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CalcMax

Expanded Form Calculator

Result

3000 + 400 + 50 + 6

Expanded form

Expanded form with powers of 10
3 × 10³ + 4 × 10² + 5 × 10¹ + 6 × 10⁰

This calculator writes a number in expanded form, as the sum of its digits, each one multiplied by the value of the place it sits in. 3456 becomes 3000 + 400 + 50 + 6 — not a different number, and not a shorter way of writing it, but the same number with its structure exposed. Reading that sum back is the whole exercise: the 3 is worth 3000 because it is in the thousands place, the 4 is worth 400 because it is in the hundreds place, and so on down to the 6 in the ones place. The exponential version writes the same sum with the place values named as powers of ten: 3 × 10³ + 4 × 10² + 5 × 10¹ + 6 × 10⁰. The two lines are the same statement, and the page prints both, because they answer different questions — the first is what a place-value chart produces, and the second is what the same chart produces once the columns are labelled with exponents. Zeros are dropped from the expansion. 205.04 is 200 + 5 + 0.04: the zero in the tens place and the zero in the tenths place contribute nothing to the sum, and writing 0 × 10 next to the terms that do contribute only makes the line harder to read. This is the one place where this page and the place value page deliberately disagree. That page keeps the zeros, because its subject is the digits — every digit has a place value, including the ones that are worth nothing — while this page's subject is the sum, and a term worth zero can be left out of a sum without changing it. A number smaller than one expands the same way, with negative exponents: 0.004560 is 0.004 + 0.0005 + 0.00006, which is 4 × 10⁻³ + 5 × 10⁻⁴ + 6 × 10⁻⁵. Note what happened at the end there: the trailing zero of 0.004560 does not appear, for the same reason the zero in 205.04 does not — this page is not preserving the written form of the number, it is taking it apart. Negative numbers are written with the sign outside a bracket: −550 is −(500 + 50), not −500 + −50. The second spelling is not wrong arithmetic, but on a page whose entire content is a sum, it reads as two separate negative numbers being added together, and it also suggests the sign belongs to the digits rather than to the number. Zero is the one input with no terms at all, and it prints as 0 rather than as an empty line. Inputs may be written as ordinary decimals or in e notation, and the e notation form is accepted for the same reason it is printed elsewhere on this site: it is the spelling a program will accept. Two neighbours are worth knowing. The place value page takes the same number apart but keeps every digit, so it is the page to read when the question is what a particular digit is worth rather than what the sum of the digits is. The standard form page moves in the opposite direction, collapsing a number into a coefficient and an exponent, which is a summary where this is a dissection.

Four numbers expanded, and the same four expansions written with powers of ten

numberexpanded formexpanded with powers of ten
34563000 + 400 + 50 + 63 × 10³ + 4 × 10² + 5 × 10¹ + 6 × 10⁰
205.04200 + 5 + 0.042 × 10² + 5 × 10⁰ + 4 × 10⁻²
12001000 + 2001 × 10³ + 2 × 10²
0.0045600.004 + 0.0005 + 0.000064 × 10⁻³ + 5 × 10⁻⁴ + 6 × 10⁻⁵

Four cases chosen to cover what the rule does: a four-digit number with no zeros in it, a number with a decimal point and a zero on each side of it, a whole number with two trailing zeros, and a value below one with a trailing zero of its own. The second and fourth rows are the ones worth reading twice. In 205.04 the zero in the tens place and the zero in the tenths place both disappear from the expansion, and in 0.004560 the trailing zero disappears as well; in every case the terms that remain add back up to the number on the left. Note also how the exponents in the third column move: they step down by one for each digit to the right, cross zero at the ones place, and continue downwards as negative numbers below the decimal point. Every cell here is computed from the same kernel that answers the input above, so a row cannot fall out of step with the panel.

