A concept regularly taught along with place value is “expanded form”, a way to write a number that displays each place separately. As we’ll see, there is considerable variation in terminology here, so parents may have to check what form a teacher wants, rather than look it up and expect a single answer!

## Whole numbers

Here is a question from 1998:

Expanded and Standard Forms

My teacher has asked me to write numbers instandard formandexpanded form. For example 4,017. What exactly does she want me to do?

Doctor Ameis replied:

Hello Mark, Expanded form is a way of writing numbersso that all that is hidden about them comes out into the open. The simplest way to write numbers in expanded form is to write them sort of in English. For 4,017, this becomes4 thousands and 0 hundreds and 1 ten and 7 ones. This can be made to look like math by changing the words to math symbols.The expanded form for 4,017, then, is: 4 x 1000 + 0 x 100 + 1 x 10 + 7 x 1 The expanded form shows what each digit is worth (for example, the 4 is worth 4 x 1000, which is the same as 4 thousands, which equals 4000).

I love that initial description. We want to “expand” a number, sort of unfurling it, to show what is hidden in the compact form we call positional notation, making explicit what is implied by the position of each digit.

He says “sort of in English” because normal English would be “four thousand seventeen”, which is not so fully unfurled! We have to make it clear that 4 thousand means 4 times one thousand, and break up the clump that is “seventeen” as one ten and seven ones. So we have $$4017 = 4\times 1000 + 0\times 100 + 1\times 10 + 7\times 1$$ We’ll discuss below whether the zero should be included.

Here is another example. 12,345 becomes 1 ten thousand and 2 thousands and 3 hundreds and 4 tens and 5 ones. Changing the words to math symbols, 12,345 in expanded form is: 1 x 10,000 + 2 x 1,000 + 3 x 100 + 4 x 10 + 5 x 1

What is standard form? It’s just the form we started with, like Mark’s 4017. What if you’re given the expanded form first?

Standard formis the reverse of expanded form. You begin with expanded form and change it tothe way we normally write numbers. For example, 3 x 1,000 + 5 x 100 + 7 x 10 + 4 x 1 can be changed to 3 thousands and 5 hundreds and 7 tens and 4 ones. This becomes: 3000 + 500 + 70 + 4 Do the addition and you get 3574 (the standard form).

Writing in standard form just means doing the arithmetic indicated by the expanded form. We’re putting the number back together again.

The intermediate form here, as we’ll see, is also sometimes called “expanded form”: $$3574 = 3000 + 500 + 70 + 4$$

But “standard form” doesn’t mean the same thing everywhere; in many countries that term refers to what we in America call “scientific notation”. As I have often commented, the least standardized word in all of math is the word “standard”.

## In between: Short word form

A closely related question is represented by this from 2002:

Short Word Form

How is 474,136 written inshort word formvs.standard form?

Doctor Sarah answered:

Hi Nicholas - thanks for writing to Dr. Math. The standard form of a number is written with commas separating every three digits. Your number 474,136 isalready in standard form. Inshort word form, the number is written using a combination of numerals and letters. Your number would be 474 thousand 136

This form focuses on the “periods” (groups of three digits, separated by commas) in a number, which represent powers of 1000 (thousand, million, billion, …).

Another form (which one teacher who asked about it called “mixed form”) looks like “3 hundreds, 4 tens, 5 ones”, which is essentially the “sort of English” form of Doctor Ameis. A student once asked about “word-and-number” form, which might be what his teacher calls either of these.

Inword form, the number is written in words: four hundred seventy-four thousand one hundred thirty-six

This could be called “long word form” or “standard word form”; it doesn’t directly relate to place value.

Inexpanded formorexpanded notation, your number is written as: 4 x 100,000 + 7 x 10,000 + 4 x 1,000 + 1 x 100 + 3 x 10 + 6 x 1

This is the same expanded form we saw before. But we’ve seen teachers use that term for a couple variations.

## Three kinds of “expanded”

A similar question (never archived) comes from 2009:

How do I write 680034987 inexpanded notation?

Doctor Ian answered, taking a slightly different approach to the form:

Hi Monique, Suppose I want to write 4098 in expanded form. I can write it as a column addition, 4000 90 + 8 ----- 4098 If I write it on a single line, 4000 + 90 + 8 then it's in 'expanded form'.

