Is 0 an integer?
Yes.
Integers are the set of whole numbers, their opposites, and zero.
Yes, it’s generally considered an integer which is neither positive nor negative.
I’d say yes, if for no other reason that C++ considered it one.
Yes, but depending on text books you go by, it may or may not be a natural number.
True but, most commonly, it is considered a natural number but not a counting number.
Ridiculous degrees of definition, mathematicians go into sometimes.
This was discussed rather extensively in a CoSR thread a while back.
Quick version:
In the standard foundations of mathematics, we assume that the empty set exists, and assume that we can construct other sets from it. We can construct two families of sets, one that satisfies the properties of the naturals with {1, 2, 3, …}, and one that satisfies the properties of {0, 1, 2, 3, …}. Both are perfectly good candidates for the natural numbers as long as we’re clear what we mean by that term.
From either of those, we can construct a family of sets that satisfies the properties of the integers, and one of those is 0. I suppose we could construct the integers without 0, but in that case, you’d have “pintegers” where the sum of two “pintegers” is not necessarily a “pinteger”. That’s not interesting.
The number systems we teach in school:
Natural numbers - the counting numbers (1,2,3,…)
Whole Numbers - the counting numbers and zero (0,1,2,3,…)
Integers - whole numbers and the negative numbers (-2,-1,0,1,2,3,…)
Rational Numbers - Integers and terminating decimals and fractions
Irrational Numbers - Non-terminating decimals and special numbers such as *e *and pi
Imaginary Numbers - i = sqrt[-1]
Thus, 0 is an integer.
Unless you went to my school(s), where it’s the other way around.
But that in no way affects the definition of “integer”.
This thread just reminds all over again of my CS prof days. I’d get junior and senior majors in Computer Science who would ask me if “0 is an even number?” Egad. CS majors! Ever heard of binary numbers??? I once asked one of my kids that when he was 4. He knew it was even. Why? “Because 1 is odd.” It ain’t that hard a concept…
As you’ve phrased it here, the rationals are a subset of the irrationals. The decimal representation of 1 is 1.00000… It doesn’t terminate, but by convention, we drop all the 0s at the end. You may want to check your definitions.
you are correct in saying it does not terminate as a decimal, but it can be represented as a fraction, which keeps my definition.
Whoops :smack: I hate it when that happens.
There are fractions which are non-terminating decimals (say 1/3, for a more standard example). According to your definitions, since 1/3 does not terminate, it is irrational. You meant to say “non-repeating” decimals.
ok - I’m talking in circles and I stand corrected
This is the way it is in all the Basic Algebra textbooks I’ve seen, but as you can see in the thread I linked to earlier, this is hardly universal.
As ultrafilter has already pointed out, non-terminating repeating decimals are rational, not irrational. (For example, 1/3 = 0.33333… By the way, repreating decimals have been discussed to death in at least one other thread, but among mathematicians, it is non-controversial and universally accepted to say that the repeating decimal 0.3333… does equal 1/3.)
Nonterminating decimals which do not just repeat the same digit or group of digits over and over are irrational. Pi and e are indeed famous irrational numbers, but not every irrational number is “special.” In fact, there are more irrational numbers than rational numbers (in the sense that the set of irrationals has a higher cardinality that the set of rationals. There’s no way to match them up one-to-one without having some irrationals left over).
The rationals and irrationals together make up the set of Real Numbers.
The set of complex numbers, of which all the aforementioned sets are subsets, is the set of all numbers of the form a + bi, where a and b are real numbers and i is the square root of -1. When someone mentions the Imaginary Numbers, they could mean all non-real complex numbers (i.e. a + bi where b is not zero), or they could mean the Pure Imaginary numbers of the form bi. But if you take all numbers of the form bi (the Imaginary Axis), you include zero, which raises the question:
Is zero an imaginary number?
“God gave us the natual numbers. All else is the work of man.”
Howzabout a perspective shift?
[ul]
[li]Natural numbers: As per the Peano axioms, this is the set you can create from 0 and the successor function S(). S(0) == 1, by our notation, but that’s not completely relevant. Here’s what’s relevant: Zero is a natural number. If a is a natural number, S(a) is a natural number. Zero is not the successor of any natural number. Any two natural numbers with the same successor are equal. If a set A contains zero and the successor of every natural number already in A, A contains all natural numbers.[/li][li]Integers: Natural numbers with a negative part. Any natural number can be made into an integer by giving it a negative part of zero.[/li][li]Rationals: The ratio of two integers. Any integer can be made into a rational number by giving it a denominator of one. (That is, 5 -> 5/1.)[/li][li]Irrationals: An integer with a fractional part. Fractional parts are infinite (or, at least, unbounded), though nobody writes everything out. Any integer can be made into a rational by giving it a fractional part of .0000… .[/li][li]Pure imaginary: An irrational multiplied by sqrt(-1).[/li][li]Complex: A tuple of irrationals, one of which is a pure imaginary number. Any irrational can be made complex by giving it a pure imaginary part of 0.000… .[/li][/ul]
That’s how I view them, anyway.
Then there are the quaternions, which finishes out the real division algebras. After that come the Cayley numbers (or “octonions”)…
Don’t even get me started on the Adèles.
Oh come now, don’t be a tease.
I need to put in my obligiatory complaint on the name ‘imaginary’ numbers.
Stupid, stupid, stupid name.
It brings to mind that these numbers don’t really exist and that they have no use.