Formula

3456 = 3000 + 400 + 50 + 6 = 3 × 10³ + 4 × 10² + 5 × 10¹ + 6 × 10⁰; 205.04 = 200 + 5 + 0.04 = 2 × 10² + 5 × 10⁰ + 4 × 10⁻²; −550 = −(500 + 50)

3456
The number being taken apart, given as an ordinary decimal or in e notation. Its digits are not changed and nothing is rounded; the number is only re-expressed, so an expansion that does not add back up to the input is a bug rather than a rounding decision
3000 + 400 + 50 + 6
The expanded form — one term per non-zero digit, each term being that digit times the value of its place. The digits appear in their original order from left to right, so the line reads the same way the number does
3 × 10³
The same term with its place value written as a power of ten. The exponent is the digit's distance from the ones place, counting left as positive and right as negative, which is why the ones place itself is 10⁰ rather than 10¹
200 + 5 + 0.04
Where the zeros go. 205.04 has a zero in the tens place and a zero in the tenths place, and both terms are left out, because adding zero to a sum changes nothing. The place value page keeps them; this page does not, and that contrast is the reason both pages exist
4 × 10⁻³ + 5 × 10⁻⁴
The part of an expansion below the decimal point, where the exponents are negative. 0.004560 gives 4 × 10⁻³ + 5 × 10⁻⁴ + 6 × 10⁻⁵ — the trailing zero of the input does not produce a term, since this page reads a value rather than preserving a spelling
−(500 + 50)
How a negative number is written. The sign applies to the whole sum and sits outside the bracket, rather than being distributed over each term as −500 + −50, which would read as a sum of two negative numbers

Expanded form is how place value is taught, and it is the step between counting and column arithmetic. A child who can write 3456 as 3000 + 400 + 50 + 6 can see why the 3 and the 4 cannot simply be added, which is the idea that column addition depends on; the same expansion underneath a subtraction explains borrowing, and underneath a multiplication it is what makes 3000 × 4 different from 3 × 4. Long after school it keeps turning up as the readable version of a number's structure. An engineer comparing two readings of the same quantity finds the expansion the quickest way to see which place a discrepancy is in, and a reader checking a spreadsheet result can expand the total and confirm each digit against the inputs rather than trusting the whole figure at once. The exponential form is the bridge to scientific notation and to computing: 3 × 10³ + 4 × 10² + 5 × 10¹ + 6 × 10⁰ is one step from 3.456 × 10³, and the same expression with powers of two instead of ten is exactly how a binary number is expanded, with each bit multiplied by the value of its position. Working in the other direction — collapsing a number back into a coefficient and a power of ten, which loses the individual digits rather than listing them — is what the standard form page does. And when the question is about one particular digit rather than all of them at once, the place value page keeps every digit in place, zeros included, instead of omitting the terms that are worth nothing.

Worked examples

  1. A four-digit number: 3456

    1. Read the digits from the left: 3, 4, 5, 6
    2. Name each digit's place: thousands, hundreds, tens, ones
    3. Multiply each digit by its place value: 3 × 1000, 4 × 100, 5 × 10, 6 × 1
    4. Add the terms: 3000 + 400 + 50 + 6
    5. Write the same four terms with the place values as powers of ten, giving 3 × 10³ + 4 × 10² + 5 × 10¹ + 6 × 10⁰

    The default, and deliberately the easy case: no zeros, no decimal point, no sign, so the shape of the answer is visible before any rule has to be applied. Adding the terms back gives 3456, which is the check worth running on any expansion — the terms must sum to the number they came from, and every digit must appear exactly once.

  2. A number with zeros and a decimal point: 205.04

    1. The digits are 2, 0, 5, 0, 4, and their places run from hundreds down to hundredths
    2. The 2 is worth 200 and the 5 is worth 5; the zeros in the tens and tenths places are worth nothing and produce no term
    3. The 4 sits two places right of the decimal point, so its place value is 0.01 and its term is 0.04
    4. The expansion is 200 + 5 + 0.04, which sums to 205.04
    5. With powers of ten, the same terms are 2 × 10² + 5 × 10⁰ + 4 × 10⁻², and the exponents 2, 0 and −2 are the distances from the ones place

    The page's central case: two zeros, one on each side of the decimal point, and neither produces a term. Compare it with the place value page, which answers 2 × 100, 0 × 10, 5 × 1, 0 × 0.1, 4 × 0.01 for the same input — five terms against three, and both are correct answers to different questions. The zero exponent in 5 × 10⁰ is worth noticing too: the ones place is a place like any other, and its exponent is zero because the digit is zero places away from itself.

  3. A negative number: −550

    1. Take the digits first and ignore the sign: 5 and 5, in the hundreds and tens places
    2. Their terms are 500 and 50, and the zero in the ones place adds nothing
    3. The sum of those terms is 550, and the original number is that sum made negative
    4. Write the sign once, outside a bracket around the whole sum: -(500 + 50)

    The sign is a property of the number, not of any one digit, and the bracket is how that is said in writing. Distributing it would give -500 + -50, which is arithmetically identical and worse to read: on a page whose output is a sum, two minus signs read as two numbers being added, and a reader checking the line has to work out whether the terms were meant to be negative digits or a negative total.