This is the simplest “expanded form”; there are a couple others:

Note that this isn't really a mathematical thing. It's a math-teacher thing. Sothe 'right' answer will depend on what it is your teacher wants to see. Some teachers might require it to be written like 4*1000 + 9*10 + 8*1 and some might require it to be written as 4*10^3 + 9*10^1 + 8*10^0 It's really a matter of taste. Any one of those would be correct, but any teacher might decide to mark only one of them as 'right'. Anyway, do you see the main idea? Can you apply it to your number? Try that, and let me know what you come up with.

We’ve seen the first of these; the second we haven’t yet. This can be called **exponential expanded form**, or expanded form with powers of ten.

Incidentally, we have had a number of teachers or students ask us about the difference between “expanded form” and “expanded notation”, and it seems that some authors do distinguish these terms (e.g. here), saying that \(400+90+8\) is “expanded form” while \(4\times 1000+9\times 10+8\times 1\) is “expanded notation”. But Common Core (here) calls the latter “expanded form”; and other sources (e.g. here) call the former “expanded notation”. I don’t think any of these naming conventions are standard. As Doctor Ian said, it’s just a matter of taste.

I’ll call the three forms “**short expanded form**“, $$498 = 4000+90+8$$

“**long expanded form**“, $$498 = 4\times 1000 + 9\times 10 + 8\times 1$$

and “**exponential expanded form**“, $$498 = 4\times 10^3 + 9\times 10^1 + 8\times 10^0$$

That’s just my choice of names, made up this minute!

## Should you include zeros?

Here is a question (from 2012) that should have been archived, because it is surely a common issue:

How would you write asingle digit numberin expanded form? For example the number 6. We know that 16 is 10+6, but we are confused about the right way to do a single digit number. My teacher asks me to do 5 different things with the date everyday. Tally marks, money, place value blocks, odd/even, and expanded form. We are all confused about the right way to write single digit numbers in expanded form.I say it's just the number itself.My friend says you would do 0+6 like you do in money or time for single digit numbers. We really want to know what the right way to do it is.

Emily is right; but her friend isn’t terribly wrong. (Note that they are using “short expanded form”.) Doctor Ian answered:

Hi Emily,A single digit number is *already* expanded.You don't have to do anything. The problem with writing 0+6 is that you could also write 0+0+6, or 0+0+0+6, and so on. There would be no end to it. Just out of curiosity, if asked towrite something like 1506 in expanded form, how would you do that?

If you require an added zero, how do you know when to stop? It seems best to stop before you start.

The new question is a good way to stimulate thought. We’ve already seen examples like this, probably chosen specifically to bring up this point.

Emily replied,

Thank you for helping us to understand math. Just to be sure I have this right, if the number is a single digit, I should justrewrite the number in the answer spot. Would it be better to leave it blank or to write,no expanded form?Emily

To answer your question 1506 in expanded form is 1000+500+6 right?I used to think I had to put a +0 in there where the tens go, but my teacher said that it wasn't needed.

Well, now that brings up another question.What if the number is 40?40 + 0?

Emily already has been told about zeros in the middle; but zeros on the left, and zeros on the right, seem different!

Doctor Ian responded first to the question about “no expanded form”:

Hi Emily, It would be better to just put the number there. For a single-digit number, the regular form and the expanded form are the same.

So there *is* an expanded form; it just isn’t different.

Next, about the zero in 1506 not being needed in expanded form:

I would agree.

The reason it’s needed in standard form is as a place-holder. In expanded form, places are explicit, so they don’t need to be “held”.

And then, on 40:

That *is* a good question. If we don't have to write the 0 for 1506, we probably wouldn't want to write it for 40, either. It might help tothink about what 'expanded notation' is good for. For example, it's useful if we want to do 'lattice multiplication'. That is, if we want to multiply 234 by 56, we can make up a table: 50 6 200 30 4 We can multiply the individual parts, 50 6 200 10000 1200 30 1500 180 4 200 24 and then add those up to get the product: 234 * 56 = 10000 + 1500 + 200 + 1200 + 180 + 24 = 13104 Don't worry if you haven't learned about this yet. You will. But let's think about doing something like multiplying 40 by 56. The table would look like 50 6 40 2000 240 0 0 0 That last row isn't making much of a contribution, is it? Sothere's no real reason to include it. That's also why we wouldn't bother to write 1506 as 1000+500+0+6. Let's look at a table for that: 1000 500 0 6 30 30000 15000 0 180 9 9000 4500 0 54 Again, that third row isn't making much of a contribution. Sothere's no real reason to include it. This is the kind of thing that leads us to the decision not to include entries for places where the digit is zero. It just takes up space, without really being useful. Does this make sense?

