The rule that a digit's value comes from where it is written rather than what it looks like. The digit 5 in 5, in 25 and in 205.04 is worth 5, 50 and 0.05 respectively, because it sits in a different place each time. Each place is worth ten times the one to its right, which is what makes ten digits enough to write a number of any size.

Why are the zeros kept in the list here?

Because a place value belongs to a position, and a position with a zero in it still holds that place open — the zero in 205 is what makes it 205 rather than 25. This page lists digits and their places, so it keeps them. The expanded form page turns the same number into a sum and leaves them out, since a term worth zero adds nothing; the two answers differ and both are right.

Why does 4.5e4 not list any place below the thousands?

Because e notation carries the size of the number in its exponent rather than in written zeros, and this page reads the digits it was given. 4.5e4 names two digits, so the list has two terms, 4 × 10000 and 5 × 1000, and the smallest place is 1000. Type 45000 instead and the same number lists every place down to the ones. It is the same quantity either way; only the number of digits written down has changed.

What are the tenths, hundredths and thousandths places?

The three positions immediately to the right of the decimal point, worth 0.1, 0.01 and 0.001. Each one is a tenth of the place to its left, and their names say the same thing their values do: the tenths place is where a digit is worth a tenth, the hundredths place where it is worth a hundredth. So 0.04 is four hundredths, not four tenths, and the 4 sits two places right of the ones.

Why does the answer print 2 × 100 instead of 2 hundreds?

Because a list of place names has to stop somewhere. Beyond the hundred-thousands and the thousandths the chart would need names most readers have never used, and a chart that runs out of words halfway is worse than one that values every position as a power of ten and never runs out. The value is the same statement either way: 2 × 100 and two hundreds are one place, described two ways.

Does the place value page tell me the sum of the digits?

No. It lists each digit with the value of its place and stops; the terms add up to the input, but the page never performs that addition. If what you want is the number written as the sum of its place values, that is the expanded form calculator, which uses the same places as terms and leaves out the ones worth zero.

Place Value Calculator

Result

2 × 100, 0 × 10, 5 × 1, 0 × 0.1, 4 × 0.01

Digit values

Largest place value: 100
Smallest place value: 0.01

Place value is the rule that a digit's meaning comes from its position rather than from its shape. The same 5 means five, fifty or five hundred depending on where it is written, and in 205.04 each of the five digits has its own value: the 2 is worth 200, the 0 in the tens place is worth nothing, the 5 is worth 5, the 0 in the tenths place is worth nothing, and the 4 is worth 0.04. This calculator lists all five of them, in order from left to right, as digit times place value: 2 × 100, 0 × 10, 5 × 1, 0 × 0.1, 4 × 0.01. The places on either side of the decimal point are mirror images of each other. To the left the values go 1, 10, 100, 1000, each one ten times the last; to the right they go 0.1, 0.01, 0.001, each one a tenth of the last. The ones place sits between them and is the pivot: its value is 1, and the exponent form of every place value is a power of ten whose exponent counts the distance from that pivot, positive to the left and negative to the right. That is the whole of the chart below, and the chart is fixed — it is the ruler the digits are read against, not a description of any one input. Zeros are kept. A place value is a property of a position, and a position with a zero in it still has a value, which is the only reason 205 and 25 are different numbers: the zero is holding the tens place open. Dropping it from the list, as the expanded form page does when it turns the same number into a sum, would hide the digit that does the holding. This is also why the two pages print different lines for the same input and neither is wrong — one is a list of digits with their values, the other is a sum of the terms that count. Two details about how the input is read are worth stating, because both are easy to mistake for bugs. The digits listed are the digits as written, so 1200 keeps its two trailing zeros and 0.004560 keeps its leading ones; the calculator does not decide that some of them are not worth mentioning. And in e notation only the coefficient is read: 4.5e4 is a way of writing 45000 that names two digits, so this page lists 4 × 10000 and 5 × 1000, giving the largest place as 10000 rather than 100000. The value of the number is the same either way; what differs is how many digits were written down, and this page can only report the ones it was given. A negative sign is not part of any digit's place value, so it does not appear in the list. The last two lines of the result answer a question the list itself makes awkward to answer by eye when the number is long: the largest and smallest place the number uses, printed as values. And what this page does not do is tell you what a digit is called. It will say the 2 in 205.04 is worth 200 and not that it is in the hundreds place, because the names run out — past the hundred-thousands the chart would need ten-thousandths, hundred-thousandths and further, and a list that stops is worse than one that never starts. If you want the terms added up rather than listed, that is the expanded form page; if you want the number collapsed into a coefficient and a power of ten, that is the standard form page.

