Exploring the Complexity of 0: Real, Imaginary, or Both?

[Isaac0427]

I was just thinking about this question, and I see 3 possible answers:
1) 0 is a purely real complex number. This seems to be the most intuitive, however the one problem is that it shows up on the imaginary numberline.
2) 0 is not real nor imaginary. I understand this one, but I have found one problem with this: the absolute value of a complex number is modeled by the equation |x+yi|=√(x²+y²), where x and y must be real, and either can be equal to 0, and therefore 0 must be real. However, 0=0i, just as 0=-0, and 0 is thought of as neither positive or negative.
3) 0 is both real and imaginary. I'm leaning towards this one, because it appears on the real and imaginary numberlines (and other degrees of imaginary numberlines), and it can satisfy the absolute value equations as 0 can be thought of as real. I am not sure about this, however, which is why I ask.
I ruled out just imaginary because it just doesn't make sense at all, but if I'm wrong tell me.
Thanks!

 

[K41]

I'm guessing it must be a real number because it obeys all the rules (not sure about the Dedeking-complete rule) which define whether a number is real or not:
https://en.wikipedia.org/wiki/Real_number#Definition

It is also a complex number because the only definition of that is that a and b (from a + bi) must be real numbers and 0 is a real number as just discussed.

So it forms part of the real numbers and it forms part of the complex numbers.

Elementary algebra
The number 0 is the smallest non-negative integer. The natural number following 0 is 1 and no natural number precedes 0. The number 0 may or may not be considered a natural number, but it is a whole number and hence arational number and a real number (as well as an algebraic number and a complex number).

https://en.wikipedia.org/wiki/0_(number)#Mathematics

 

[Svein]

The real numbers are a subset of the complex numbers, therefore it does not matter what you call it.

 

[mgkii]

Hi

Here's a quick extract from Wiki - never guaranteed to be accurate, but generally a good start! Be careful when using the word "real" as you'll see from the first line, Real numbers include "types" of number such as natural and rational; and indeed the Complex plane includes the Real numbers!

Elementary algebra
The number 0 is the smallest non-negative integer. The natural number following 0 is 1 and no natural number precedes 0. The number 0 may or may not be considered a natural number, but it is a whole number and hence a rational number and a real number (as well as an algebraic number and a complex number).

The number 0 is neither positive nor negative and appears in the middle of a number line. It is neither a prime number nor a composite number. It cannot be prime because it has an infinite number of factors and cannot be composite because it cannot be expressed by multiplying prime numbers (0 must always be one of the factors).[39] Zero is, however,even.

Other branches of mathematics

• In set theory, 0 is the cardinality of the empty set: if one does not have any apples, then one has 0 apples. In fact, in certain axiomatic developments of mathematics from set theory, 0 is defined to be the empty set. When this is done, the empty set is the Von Neumann cardinal assignment for a set with no elements, which is the empty set. The cardinality function, applied to the empty set, returns the empty set as a value, thereby assigning it 0 elements.
• Also in set theory, 0 is the lowest ordinal number, corresponding to the empty set viewed as a well-ordered set.
• In propositional logic, 0 may be used to denote the truth value false.
• In abstract algebra, 0 is commonly used to denote a zero element, which is a neutral element for addition (if defined on the structure under consideration) and an absorbing element for multiplication (if defined).
• In lattice theory, 0 may denote the bottom element of a bounded lattice.
• In category theory, 0 is sometimes used to denote an initial object of a category.
• In recursion theory, 0 can be used to denote the Turing degree of the partial computable functions.

 

[PeroK]

I was just thinking about this question, and I see 3 possible answers:
1) 0 is a purely real complex number. This seems to be the most intuitive, however the one problem is that it shows up on the imaginary numberline.
2) 0 is not real nor imaginary. I understand this one, but I have found one problem with this: the absolute value of a complex number is modeled by the equation |x+yi|=√(x²+y²), where x and y must be real, and either can be equal to 0, and therefore 0 must be real. However, 0=0i, just as 0=-0, and 0 is thought of as neither positive or negative.
3) 0 is both real and imaginary. I'm leaning towards this one, because it appears on the real and imaginary numberlines (and other degrees of imaginary numberlines), and it can satisfy the absolute value equations as 0 can be thought of as real. I am not sure about this, however, which is why I ask.
I ruled out just imaginary because it just doesn't make sense at all, but if I'm wrong tell me.
Thanks!

