Is zero a natural number?

Some argue that 0 is a natural number; others say it isn’t. Who is right?

[packedbits]

Fun question. While sub-domains can always have their own conventions. But for general world-wide communication ISO standards engineering standards usually prevail. So 0 is a very natural number :) Thus that is what I teach my students. Though I let them in on the history

[agnijobanerjee]

I agree, and that's why I would recommend using the terms "positive integer" or "non-negative integer" when you need to be precise about whether 0 counts. My supervisor, Prof. Imre Leader, said that if you are doing number theory you should exclude 0 from natural numbers (for instance, it makes the Fundamental Theorem of Arithmetic fail) but if you are doing set theory then 0 should be included (as the natural numbers can be naturally defined as the finite ordinals or, as you did here, their corresponding cardinals).

[Synthetica]

The intersection of inductive sets definition works just as well when you start at zero (or two or any other real number for that matter), while the first definition does not (you could specify "non-empty", but that's a more complicated definition than the original definition while replacing the one with zero does not add Complication)

[humblehotpockets]

But the real question is, will pearson count it wrong if I put 0 as a natural number in the answer? Because I'm not about to get another question wrong on my test because pearson considers 0 as the opposite of what I consider it.

[toryglen]

I agree it is a "place holder" (initially)

I would say it is a natural number when measuring, as in addition or deduction:
-5 -4 -3 -2 -1 0 1 2 3 4 5
If you deduct 1 from 5 or -5 on "repeat", the equation soon reaches 0.
you cannot jump from -1 to 1 or vice versa, as that would be a deduction (or addition) of 2.

When it is used as a fulcrum, as in a see-saw and using multiplication (moments),
Where one side is balanced with the opposing side:
5 metres x 2 newtons is equal to 2 metres x 5 newtons
then zero is a place holder.

Even at the fulcrum point a force of say 10 newtons would equal 10 newtons x 0, it would have no effect.
so therefore, I would have to correct myself and say that 0 is a natural number.


What do I know?
I am just rambling.

[ewthmatth]

2:07 "a different definition of natural numbers starting from real numbers..."

What was the connection to real numbers? I don't think you demonstrated one in this part of the video.

[dennisthegreek]

Natural numbers are these used to describe or explain natural objects. To count things/items/animals we meet in nature. For example, "the cat gave birth to 4 kittens", or "9 apples fell off the apple tree". That's why only positive integers are natural (not the negative integers, nor the decimals etc).
Zero is never used to count natural items, ie you can't naturally use zero to count (?!?) and say that "there are 0 oranges on the table". And why say 0 oranges and not 0 apples?
Since you never use 0 to count natural objects, it can't be considered a natural number.

[WhCrsOrochi]

It is easy. Absence of something is a natural condition. e.g. “There is no apple left” is a natural condition. With this logic we must say zero is a natural number.

[cainthebraindrain]

Can someone link something about the church being against the number zero, that sounds like interesting stuffk but i couldn't find anything.

[paulromsky]

This is how I teach it in my STEM class:

Zero is usually considered to be a Natural number except in certain types of math.
To allow for both, Zero has its own root Set.

The Zero set O (the letter O for origin) is just zero: O = {0}

The Natural number set N is all the positive ordinal (counting numbers): N = {1, 2, 3, ...}

The Whole number set W is the union of sets O and N: W = {0, 1, 2, 3, ...}

Note: In most math, W is known as Natural Set N (the N set including the zero). This ok as long as it is used consistently and the math does not "break down".

The Negative number set L (Less than zero) is all of the ordinals with negative values: L = {..., -3, -2, -1}

The Integer number set Z (because it includes Zero) is the union of the L and W sets: Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

The Pure Fractional number set F is all numbers that can only be represented accurately as a fraction (ratio and not an Integer) of two non-zero integers:  F = {..., -2/3, ... 1/3, ... 1/2, ... 2.25, ...} 

Note: Ordinal numbers like 1, 2, 3, ... can be represented as fractions like 2/1 but because they can also be represented as their ordinal number, they are are not Pure Fractional numbers, hence this set includes numbers that can "ONLY be represented accurately as a fraction". Decimal numbers like 2.25 resolves to all zeros past the fractional part and is the same as 2 1/4 or 9/4 (Pure Fractional). However, 1/3 = 0.3(viculim or bar over the 3) or .3... repeating forever is still Pure Fractional because it can only be written accurately as a fraction of two non-zero numbers: 1 and 3.

The Rational number set Q (quotient) is the union of sets Z and F: Q = {..., -3, ... -2, ... -2/3, ... -1, ... 0, ... 1/3, ... 1, ... 1/3, ... 1/2, ... 2, ... 2.25, ... 3, ...}

The Irrational number set S (specular) are numbers that cannot be represented accurately as any fraction of two non-zero integers: S = {..., -pi, ... e, ... sqrt(2), ... pi/2, ... pi, ...}

Note: Irrational numbers do not resolve to all zeros somewhere past the decimal point, and in fact have a numeric pattern that never repeats.

