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Standard Form Calculator

Result

4.5 × 10⁴

Standard form

Ordinary notation
45000

Standard form is a number written 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. It is the same form that American usage calls scientific notation, and this calculator converts between it and the ordinary decimal in either direction: type 45000 and read 4.5 × 10⁴, or type 4.5e4 and read 45000 back. The conversion is a decimal-point move and a count. Put the point after the first non-zero digit, count how many places it travelled, and that count is the exponent — positive when the number is ten or more, negative when it is below one. 45000: the point goes from after the last zero to after the 4, four places left, so the exponent is 4 and the coefficient 4.5. 0.00045: five places right, so the exponent is −5 and the coefficient is 4.5, except that the leading zeros are not counted as part of the five — the count is to the first non-zero digit, which gives 4.5 × 10⁻⁴. The rule that catches people is the one about zeros. In a written number, a zero can mean two different things: it can be part of the size of the number, or it can be a digit someone deliberately wrote down. Standard form keeps that distinction, because it is the difference between two numbers of different precision. 1200 becomes 1.2 × 10³ — the two zeros at the end of an integer only say that the number is in the thousands. 1200. becomes 1.200 × 10³ — the decimal point says those zeros were written, so they are kept, and the coefficient is 1.200 rather than 1.2. 0.004560 behaves the same way from the other side: the zeros after the decimal point are kept, giving 4.560 × 10⁻³, which is a different number from 4.56 × 10⁻³ only in what it claims to know. Two special cases are worth stating rather than discovering. Zero prints as 0 in both forms, because attaching a power of ten to a number with no magnitude is a claim the number cannot support. And a negative sign belongs to the coefficient, not to the exponent: −45000 is −4.5 × 10⁴ and never 4.5 × 10⁻⁴. Inputs are accepted as ordinary decimals or in e notation, and the written form 4.5 × 10⁴ is refused — recognising it would mean accepting several spellings of the same idea, and the e notation this page prints is written so that it can be pasted back in. What this calculator does not do is arithmetic. Combining two numbers in this form is the scientific notation page, which is why that page exists separately; and deciding how many significant figures the coefficient should carry is a third question, handled by the significant figures page. Here the number goes in and comes out unchanged, with only its spelling settled.

Six values written as ordinary numbers and in standard form

numberstandard form
0.0000011 × 10⁻⁶
0.000454.5 × 10⁻⁴
0.55 × 10⁻¹
454.5 × 10¹
450004.5 × 10⁴
6020000000000000000000006.02 × 10²³

Six conversions that cover the shapes this page gets asked about: a value below one, a value below one with more digits, a decimal just under one, a two-digit number, a large integer, and Avogadro's number. Read the left column as what you type and the right column as what comes back. Note the two rows in the middle: 45 becomes 4.5 × 10¹, and the exponent 1 is written rather than dropped — it is what tells the coefficient where the point belongs, and leaving it off would turn a two-digit number into a one-digit one. The last row is the one that shows why the form exists at all; the ordinary form of that number runs to 24 characters and is printed without separators, so it can only be read by counting. This table does not change when the input above it changes. It is a fixed set of examples to compare an answer against, and every cell in it is computed from the same kernel that produces the answer, so a row can never contradict the panel.

Formula

45000 = 4.5 × 10⁴; 0.00045 = 4.5 × 10⁻⁴; 1200 = 1.2 × 10³ but 1200. = 1.200 × 10³; 6.02e23 = 6.02 × 10²³ = 602000000000000000000000

