Standard and Expanded Notation

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A mom recently wrote in to ask this question about standard and expanded notation.

“How do you know when you are writing in standard form, expanded form? For example, is the expanded for of 30,048

30000 + 40 + 8 ?

Or for 29,486, the expanded form = 20000 + 9000 + 400 + 80 + 6 ?”

Professor Homunculus replies:

Precisely! Oddly enough, though, there seems to be no “standard” for “expanded.” What I mean is, for 30,048 the expanded form could also be considered:

3(10,000) + 4(10) + 8(1)

as well as

3(10⁴) + 4(10¹) + 8(10⁰).

To make it even more confusing, some teachers might require you to include the empty columns. That would mean they might require you to write:

30000 + 0 + 0 + 40 + 8

or

3(10,000) + 0(1000) + 0(100) +4 (10) + 8(1)

or

3(10⁴) + 0(10³) + 0(10²) + 4(10¹) + 8(10⁰)

Even though it seems silly to be that “complete” (I can’t imagine having to include the empty columns in “the real world”) it isn’t bad to teach it like that in order for students to get the feeling for the whole concept. It helps solidify it in one’s mind. It’s just when kids get tested on it and writing out the whole thing is required, that it gets a little ridiculous.

They may be other ways to express “expanded notation.” Of course, it depends on what one mean’s be “expanded.”

One of the problems with the way schools teach is that they usually teach you one way and imply that is the “official” way. Most often, the teachers are not even aware that they could be confusing students by not at least making them aware that there are other ways to do “standard” things.

Schools do that with many things in math, and in other subjects as well. I think that is an educational crime, because it encourages students to stop thinking and accept the narrow information they are given as the “absolute truth,” which, of course, it never is.

One thing you may want to keep in mind, is that although school lessons on “standard and expanded notation” tend to be boring and leave you with the feeling that those things are not important, and just basically fodder for tests, that is not necessarily so. To find out more about expanded notation, and why you should understand it, check out:

http://mathmojo.com/interestinglessons/expanded_notation/expanded_notation.html

There is also something called scientific notation which is often confused with expanded notation. They are not the same. For a good explanation of scientific notation, refer to:

http://www.nyu.edu/pages/mathmol/textbook/scinot.html

“Mom” also asked:

“3205; is this number writen in standard form?”

Prof. Hunc sez:

Yes. You could also have written it with the comma, as in 3,205. (You don’t have to write the comma in the real world, but sometimes schools require it.)

Thanks for writing in. Reader input is what drives Math Mojo. Keep it coming!