The Algebraic Real number set A (or R sub A) is a subset of Irrational numbers that are a root of a polynomial, and are often used in Algebra and related fields like Trigonometry, Physics, and more: A = {..., -sqrt(2), ... sqrt(3), ... (1+sqrt(5))/2, ...}

The Transcendental number set T is a subset of Irrational numbers that are not a root of polynomial, they often represent common aspects in the natural universe: T = {... -pi, ... -e, ... e, ... pi, ...}

The Real number set R is the union of the Q and S sets: R = {all of the pior numbers mentioned}

The Imaginary number set I is any number that has a non-zero factor of the sqrt(-1) or i (the Imaginary number): I = {..., -sqrt(-1), ... 2i, ... (pi)i}

The Complex number set C is the union of the R and I sets: C = {all numbers in the Complex Plane in the form: x + yi, that is: Real Part + Imaginary Part}

Note: All numbers are actually Complex numbers but we don't normally need to write the Imaginary part when it's Imaginary factor is zero. Example: 2 = 2+0i so we normally just write "2".

There can actually be higher dimensions of number sets that extend to the Complex Space (x + yi + zj), and more.

[anthonym]

Just my random thoughts ... I would argue that "zero" is an initial condition, rather than a count value. If I look in a basket and see 5 apples; then 5 is the initial condition: 0 == 5.

I can throw in 2 apples: 0+2 == 5+2. Or, I can throw in no apples: 0+0 == 5+0

Zero is the initial condition, it is what results when no change occurs.

What about multiplication? Spose I have 7 snails, and I modify my collection by buying 3 new snails every friday.

7 + 3(0); the initial condition before adding any group of 3 snails.
7 + 3(1); the condition after I start adding snails to the collection.
7 + 3(2); the condition after I added a second group of snails.
etc

But then what is division by zero? Division is "partitioning" of a unit metric.

12/0 = ; the initial condition of 12 units; no partitioning has occured.
12/1 = the partitioning of 12 into single units.
12/2 = {11} {11} {11} {11} {11} {11}; the partitioning of 12 into double units.
12/3 = {111} {111} {111} {111}; the partitioning of 12 into triple units.
....
12/5 = {{11}111}; you need 3 partitions of 5 units each to get to 12; 2 whole partitions, and a partial [what is a fraction?]
...
12/12 = the partitioning of 12 into twelvple units? lol.

So yeah, what is a fraction; it is a partitioning of a unit metric. 0/7 to 7/7

0 out of 7: { }; initial condition, no partitioning has occured.
1 out of 7: { }; the partitioning has begun
2 out of 7: { }
3 out of 7: { {111}1111 }
....
7 out of 7: { }; they chose all of it :/

[paulromsky]

Zero is a "strange" number, and as the infinitesimal approaches zero the infinite approaches infinity. I normally consider 0 to be in the Natural set as MOST math will define something using the set N that includes Zero. But in the abstract form of math, zero is in the O (Origin set on its own) and thus in the Whole number set W not the Natural number set - in that case: The Whole set W is the union of the O and N set, which makes it clear if zero is included or not, W it is and N it is not - but alas, that is not the general consensus. So, at: Universe, Virgo Supercluster, Milkyway Galaxy, Solar Sytem, Earth, 2022 Jun 29 16 hours 8 minutes 0 seconds UTC, the Natural number set includes 0 unless otherwise specified. Yes, which root set in where zero falls has flipped around over the centuries.

[ojasgoel]

I my opinion zero is not a natural number

[iyatomas]

This video helps me a lot to understand 0 in my number theory subject this term. Thank you for discussing this subject.

[robbiedenham]

Well if you apply it to a set of 1 + 0 = 10.
Or if you can apply a 0 after the 1 = 10 It was a whole number

[justusobi]

I think it's safe to say that natural numbers are a subset of whole numbers which include 0. so therefore 0 won't be regarded a natural number.

[LrCloud]

Is it right to say that 0! = 1 using the reductive proof of patterns 5!, 4!, 3!, 2!, thus claiming that 0! must equal 1 solely based on the extrapolation of what is observed for the other actual numbers..?
I argue that it is a false logic to use integers 5, 4, 3, 2, 1 and then throw in 0 to reach for a proof that 0! = 1 when it can be argued that it is really a concept rather than an actual number.

[math_travel]

it depends on the definition you adopt! short but correct answer! thanks~~

[ex]

This video has been dubbed in Cantonese. Can we get the English version please.

[captainmisan]

Then where does zero belong to? R+ or R- ?