A lot of math comes down to a question of what is useful.

## Expanded form with decimals

A couple new issues arise when we include decimals, which are only mentioned in the archive in this question from 2001:

What is Expanded Notation?

We get more details in this unarchived question from 2010 explicitly about decimals:

My math assignment says I have to writeexpanded notation with numbers under 1, for example 0.05 and 0.7805

How do you do it? Especially the zero's. 0.00+ 0.05 0.000+ 0.7805

Thanks Katie

The leading zeros might make a student trip up; it appears that Katie is trying to include terms for them, which would be confusing. I answered:

Weren't you given any examples in class? Or is this an assignment to test what you already know, and you have never been taught this before? I'd like to have seen some attempt (or an example of how you would write expanded notation for a number greater than 1) in order to give me a better idea where to start.

Students do sometimes ask questions about topics they missed, or on tests not associated with what they have already learned; it’s helpful when we can get such background information. Also, as we’ve seen, an example would clarify what kind of expanded form they are (or will be) learning. Lacking that information, I offered several options:

There areslightly different waysto write "expanded notation", but one form is to show the value of each digit in the number, like this: 123 = 100 + 20 + 3 or 123 = 1x100 + 2x10 + 3x1 That is, the 1 means 1 hundred, so we write either 100 or 1x100. The 2 means 2 tens, so we write 20 or 2x10; and the 3 means 3 ones, so we write 3 or 3x1.

These are my “short” and “long” extended forms, so far applied only to whole numbers. Now we extend the idea, using fractions:

The digit after the decimal point is the number of tenths, and the one after that is hundredths. So we would write 1.23 as 1.23 = 1 + 2/10 + 3/100 or 1.23 = 1x1 + 2x1/10 + 3x1/100 Does that remind you of something you've been taught? Can you tell whether one of these forms, or something like them, is the way you have been taught to write expanded form?

The short extended form is $$1.23 = 1\times 1 + \frac{2}{10} + \frac{3}{100}$$ and the long extended form is $$1.23 = 1\times 1 + 2\times\frac{1}{10} + 3\times\frac{1}{100}.$$

It looks like you may have been trying to write something like my first form, but using decimals rather than fractions: 1.23 = 1 + .2 + .03. That's a possible form, but I don't think it's very useful, and so I doubt it is what you would be expected to do. But you'd have to show me an example of what your teacher means by expanded notation in order for me to be sure.

I’m not sure whether anyone teaches this form, $$1.23 = 1\times 1 + 2\times .1 + 3\times .01.$$

## How many ways to name a number?

Occasionally, we got questions like this (unarchived) from 2007:

What are the five ways to name a number? e.g. 127

I have

1. pictorial (drawing blocks)

2. words (one hundred twenty-seven)

3. expanded notation (100 + 2 10's + 7 1's) I suppose 127 is called standard notation. If so I have four of the five ways to name a number.

Doctor Rick answered:

Who says there are just five ways to name a number? That must be some educator's pronouncement, in which case you'll have to read the educator's book to get the full list. You can use scientific notation: 1.27*10^2. Or Roman numerals: CXXVII. Or hexadecimal: 7F. Or binary: 1111111. Or ancient Egyptian hieratic numerals. Or modern Arabic numerals. Or ancient Babylonian base-60 numerals. Or French words ... Honestly, I don't know what the answer is supposed to be.

Someone else asked about the *four* ways to name a number. I imagine each of these was really “the ___ ways we discussed in class”; such questions are particularly frustrating to parents when they don’t know what was taught, and assume it must be something universal.

*Why* so many ways to name a number?

Let’s look at one more unarchived question, this one from 2009:

Why are there so many forms or notations for numbers?

Like Standard notation = 782

But there's also

Expanded notation = 782 = (7*100)+ (8*10)+(2*1)

Scientific notation = 1,000,000 = 10^6 (ten to the sixth power) * 1 = 1,000,000

Word form = seven hundred and eighty two

Word and Number form = (1,000) 1 thousand

It's just my teacher has taught us all these and I just find that there are A LOT.

Like here's an example of a question: "Write 107 in word form"

Answer: one hundred seven

We see here (long) expanded form, (long) word form, the “mixed” form I’ve mentioned, and also scientific notation, which uses a single power of ten. Doctor Ian answered:

I'll bet you have more than one pair of shoes, right? How come? Because different kinds of shoes are good for different situations, right? The shoes you wear to church aren't so good for the beach, and shoes for going out in the snow aren't so good for playing soccer, and so on.Each form for writing a number has its own uses, things it's good for. And it's not so good for other things!

