as 0 in all three forms: 0 × 10⁰ would be a magnitude attached to a number that has none. Results are rounded to fifteen significant digits and their trailing zeros are dropped, so the answer carries no information about precision and the page makes no claim about it; that question belongs to the significant figures page. Because multiplication, division, addition and subtraction are done in double precision, a result is exact only to fifteen significant digits — enough for the inputs this page accepts, and not enough to be treated as exact arithmetic on longer values, which are refused for that reason. Finally, the reference table below is a fixed list of powers of ten and does not follow your input; it is there to be read beside the answer, not to be computed from it.

Frequently asked questions

What is scientific notation?
A way of writing a number as a coefficient between 1 and 10 multiplied by a power of ten: 45000 is 4.5 × 10⁴ and 0.00045 is 4.5 × 10⁻⁴. The size of the number moves into the exponent, which is what makes multiplying and dividing in this form a matter of handling small integers instead of long runs of zeros. British usage usually calls the same thing standard form.
What is e notation, and why does the page print it?
e notation is the same number in plain characters: 4.5e4 means 4.5 × 10⁴, and 4.5e-4 means 4.5 × 10⁻⁴. It is what a program, a spreadsheet or a search box will accept, so the page prints it as one of the three forms — it is the version of the answer that can be pasted straight back into an input.
Why does 0.1 + 0.2 give exactly 0.3 here?
Because the result is rounded to fifteen significant digits before it is printed, and the stored sum is 0.30000000000000004 — the difference is at the sixteenth and seventeenth digit. Fifteen digits is not a round number chosen for looks: it is the widest display width for which every value the inputs accept is printed exactly, so 123456789012345 + 0 still comes back as 123456789012345 rather than being rounded to something shorter.
Can I type 4.5 × 10⁴ into the inputs?
No, and the page will say so. Recognising that form means accepting several spellings of it — a multiplication sign, an x, an asterisk, a superscript exponent, a caret — and reading one of them wrongly means computing with the wrong number. Type 4.5e4 or 45000 instead; both are the same value, and the e notation line of every answer is written to be pasted back in.
How do I multiply and divide in scientific notation?
Multiply the coefficients and add the exponents for a product; divide the coefficients and subtract the exponents for a quotient. Then check that the coefficient is between 1 and 10 and move a factor of ten into the exponent if it is not: 3.2 × 10⁵ times 4 × 10⁻³ is 12.8 × 10², which is written as 1.28 × 10³. Adding and subtracting need the exponents aligned first, which is the step the notation makes easy to skip.
Does the answer say how many significant figures it has?
No. The answer is a computed number, and no display width can tell you how many of its digits were measured — that depends on the precision of the numbers that went in, and on whether they were measurements at all. Trailing zeros are dropped for the same reason: keeping them would suggest a precision the arithmetic does not have. Working out how many significant figures an answer deserves is the subject of the significant figures page.

References

Related calculators

Standard Form CalculatorConvert between standard form and ordinary numbers.Sig Fig CalculatorCount significant figures and round to a set number.Exponent CalculatorRaise a number to any power, including negative exponents.Rounding CalculatorRound to any place value or number of significant figures.