This sort of question is answered by looking at the precise definitions. How would you define the real and imaginary subsets of the complex numbers? The most obvious way would be:

A real number is a complex number whose imaginary part is 0.
An imaginary number is a complex number whose real part is 0.

What does that make 0?

 

[Silicon Waffle]

...
3) 0 is both real and imaginary. I'm leaning towards this one, because it appears on the real and imaginary numberlines (and other degrees of imaginary numberlines), and it can satisfy the absolute value equations as 0 can be thought of as real. I am not sure about this, however, which is why I ask.
I ruled out just imaginary because it just doesn't make sense at all, but if I'm wrong tell me.
...

I am with you on this (3) that 0 can be considered both real and imaginary. And I think the complex plane supports your thought of this; zero is where both real and imaginary axes meet.

 

[Number Nine]

I was just thinking about this question, and I see 3 possible answers:
1) 0 is a purely real complex number. This seems to be the most intuitive, however the one problem is that it shows up on the imaginary numberline.

Actually, $0+0i$ shows up on the imaginary number line. This normally isn't such an important distinction, since the two objects are, for some purposes, "the same", but it pays to be pedantic here, since it seems to be causing confusion.

2) 0 is not real nor imaginary. I understand this one, but I have found one problem with this: the absolute value of a complex number is modeled by the equation |x+yi|=√(x²+y²), where x and y must be real, and either can be equal to 0, and therefore 0 must be real. However, 0=0i, just as 0=-0, and 0 is thought of as neither positive or negative.

Zero is real. It is an element of the real numbers. The claim that $0 = 0i$ is technically false, because one is real and one is complex. There is an isomorphism between the real numbers and a set of complex numbers which maps zero to $0+0i$, but they are not technically "the same".

3) 0 is both real and imaginary. I'm leaning towards this one, because it appears on the real and imaginary numberlines (and other degrees of imaginary numberlines), and it can satisfy the absolute value equations as 0 can be thought of as real. I am not sure about this, however, which is why I ask.
I ruled out just imaginary because it just doesn't make sense at all, but if I'm wrong tell me.
Thanks!

The answer, again, is that $0$ is real and $0+0i$ is complex.

 

[Isaac0427]

This sort of question is answered by looking at the precise definitions. How would you define the real and imaginary subsets of the complex numbers? The most obvious way would be:

A real number is a complex number whose imaginary part is 0.
An imaginary number is a complex number whose real part is 0.

What does that make 0?

I have thought about that too, but I have heard this definition that supports number 2:
A real number is a complex number whose imaginary part is 0 and whose real part is not zero.
An imaginary number is a complex number whose real part is 0 and whose imaginary part is not zero.

These two conflicting definitions (when it comes to zero) are one of the sources of my confusion, and then there is

I'm guessing it must be a real number because it obeys all the rules (not sure about the Dedeking-complete rule) which define whether a number is real or not:

 

[Samy_A]

I have thought about that too, but I have heard this definition that supports number 2:
A real number is a complex number whose imaginary part is 0 and whose real part is not zero.

Do you have any serious reference for this "definition"?

 

[Isaac0427]

Zero is real. It is an element of the real numbers. The claim that 0=0i0=0i0 = 0i is technically false, because one is real and one is complex.

I'm pretty sure that technically, both are complex and 0=0i=0+0i.

 

[Number Nine]

The difficulty here is that completely answering your question needs tools that you probably haven't studied yet.
People say "real numbers are complex numbers with zero imaginary part", but this technically isn't true -- it's just a casual shorthand way of saying "the real numbers are isomorphic to the subfield of complex numbers with zero imaginary part". What this means is that the complex numbers with zero imaginary part behave "just like" the real numbers, and so people often talk about them as if they're "the same". They are not technically the same, though, in the sense that, if you're working with the real numbers, there's not necessarily any notion of complex or imaginary at all; the complex numbers and the real numbers are entirely different sets.