The place value of each position, from the millions down to the thousandths

place value	number
1 × 10⁶	1000000
1 × 10⁵	100000
1 × 10⁴	10000
1 × 10³	1000
1 × 10²	100
1 × 10¹	10
1 × 10⁰	1
1 × 10⁻¹	0.1
1 × 10⁻²	0.01
1 × 10⁻³	0.001

Ten positions, written the way each one is named and the way each one is valued: the millions, hundred-thousands and thousands on the left, the ones place in the middle, and the tenths, hundredths and thousandths on the right. Read it as the ruler the answer above is measured with — the 2 in 205.04 sits against the 100 row, and the 4 against the 0.01 row. The two rows either side of the middle are the ones that catch people out: 1 × 10⁰ is 1, the ones place, so the exponents do not start at 1; and 1 × 10⁻¹ is 0.1, the tenths place, where the minus sign is doing the work of moving the digit to the other side of the decimal point. The chart stops at the thousandths on the right and the millions on the left, which is not a limit on the calculator — a longer number simply gets a longer list, with the exponents continuing in both directions. Every value here is computed by the same kernel that produces the answer, so a row can never disagree with it.

Formula

205.04 = 2 × 100 + 0 × 10 + 5 × 1 + 0 × 0.1 + 4 × 0.01; place values run 10ⁿ with n = 2, 1, 0, −1, −2; 4.5e4 lists 4 × 10000 and 5 × 1000, largest place 10000, smallest place 1000

205.04: The number being read, given as an ordinary decimal or in e notation. Each of its digits is listed in the order it is written, so the position of every term in the answer matches the position of the digit in the input
0 × 10: A zero digit's term, kept rather than dropped. The tens place in 205.04 has a zero in it and that zero is what makes the number 205 rather than 25, so its place value is listed along with the rest, even though the digit contributes nothing to the total
10ⁿ: The place value of any position, where the exponent counts steps from the ones place: positive for the tens, hundreds and thousands, zero at the ones place itself, and negative for the tenths, hundredths and thousandths. The exponent is the place's address
× 10: The multiplication sign in each term. The line reads digit times place value all the way across, which is what makes it clear that the digit 5 in 205.04 is worth five and not five hundred, even though the two are written next to each other in the answer
10000 and 1000: The largest and smallest place the number uses, printed as their values. In 4.5e4 they are 10000 and 1000 rather than 10000 and 1, because e notation names only the digits of the coefficient and the trailing zeros of the written-out number are not among them
0.1 and 0.01: Two place values from the right of the decimal point, printed as decimals rather than in exponent form because that is how they are read aloud. 0.1 is the tenths place and 0.01 the hundredths place, and each is a tenth of the place to its left

Place value is the first idea in arithmetic and the one that everything after it depends on. It is what a child learns with counters on a chart, and it is what is being appealed to years later in every line of column addition, subtraction, long multiplication and long division: those methods work only because digits in the same column are worth the same amount, and they fail for anyone who does not believe it. The same idea is the whole of the decimal system's compactness — ten digits are enough to write any number of any size, because position carries the rest of the information — and it is the first thing that has to be relearned, rather than merely extended, when a different base arrives: a binary number works exactly the same way with place values of 1, 2, 4, 8 instead of 1, 10, 100, 1000, which is why the binary conversion pages and this one describe the same mechanism. In measurement and in money the chart is the daily tool. The tenths and hundredths places are what a reading to two decimal places means, and knowing that the digits after the decimal point are worth progressively less is the difference between reading 0.04 as four hundredths and as four. The two lines below the list, the largest and smallest place, are the quick answer to how big and how precise a number is: a measurement recorded to the thousandths place claims a different precision from one recorded to the ones place, and that claim is made by where its last digit sits. When the digits need to be added up rather than listed, the expanded form page does that. When they need to be summarised as a coefficient and an exponent, it is the standard form page.