45000
The number being converted, given either as an ordinary decimal or in e notation. It may be an integer or carry a decimal point, and it may be negative; the sign is carried through to the coefficient and never to the exponent
4.5
The coefficient — or mantissa — which has to sit between 1 and 10. It is the original digits with the decimal point moved, so the digits themselves never change: 45000 and 4.5 × 10⁴ hold the same digits 4 and 5, and only the position of the point differs
10⁴
The exponent, which counts the places the decimal point moved and carries the sign of the move. Four places left gives 4 and a large number; four places right gives −4 and a small one, which is why 0.00045 becomes 4.5 × 10⁻⁴ rather than 4.5 × 10⁴
1200 vs 1200.
Where the trailing zeros come from. Written as an integer, the two zeros only indicate magnitude, so the coefficient is 1.2. With a decimal point after them they were written down deliberately, so they are kept and the coefficient is 1.200. The same number of digits, two different claims about precision
0
The one value with no exponent attached. Zero prints as 0 in both forms, because any power of ten written beside it would state a magnitude that zero does not have
602000000000000000000000
The ordinary form of 6.02 × 10²³, which is twenty-four characters long and unreadable without counting. The e notation 6.02e23 that produced it is what you would type; the notation exists so that this line never has to be written out

Converting in this direction is what you do when a number has to be written down for someone else. A measurement taken in a laboratory or a datasheet comes out as 0.00045 and has to appear in a report as 4.5 × 10⁻⁴, because the second form states the precision of the coefficient and the first leaves it to be counted. The reverse conversion happens just as often for a different reason: a value arrives as 6.02 × 10²³ and has to be entered into something that only accepts digits, so it is written out or, better, pasted as e notation. In a classroom the conversion is the exercise itself. Moving the decimal point and counting the places is the one mechanical skill in this topic, and the trailing-zero rule is the part of it that is actually about measurement rather than about notation — 1200 and 1200. are the same quantity and not the same claim, and a coefficient that keeps those zeros is saying that someone measured down to the ones place. Exams use this form because it is checkable: a coefficient outside the range 1 to 10 is wrong at a glance, in a way that a misplaced zero in a long decimal is not. Two neighbours are worth knowing about. If the number is being combined with another one rather than merely rewritten, the scientific notation page does that arithmetic. If the question is how many digits the coefficient should be allowed to carry, the answer comes from the significant figures page, and it comes from the provenance of the number rather than from the notation.

Worked examples

  1. A whole number: 45000

    1. Find the first non-zero digit, which is the 4
    2. Put the decimal point after it: 4.5
    3. Count the places the point moved from its original position at the end of the number: four
    4. The number is larger than 10, so the exponent is positive: 10⁴
    5. The result is 4.5 × 10⁴, and the ordinary form printed beside it is the 45000 that went in

    The default, and the simplest shape of the conversion: the digits are already 4 and 5, so only the point moves. The exponent is the number of places it moved, which is the same as the number of digits after the first one when the number is an integer with no trailing zeros to be argued about.

  2. A value below one: 0.00045

    1. Find the first non-zero digit in 0.00045, which is the 4 in the fourth decimal place
    2. Put the point after it: 4.5
    3. Count the places it moved to get there from after the leading zero: four
    4. The number is below one, so the exponent is negative: 10⁻⁴
    5. The leading zeros do not appear in the coefficient, which is 4.5 and not 0.00045

    The direction that gets miscounted. The coefficient is 4.5 and the exponent is −4, so the answer is 4.5 × 10⁻⁴; writing 4.5 × 10⁻⁵ is the usual error, and it comes from counting the leading zero as a place. The check is to count back: 4.5 × 10⁻⁴ moves the point four places right, which lands on 0.00045.

  3. Two ways of writing 1200

    1. The trailing decimal point is not a typo: it says that the two zeros after the 2 were written deliberately
    2. Put the point after the first digit: 1.200
    3. Count the places: three, so the exponent is 3
    4. The coefficient keeps its zeros, because they were written: 1.200 × 10³
    5. Now type 1200 without the point and the same calculator answers 1.2 × 10³ — same value, different claim

    The rule that makes standard form useful rather than merely tidy, and the one place on this page where the input's spelling changes the output. Both answers describe the value 1200; they differ in how many digits are being asserted. A coefficient of 1.200 says the ones place was measured or at least written down; 1.2 does not. Note what the calculator is not doing: it is not guessing which one you meant, and it is not counting significant figures for you — it is reading what you typed, which is the only honest thing it can do.

