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My brain is standing completely still on this. I'm sure I've known this in the past. The difference between the number:

123

and

123.25

is obviously that the first number number is a "full" number. There is no fraction of a number that requires that the number be written with a period symbol and more numbers describing the fraction.

What is a number called when it can be written without a period, and what is it called otherwise? Also, what are the numbers behind the period symbol called? (So really this is a request for three words, not one. I hope that flies.)

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You also need a fourth term to describe 123. -- a number which has a decimal part but has been rounded to the nearest integer value. You might also want to distinguish 123.000. –  Fortiter Jan 18 '13 at 4:15
    
"Whole number": A number without fractions; an integer; and "Fraction": A numerical quantity that is not a whole number. Don't split hairs. HTH –  Kris Jan 18 '13 at 6:17
    
I think it depends on the context a bit too, in computer science 123. is usually a very different number from 123 –  wim Jan 18 '13 at 6:34
    
@wim: The main reason for me asking this question was actually a comment I was writing in a Python source file, where I was trying to explain that if the input float variable was "round", it would be converted to an integer. –  Codemonkey Jan 18 '13 at 8:38
    
@codemonkey Just with the new context, you could just be explicit in this case with something like if the input float variable is a whole number (i.e. there's nothing after the decimal point) it'll be converted to an int to avoid any confusion? And mild curiosity, can I get a pointer at the source? Am just wondering how it's implemented :) –  tanantish Jan 18 '13 at 8:51
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4 Answers

up vote 7 down vote accepted
  1. Counting numbers = {1, 2, 3, ...}
  2. Whole numbers = {0, 1, 2, 3, ...}
  3. Natural numbers is often synonymous with counting numbers. But the term is sometimes used to mean whole numbers.
  4. Integers = {..., -3, -2, -1, 0, 1, 2, 3, ...}
  5. ½ is an example of a fraction. is an example of a mixed number and specifically a mixed fraction.
  6. 123.25 is also an example of a mixed number, specifically a mixed decimal, and usually just called a decimal number or simply a decimal.
  7. 123.25 is also an example of a rational number. It can be written as a ratio (hence, rational) of two integers. Both ½ and (which has a repeating decimal) are rational for the same reason.
  8. π (pi, 3.1415...) (which has an infinite, non-repeating decimal part) is an example of an irrational number. It cannot be written as a ratio of two integers. While the fraction 22/7 is often used to do so, it is only an approximate representation.
  9. Real numbers comprise both rational and irrational numbers.
  10. Imaginary numbers are those numbers which are not real. Complex numbers are those that have both real and imaginary components. Consequently, all numbers, real or imaginary, can be represented as complex numbers.

The part before the decimal point is a numbers integral (or integer) part and after it is its decimal part. In terms of mathematical terminology, the before and after parts are sometimes referred to as the characteristic and mantissa respectively. As Wiktionary states, these two terms are often required when working with logarithms.

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What about a mixed number with an infinite repeating decimal part? Would that still be a rational number? –  Codemonkey Jan 18 '13 at 5:36
    
@Codemonkey I've updated my answer to address this. But, in short, yes. –  coleopterist Jan 18 '13 at 5:49
3  
π is irrational, but more commonly is called transcendental because it is not an algebraic number. Square roots of integers are algebraic, and if not integral are irrational. Also, item 10 is incorrect. The real numbers are a subset of the complex numbers; every real is a complex number with its imaginary part being zero. –  jwpat7 Jan 18 '13 at 5:50
    
@jwpat7 Re: complex numbers, good point. I've edited the answer to address this. Please check. –  coleopterist Jan 18 '13 at 6:06
    
“Imaginary numbers are those numbers which are not real” is incorrect. Imaginary numbers are complex numbers whose real part is zero. Complex numbers with nonzero imaginary and nonzero real parts are neither purely imaginary nor real. In the usual Cartesian-coordinates visualization of complex numbers, reals are those complex numbers that lie on the x-axis, and imaginaries are those complex numbers that lie on the y-axis. BTW, if complex numbers are relevant here, surely you should go ahead and explain how they fit into quaternions. –  jwpat7 Jan 18 '13 at 6:20
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The first one is called a natural number (if it's positive) or an integer (if it's either positive or negative), the other one is often called a rational number (if the decimal string terminates or repeats in a pattern) or a real number (if there is no restriction on how long or regular the decimal string is). To the left of the decimal is the integer part and to the right of the decimal is the fractional part.

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You could also call the first one a whole number, generally a synonym for integer. –  Mark Beadles Jan 18 '13 at 2:12
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@Codemonkey round number would not be wrong and will be understood. it would depend on the audience whether it's more appropriate than integer. –  jlovegren Jan 18 '13 at 2:35
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@Codemonkey If you're not talking mathematical definitions then I'd say avoid 'round number' as it's a bit of a fluid term more closely related to the degree of precision. For example if something is expected to be in the range 0-99 with a fractional part, it sounds natural. If you move into another context such as forex trading (there's whole strategies on round number forex trading), it'd mean roughly numbers zeroing the last two sig figs, not all fractional components. –  tanantish Jan 18 '13 at 3:12
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@tanantish: I'm looking for words that the layman will understand. I'm comfortable with using "round number" for an integer, but what would be a common, not necessarily technically correct, word for a number that is not an integer? A word you would use in a casual conversation? –  Codemonkey Jan 18 '13 at 3:38
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@Codemonkey the first comment suggests whole number, which is the word they teach you in grade school, so that will be best for the layman. I hadn't even realized there was a technical sense of round number until @ tanantish pointed it out. –  jlovegren Jan 18 '13 at 4:37
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A number with no decimal values is called an integer.

If there are a limited number of decimal digits or it contains an infinite set of repeating digits, it is a rational number. If there are an infinite amount of decimal digits, it is an irrational number. The set of rational and irrational numbers are real numbers.

If there is a decimal point, the numbers to the left are the integer part and the numbers to the right are called the fractional part.

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Do you claim that 0.112211122211112222..., with its infinite sets of repeating digits, is a rational number? –  jwpat7 Jan 18 '13 at 5:41
    
en.wikipedia.org/wiki/… Repeating decimals can be converted to fractions as shown above. –  Josh Jan 21 '13 at 2:50
    
Merely referring to "repeating decimals" is too simple-minded. For example, numbers with repeating patterns like 0.01011011101111011111... are irrational. You need to include the word periodic or cyclic in the definition. See wikipedia article. –  jwpat7 Jan 21 '13 at 5:19
    
Exactly, see wikipedia article. They are called repeating decimals, not periodic decimals or cyclic decimals. Your examples do not repeat. –  Josh Jan 24 '13 at 1:35
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mixed decimal - a combination of a whole number and decimal, such as 59.8, 810.85

For some, that's as opposed to a decimal fraction, defined there by thefreedictionary as a decimal having no digits to the left of the decimal point except zero, such as 0.2 or 0.00354

Others, such as Merriam-Webster (and me!) define a decimal fraction as

a fraction (as .25 = 25⁄100 or .025 = 25⁄1000) or mixed number (as 3.025 = 325⁄1000) in which the denominator is a power of 10 usually expressed by use of the decimal point. [italics mine]

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