So, to answer your question, zero is a real number, but the complex numbers contain an element $0+0i$ that behaves "just like" zero.

I'm pretty sure that technically, both are complex and 0=0i=0+0i.

As I explain above, this technically isn't true, they they're members of completely different sets.

 

[PeroK]

The difficulty here is that completely answering your question needs tools that you probably haven't studied yet.
People say "real numbers are complex numbers with zero imaginary part", but this technically isn't true -- it's just a casual shorthand way of saying "the real numbers are isomorphic to the subfield of complex numbers with zero imaginary part". What this means is that the complex numbers with zero imaginary part behave "just like" the real numbers, and so people often talk about them as if they're "the same". They are not technically the same, though, in the sense that, if you're working with the real numbers, there's not necessarily any notion of complex or imaginary at all; the complex numbers and the real numbers are entirely different sets.

So, to answer your question, zero is a real number, but the complex numbers contain an element $0+0i$ that behaves "just like" zero.
As I explain above, this technically isn't true, they they're members of completely different sets.

Once you've defined and proved your isomorphism, you can drop the formal distinction. It's not a casual shorthand.

 

[Number Nine]

Once you've defined and proved your isomorphism, you can drop the formal distinction. It's not a casual shorthand.

The distinction is useful here, where the OP is confused about the relationship between real and complex numbers. The p-adic numbers contain a subfield isomorphic to the rationals, but I would still correct a student if they came to class confused because they didn't know if zero was real or rational or complex or p-adic or neither or all at once because of something they read on wikipedia.

 

[PeroK]

The distinction is useful here, where the OP is confused about the relationship between real and complex numbers. The p-adic numbers contain a subfield isomorphic to the rationals, but I would still correct a student if they came to class confused because they didn't know if zero was real or rational or complex or p-adic or neither or all at once because of something they read on wikipedia.

Without wanting to get into an argument, you now have to explain isomorphisms to the OP, which isn't really necessary to get a feel for complex numbers.

 

[fresh_42]

Maybe it can be answered this way:
There is a neutral element $0_ℝ$ of real addition and a neutral element $0_ℂ$ of complex addition.
The moment you regard $ℝ ⊂ ℂ$ as a subfield or field extension you have to identify the two: $0_ℝ ≡ 0_ℂ$.

 

[Isaac0427]

Once you've defined and proved your isomorphism, you can drop the formal distinction. It's not a casual shorthand.

Ok, I'm confused. Is this statement true: 0=0i=0+0i?

 

[PeroK]

Ok, I'm confused. Is this statement true: 0=0i=0+0i?

Yes.

 

[mgkii]

@Isaac0427 No argument with @PeroK on this but be careful on the question you ask...

Does 0 = 0i ?
Yes - if you evaluate both sides, then you get the same answer; and you could substitute 0i for 0 in an equation and get the same answer; etc. It's s similar question to
Does 4 = (4 + 0i) ? Yes.

However...
Is 0 the same as 0i ? Well you're back to your original question! No... 0 is a whole number in the set of real numbers; 0i is a complex number.

 

[PeroK]

@Isaac0427 No argument with @PeroK on this but be careful on the question you ask...

Does 0 = 0i ?
Yes - if you evaluate both sides, then you get the same answer; and you could substitute 0i for 0 in an equation and get the same answer; etc. It's s similar question to
Does 4 = (4 + 0i) ? Yes.

However...
Is 0 the same as 0i ? Well you're back to your original question! No... 0 is a whole number in the set of real numbers; 0i is a complex number.

If you really believe that, then are not the whole numbers only isomorphic to a subset of the real numbers? How are you going to denote $0$ the integer, $0$ the rational number and $0$ the real number? It seems to me that you have three more isomorphisms there that you are forgetting about.