Worked examples

A number with a digit on each side of the point: 205.04
1. Read the digits from left to right: 2, 0, 5, 0, 4
2. The 2 is three places left of the ones place: 2 × 100
3. The 0 in the tens place keeps its term: 0 × 10
4. The 5 is in the ones place, whose value is 1, so its term is 5 × 1
5. The 0 in the tenths place keeps its term: 0 × 0.1
6. The 4 is two places right of the ones place, in the hundredths: 4 × 0.01
7. The largest place used is 100 and the smallest is 0.01, so the number runs from the hundreds to the hundredths
The default, and the case with a zero on each side of the decimal point, so both directions of the mirror are visible at once. Compare it with the same input on the expanded form page, which answers 200 + 5 + 0.04: three terms against five. The two terms that differ are exactly the two zeros, and the reason is that one page lists positions while the other adds up amounts.
A whole number with trailing zeros: 1200
1. The digits are 1, 2, 0, 0 and the places run thousands, hundreds, tens, ones
2. The 1 is worth 1000 and the 2 is worth 200
3. Both zeros are kept as terms, because the positions they occupy still exist: 0 × 10 and 0 × 1
4. The largest place used is the thousands and the smallest is the ones place, whose value is 1
The zeros are the point here. Dropping them would give 1 × 1000 and 2 × 100, which is what the expanded form page does and is a perfectly good answer there, since a sum does not need terms worth zero. Here they are kept because the tens and ones places are real positions in the number 1200 and a reader checking a column subtraction needs to know they are occupied. Note also that the smallest place is 1 and not 0.01: the number has no decimal point, so its last digit sits in the ones place and that is as fine as it gets.
A value written in e notation: 4.5e4
1. 4.5e4 is a written form of 45000, but it names only the digits 4 and 5
2. The 4 is the first digit of the coefficient, so it lands in the ten-thousands place: 4 × 10000
3. The 5 is the second digit, so it lands in the thousands place: 5 × 1000
4. The three zeros that make up the rest of 45000 are not written in the input, so they are not listed
5. The largest place is 10000 and the smallest is 1000 — not 1
The case that looks like a bug and is not. Type 45000 and the list runs down to the ones place; type 4.5e4 and it stops at the thousands, because e notation carries its size in the exponent rather than in written zeros, and this page reads the digits it was given. Both inputs describe the same number, and the second one simply names fewer of its digits. The exponent inside the notation is not expanded into place terms either, since a term such as 0 × 100 would be this page inventing digits for an input that never wrote them.

Limitations

The list is as long as the number of digits written, not as long as the number's full width. 4.5e4 produces two terms rather than five, and its smallest place is 1000 rather than 1, because the input names two digits. Type the number out as 45000 to see every place down to the ones. Place names are not printed. The answer says the leading digit of 205.04 is worth 200 and never that it is in the hundreds place, because a list of place names would have to stop: it would need ten-thousandths and hundred-thousandths past a certain point, and a chart that runs out of words partway is worse than one that uses exponents and never does. Written forms are refused, so 2 × 10² is not an input; use 200 or 2e2. Thousands separators are refused as well, so 1,200 is an error rather than 1200. Inputs are limited to fifteen significant digits and to exponents within ±99, and a leading zero in a written integer is not preserved — 007 is read as 7, whose only place is the ones. The negative sign does not appear in the list, because it is a property of the number rather than of a digit: −205.04 produces the same five terms as 205.04. Nothing is summed and nothing is rounded, so the answer never tells you what the digits add up to; that is the expanded form page. And the chart below is a fixed set of powers of ten, not a description of the input above it.

Frequently asked questions

What is place value?: The rule that a digit's value comes from where it is written rather than what it looks like. The digit 5 in 5, in 25 and in 205.04 is worth 5, 50 and 0.05 respectively, because it sits in a different place each time. Each place is worth ten times the one to its right, which is what makes ten digits enough to write a number of any size.
Why are the zeros kept in the list here?: Because a place value belongs to a position, and a position with a zero in it still holds that place open — the zero in 205 is what makes it 205 rather than 25. This page lists digits and their places, so it keeps them. The expanded form page turns the same number into a sum and leaves them out, since a term worth zero adds nothing; the two answers differ and both are right.
Why does 4.5e4 not list any place below the thousands?: Because e notation carries the size of the number in its exponent rather than in written zeros, and this page reads the digits it was given. 4.5e4 names two digits, so the list has two terms, 4 × 10000 and 5 × 1000, and the smallest place is 1000. Type 45000 instead and the same number lists every place down to the ones. It is the same quantity either way; only the number of digits written down has changed.
What are the tenths, hundredths and thousandths places?: The three positions immediately to the right of the decimal point, worth 0.1, 0.01 and 0.001. Each one is a tenth of the place to its left, and their names say the same thing their values do: the tenths place is where a digit is worth a tenth, the hundredths place where it is worth a hundredth. So 0.04 is four hundredths, not four tenths, and the 4 sits two places right of the ones.
Why does the answer print 2 × 100 instead of 2 hundreds?: Because a list of place names has to stop somewhere. Beyond the hundred-thousands and the thousandths the chart would need names most readers have never used, and a chart that runs out of words halfway is worse than one that values every position as a power of ten and never runs out. The value is the same statement either way: 2 × 100 and two hundreds are one place, described two ways.
Does the place value page tell me the sum of the digits?: No. It lists each digit with the value of its place and stops; the terms add up to the input, but the page never performs that addition. If what you want is the number written as the sum of its place values, that is the expanded form calculator, which uses the same places as terms and leaves out the ones worth zero.

References

Related calculators

Expanded Form CalculatorWrite a number in expanded form, digit by digit.Sig Fig CalculatorCount significant figures and round to a set number.Standard Form CalculatorConvert between standard form and ordinary numbers.Binary to Decimal ConverterConvert binary to decimal, with the place values.