  4. A value already in e notation: 6.02e23

    1. The input is already acceptably formed: 6.02e23 means 6.02 × 10²³
    2. The coefficient 6.02 is between 1 and 10, so no adjusting is needed
    3. The exponent 23 is positive, so the ordinary form is a large integer
    4. Writing it out gives 602 followed by 21 zeros — 24 characters in all

    Avogadro's number, and the standard argument for the notation. The ordinary form is not wrong, it is unreadable: 602000000000000000000000 has to be counted to be believed, and a reader who miscounts is out by a factor of ten with no way to notice. The e notation that goes in and the coefficient-and-exponent form that comes out both carry the size in two short pieces. This is also the case that shows the conversion runs both ways on one page: the input was in e notation, the input's ordinary form was never typed, and the calculator prints it anyway.

Limitations

The written form is not an input. 4.5 × 10⁴ and 4.5×10^4 are both refused; use 4.5e4 or 45000. Thousands separators are refused too, so 45,000 is an error rather than 45000 — a comma is the decimal mark in several of the languages this site serves, and reading it as a separator would silently change the value. Zero prints as 0 in both forms, with no power of ten attached and no sign. Numbers are held to fifteen significant digits, and an exponent beyond ±99 is refused. The trailing-zero rule depends on the spelling of the input, not on the value: 1200 gives 1.2 × 10³ and 1200. gives 1.200 × 10³, and there is no third answer, because the calculator has nothing else to go on. That also means this page does not compute significant figures. It keeps or drops trailing zeros according to how the number was written; deciding how many of the remaining digits were actually measured is a question about where the number came from, and that belongs to the significant figures page. The ordinary form is printed without separators at any length, so the ordinary form of 6.02e23 is a run of 24 characters and the ordinary form of 1e99 would be a hundred of them; that is deliberate, since an ungrouped string can be read or pasted anywhere, while 602,000,000,000,000,000,000,000 cannot. Finally, no rounding takes place. The digits that go in are the digits that come out, repositioned and nothing more, so a coefficient is never trimmed to look neater than the number it came from.

Frequently asked questions

What is standard form?
A number written 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. It is the name used in British schools and exam boards for what American usage calls scientific notation. Nothing about the form differs between the two names; only the convention for writing the exponent does, and both are printed here.
How do I convert a number to standard form?
Move the decimal point to sit just after the first non-zero digit, then count how many places it moved. That count is the exponent, and it is positive for a number of 10 or more and negative for a number below 1. 45000 gives 4.5 × 10⁴, because the point moved four places left; 0.00045 gives 4.5 × 10⁻⁴, because it moved four places right. The digits themselves never change.
Why does 1200 give 1.2 × 10³ but 1200. give 1.200 × 10³?
Because a trailing zero in a written number means one of two things, and standard form keeps the difference. In the integer 1200 the two zeros only say that the number is in the thousands, so the coefficient is 1.2. Adding the point after them says they were written deliberately, so they are kept and the coefficient is 1.200. The value is the same; the claim about how much of it is known is not.
Why can't I type 4.5 × 10⁴ into the input?
Because that form has several spellings — a multiplication sign, an x, an asterisk, a caret, a superscript exponent — and each one has to be recognised exactly, since reading it wrongly means converting the wrong number. Type 4.5e4 or 45000 instead. They are the same value and the calculator prints the e notation form of any number you give it, so an answer can always be pasted back into the input.
Does this calculator tell me how many significant figures the coefficient has?
No, and it would be guessing if it did. It keeps or drops trailing zeros according to how the number was spelled, and stops there. How many digits of a number were actually measured is a question about where the number came from — whether 1200 was counted or estimated — and that is the subject of the significant figures page.
What is the difference between this page and the scientific notation calculator?
This one converts. A number goes in and the same number comes out written the other way, with no arithmetic and no rounding. The scientific notation calculator takes two numbers and an operation, does the arithmetic in this form, and prints the answer three ways. If you are rewriting a value, you are on the right page; if you are combining two of them, that is the other one.

References

Related calculators

Scientific Notation CalculatorConvert to and from scientific notation, with arithmetic.Expanded Form CalculatorWrite a number in expanded form, digit by digit.Place Value CalculatorThe place value of every digit in a number.Exponent CalculatorRaise a number to any power, including negative exponents.