# What is a decimal expansion?

I'm an adult trying to get back to school after a few years; I'm trying to prepare for college. In reading I have come across the terms 'decimal expansion' and 'decimal representation', which by the way, are synonymous as far as I can see. I have tried to understand it on my own and I feel that I have become even more confused. From what I could understand, decimal expansion is the expression of a number in the decimal numbering system as 1, 15, 359, 18.7, and 3.14159. I also read, that when we write for example the number 3264, we are saying that that number is the result of adding 3 x 10*3, 2 x 10^2, 6 x 10^1, and 4, or 4 x 10^0. So we say that 3264 is the base 10 expansion of n, or the decimal expansion of n. From what I understand, then the expansion of 3264 is 3264, the number itself. So why is it that when we talk about decimal expansion (decimal representation), we generally refer to numbers that explicitly have a decimal point? That is, an integer and a fractional number. Because as I understand it, a number like 42 would itself be a decimal expansion, but every number is said to have an infinite decimal expansion, in this case 42.00000? Another related example would be the number 1, which has 2 infinite decimal expansions: 1.00000..., and 0.99999.... But if we can also express a number in decimal notation as for example: 4567 = 4x10^3 + 5x10^2 + 6x10 + 7 = 4567(10). So are both decimal expansions (decimal representations) or just one of them? 4567 is the short form and 4x10^3 + 5x10^2 + 6x10 + 7 = 4567(10) is the expansion? Now, in a dictionary I found this definition:

decimal representation: Any real number a between 0 and 1 has a decimal representation, written .d1 d2 d3 …, where each di is one of the digits 0, 1, 2, …, 9; this means that a = d1 × 10^–1 + d2 × 10^–2 + d3 × 10^–3 + …. This notation can be extended to enable any positive real number to be written as cncn–1 … c1c0. d1d2d3 … using, for the integer part, the normal representation cncn–1 … c1c0 to base 10 (see BASE). If, from some stage on, the representation consists of the repetition of a string of one or more digits, it is called a recurring or repeating decimal. For example, the recurring decimal .12748748748 … can be written .12748 , where the dots above indicate the beginning and end of the repeating string. The repeating string may consist of just one digit, and then, for example, .16666 … is written .16 . If the repeating string consists of a single zero, this is generally omitted and the representation may be called a terminating decimal. The decimal representation of any real number is unique except that, if a number can be expressed as a terminating decimal, it can also be expressed as a decimal with a recurring 9. Thus .25 and .249 are representations of the same number. The numbers that can be expressed as recurring (including terminating) decimals are precisely the rational numbers.

And now I wonder, what is any real number between 0 and 1? And back to the beginning, what is a decimal expansion? Does expansion mean 'stretching' a number? a number? But that doesn't make much sense with its synonym; decimal representation. So what is a decimal expansion (decimal representation)?

I hope I have explained myself well enough, and not to be hopeless and very silly for you; I am struggling to understand and although I have little time (due to my job and other occupations) I want to learn.

I wish someone could be so but so kind to help me, please. I am a bit desperate. Thank you very much. :)

PS: Sorry for my English, it is not my native language.

• A lot of questions here. Unfortunately, broad questions like these aren't really well received here, but I'll try to address the major points. Commented Mar 4, 2022 at 3:06
• This post is barely readable: a total lack of quote formatting and large chunks of text just thrown together without a clear connection. What is readable contains multiple questions that should be split out separately.
– Nij
Commented Mar 4, 2022 at 3:18
• @RushabhMehta That is not what comments are for. If you need to do a twitterstorm of comments, then they should be posted as an answer. Especially if you are addressing multiple points at length. Commented Mar 4, 2022 at 4:00
• It looks like you're confused over the conceptual difference between the actual value of a number, and how we write (represent) that value in various ways. Commented Mar 4, 2022 at 4:43
• The decimal expansion of a real number is just the usual base $10$ expression for that number. See this for instance.
– lulu
Commented Mar 4, 2022 at 4:45

## 2 Answers

You will most probably see "decimal representation" in the context of integers, when the book is talking about different number bases.

That is, decimal representation will be discussed so as to compare it with, for example, binary representation.

So

• "Let us express in binary the number whose decimal representation is $$42$$: $$42 = 2^5 + 2^3 + 2^1$$ and so $$42$$ is $$101010$$ in binary."

You will most probably see "decimal expansion" in the context of fractions, when the book is talking about writing a fraction using a decimal point and so on:

• "Let us express $$\dfrac 3 4$$ using its decimal expansion: $$\dfrac 3 4 = \dfrac {3 \times 25} {4 \times 25} = \dfrac {75} {100} = 0.75$$."

But in fact "decimal representation" and "decimal expansion" both mean the same thing. It's just that "expansion" is more appropriate to fractions after the decimal point (especially when they are recurring or otherwise non-terminating) because the list of digits "expands" after the decimal point.

I hope this helps. The distinction is merely one of context.

• Thank you very much, you have helped me a lot. I wonder very much why in many basic books that deal with topics like this, they don't explain those kind of subtleties; this kind of confusion can happen. Commented Mar 15, 2022 at 23:52

Trying to summarize what is essential to prepare for college:

• good to read (thanks to lulu): is this

• it is pretty obvious and correctly observed by you that an integer, say, $$4567$$ has a trivial infinitude of decimal expansions (representations): $$4567.000...$$

• same holds for rational numbers whose exact expansion is finite, for example $$1/2=0.50....$$, so that again zeroes can be padded at the end at will

• as you correctly observed: $$0.999999...$$ represents the same number as $$1.0000...$$.

• in a comment the remark was made that there is a conceptual difference between the value of a number and how we write it in various ways.

• some further comments gave valuable hints for you to dive deeper but I don't think that's required for college - except perhaps binary representations since they will become very important should you get interested in computer science.

• "same holds for rational numbers since their exact expansion is finite" -- you may wish to expand on this, as most rational numbers have a recurring part -- and so not all rational numbers can be padded out with zeroes. Commented Mar 4, 2022 at 14:49
• Thanks for pointing out that crap. Commented Mar 4, 2022 at 14:52
• @PrimeMover I read something about it in one of the many books I looked in: Commented Mar 16, 2022 at 0:02
• @PrimeMover It reads as follows: "The rational numbers are precisely the real numbers with decimal expansions that are either terminating (ending in an infinite string of zeros), for example, 3/4=0.75000... = 0.75 or eventually repeating" Commented Mar 16, 2022 at 0:03
• @PrimeMover It is from "University Calculus, Global Edition, 3rd Ed, by Hass, Weir, Thomas, Jr. from Pearson Publishing. In the appendix of Real Numbers, p. 928. Commented Mar 16, 2022 at 0:09