  4. A value below one with a trailing zero: 0.004560

    1. Find the significant digits, which are 4, 5 and 6; the leading zeros only position the decimal point
    2. The 4 sits in the thousandths place, so its term is 0.004
    3. The 5 is in the ten-thousandths place, giving 0.0005, and the 6 in the hundred-thousandths place, giving 0.00006
    4. The last digit of the input is a zero in the millionths place; it contributes nothing and produces no term
    5. The terms sum to 0.00456, which is the value of the input

    The trailing zero is the interesting part, and it is where this page and the significant figures page part company. Here the zero is simply not a term, because a sum does not need it. There it is a measured digit and it counts towards the four significant figures that 0.004560 carries. Both readings come from the same input and neither is a mistake — one is about the value of the number, the other about the precision of the measurement.

Limitations

Zeros produce no terms. 205.04 expands to 200 + 5 + 0.04 and not to 200 + 0 + 5 + 0 + 0.04, so the expansion is not a digit-by-digit transcription of the input; it is the input written as a sum, and terms worth nothing are left out. If you want the zeros kept, the place value page keeps them. The written form is not an input: 4.5 × 10⁴ is refused, and so are thousands separators, so 3,456 is an error rather than 3456. Inputs are limited to fifteen significant digits and to exponents within ±99. A leading zero in a written integer, as in 007, is not preserved either — it has no place value at all, and the expansion of that input is 7. Zero is a special case at both ends: the input 0 expands to 0 with no terms, and the expansion of any number ends at its last non-zero digit rather than at the end of the written form. Nothing is rounded and nothing is converted, so an expansion of 9.99 is 9 + 0.9 + 0.09 and never 10. Finally, this page lists digits; it does not add them up for you, decide how many of them are significant, or write the number as a coefficient and a power of ten. Those are three separate jobs, and they belong to the place value, significant figures and standard form pages respectively.

Frequently asked questions

What is expanded form?
A number written as the sum of its digits, each multiplied by the value of its place: 3456 is 3000 + 400 + 50 + 6. It is not a different number and not a shorter way of writing one; it is the same number with the structure of its places made visible, which is why it is the form place value is first taught in and the form that explains why column arithmetic works.
Why are the zeros left out of the expansion?
Because a term worth zero can be dropped from a sum without changing it, and including it makes the line harder to read: 205.04 is 200 + 5 + 0.04 rather than 200 + 0 + 5 + 0 + 0.04. The place value page takes the opposite view and keeps every digit, because its subject is the digits rather than the sum. Both pages are correct about the same number.
What is the difference between expanded form and expanded notation?
The second one writes the place values as powers of ten: 3 × 10³ + 4 × 10² + 5 × 10¹ + 6 × 10⁰ instead of 3000 + 400 + 50 + 6. This page prints both, one under the other, because the first is what a place value chart gives you and the second is what the same chart gives once its columns are labelled with exponents.
How is a decimal expanded?
The same way, with the places to the right of the decimal point taking negative exponents: 205.04 is 2 × 10² + 5 × 10⁰ + 4 × 10⁻², since the 4 sits two places right of the ones place. 0.004560 expands to 4 × 10⁻³ + 5 × 10⁻⁴ + 6 × 10⁻⁵, and its trailing zero contributes no term at all.
Why is -550 written as -(500 + 50)?
Because the sign belongs to the whole number rather than to any one digit. Distributing it would give -500 + -50, which is the same value but reads as two negative numbers being added together — confusing on a page whose output is a sum. Writing it once, outside a bracket, says that one number is negative and what that number's parts are.
Does expanded form tell me how many significant figures a number has?
No. This page lists the digits and their place values and stops there; the trailing zero of 0.004560 is simply not a term. Whether that zero was measured is a separate question, and it is the one the significant figures page answers — there it counts, and the number has four significant figures.

References

Related calculators

Place Value CalculatorThe place value of every digit in a number.Standard Form CalculatorConvert between standard form and ordinary numbers.Base CalculatorAdd, subtract, multiply and divide in any number base.Exponent CalculatorRaise a number to any power, including negative exponents.