This is the standard way to think about any such question. How about some specifics?

For example, we useword formwhen writing checks. If I wrote you a check for, say, 15.00 dollars, it would be easy for you to put some numbers in front of that when you go to cash it. You might change it to 915.00, for example. But if I write out 'fifteen and no/100', it's a lot harder to alter that. It's also useful for catching errors, e.g., some people write their sevens the way other people write their ones. So if you just see '15', that might mean fifteen, or it might mean seventy-five, depending on who wrote it. But if the words are written out, it's hard to confuse the two, isn't it?

These two reasons are fraud prevention (which was also one reason Europe held on to Roman numerals for so long!) and error detection (as when people write something like “two (2) copies”).

(We also use word form just to read numbers aloud, of course.)

Scientific notationmakes really big numbers (like 602,000,000,000,000,000,000,000) or really small numbers (like 0.00000000056) easier to compare with other numbers, because we can just look at the exponents. It also makes them easier to work with, because we can manipulate the exponent and non-exponent parts independently. For example, to multiply 602,000,000,000,000,000,000,000 * 0.00000000056 --------------------------------- would be really hard to do, even with a calculator, without making a mistake of some kind. But if we write it as 6.02 * 10^23 * 5.6 * 10^-10 ----------------- we can just multiply 6.02 and 5.6, to get 33.7, and add the exponents, 6.02 * 10^23 * 5.6 * 10^-10 ----------------- 33.7 * 10^13 and adjust it back to scientific notation (i.e., so the first part is between 1 and 10), 6.02 * 10^23 * 5.6 * 10^-10 ----------------- 33.7 * 10^13 = 3.37 * 10^14 and it's pretty straightforward.

Each has its place.

Now, just as you don't want to wear your snow boots everywhere, using scientific notation is a bad idea for numbers that aren't very large, or very small! But for the right situation, it's a great tool to have. So basically, if mathematicians have invented a way of writing a number, it means that there is at least one situation where it solves a problem for them. Does this make sense?

Finally:

An exception is 'expanded notation'. Thisisn't used anywhere except in classrooms, as a way of teaching about place value. It's really not used outside of school.

And this is why the names and notations for expanded form(s) are not standardized: They are just teaching tools, not concepts mathematicians or scientists have to use (and therefore agree on).

Next time, we’ll look at some details of “word form”.

Pingback: Working Reform Wednesday—Math in the Real World: Understanding Expanded, Standard, and Word Forms – The Mixed Chocolate Experience

The 4Chan TellerYou can simultaneously be in expanded form WHILE being concise : so instead of

for [ 474,136 ], just write it as

This way all the digits remain grouped together, and it fully showcases exponents without actually having to use the power operator (or big chunks of 100,000,…….). So many textbooks keep saying this nested form is only used for polynomials, when in fact, it’s just as good, if not better, way to present numbers in any base as a composite of coefficients and powers of the base.

And all you need is 5 multiplies and 5 adds spanning a total of 11 numeric operands, instead of

which is 5 multiplies, 5 adds, 4 exponentiations, and 15 numeric operands.

To find out the largest exponent in the nested form, just count # of copies of “10 x” on the left ( and also for each digit ). A similar approach can also be used for so many other scenarios :

Time to seconds conversion becomes

17 : 43 : 29

60 * (60 * ( 17 ) + 43 ) + 29

————————————

6 3, 8 0 9

Dave PetersonSince the point of this post is largely to show the variety of forms that are called “expanded”, it’s entirely reasonable to add another! I suppose we can call it “nested expanded form”.

Of course, then we have to think about when it might be useful. The ones we discussed are primarily intended to help young students understand the meaning of place value and standard notation, and this probably is not very useful as a notation at that level (as it is less than obvious what it means, especially for students who have not yet learned about parentheses); but it is very useful in practice, both for evaluating polynomials efficiently, and for base conversion.

The latter is where I have indeed seen this form used heavily. Your conversion of time units is an example of this. As another example, we can convert 3D1C in hexadecimal to decimal by expanding it as \(10(10(10(3)+D)+1)+C\) in hexadecimal, or, in decimal, \(16(16(16(3)+13)+1)+12=15,644\).

This form is also equivalent to synthetic division, used to evaluate polynomials (synthetic substitution); and we can similarly use synthetic division to efficiently evaluate numbers in non-decimal bases, too.

Ultimately, I don’t tend to think of this as a

notation, but it is an excellent way toapplythe concepts.