The point is: once you have defined and proved the properties of your isomorphism, you can omit it. You don't have to persevere with the isomorphic relationship. Otherwise, you'd never get clear of the theoretical clutter.

 

[fresh_42]

To distinguish between $ℝ⊂ℂ$ and an embedding $φ : ℝ → ℂ$ or $ℂ ≅ ℝ[x]/(x^2+1)$ is more than artificial when dealing with no other properties than pure numbers.

 

[micromass]

Actually, $0+0i$ shows up on the imaginary number line. This normally isn't such an important distinction, since the two objects are, for some purposes, "the same", but it pays to be pedantic here, since it seems to be causing confusion.
Zero is real. It is an element of the real numbers. The claim that $0 = 0i$ is technically false, because one is real and one is complex. There is an isomorphism between the real numbers and a set of complex numbers which maps zero to $0+0i$, but they are not technically "the same".
The answer, again, is that $0$ is real and $0+0i$ is complex.

What the hell are you talking about?? $0 = 0+0i = 0i$. This is completely true and not "technically false".

 

[mgkii]

I'm not totally sure I follow your logic there @PeroK. Surely the fact that you're able to identify an isomorphism means that you must be starting with different mathematical objects; zero in the real numbers and zero in the complex numbers? Completely agree that when you've proven the isomorphism you can move on. Maybe that's what I should have done a few posts ago :-)

 

[Isaac0427]

@Isaac0427 No argument with @PeroK on this but be careful on the question you ask...

Does 0 = 0i ?
Yes - if you evaluate both sides, then you get the same answer; and you could substitute 0i for 0 in an equation and get the same answer; etc. It's s similar question to
Does 4 = (4 + 0i) ? Yes.

However...
Is 0 the same as 0i ? Well you're back to your original question! No... 0 is a whole number in the set of real numbers; 0i is a complex number.

If 0*i=0, how could they not be the same thing? Aren't real numbers complex too? 0 would have to be complex right? Not just 0i.

 

[micromass]

Seriously people? Isomorphisms and embeddings? The answers here got to be the worst answers to such a question ever received on this forum. Never mind that the OP is still in high school. What are you people thinking?

For the formalists who wish it: I construct $\mathbb{C}$ as usual and (re)define $\mathbb{R}$ as a subset of $\mathbb{C}$. Boom. No problem with isomorphisms anymore.

Anyway, to actually answer the question of the OP. Yes, $0$ is a real number. Yes, $0$ is a complex number. Whether $0$ is an imaginary number is a bit difficult though since there are various definitions for imaginary numbers. One defines imaginary numbers as being of the form $bi$ with $b\neq 0$, others do allow $b=0$. So it all depends on the author.

 

[micromass]

If 0*i=0, how could they not be the same thing? Aren't real numbers complex too? 0 would have to be complex right? Not just 0i.

Yes, they are the same thing. Yes, real numbers are complex. Ignore the rest of these horrible replies.

 

[Isaac0427]

Anyway, to actually answer the question of the OP. Yes, 000 is a real number. Yes, 000 is a complex number. Whether 000 is an imaginary number is a bit difficult though since there are various definitions for imaginary numbers. One defines imaginary numbers as being of the form bibibi with b≠0b≠0b\neq 0, others do allow b=0b=0b=0. So it all depends on the author.

Ok, so we can rule out number 2 completely, correct? Are you saying that whether 1 or 3 is correct is not fact, and it depends on the person?

 

[micromass]

Ok, so we can rule out number 2 completely, correct? Are you saying that whether 1 or 3 is correct is not fact, and it depends on the person?

(2) is definitely false as $0$ is definitely a real number, right. Whether it is imaginary or not depends on the particular definition used. I think most math books would follow the definition that $0$ is an imaginary number. A minority would accept the definition that it's not.

 

[Isaac0427]

(2) is definitely false as $0$ is definitely a real number, right. Whether it is imaginary or not depends on the particular definition used. I think most math books would follow the definition that $0$ is an imaginary number. A minority would accept the definition that it's not.

Thank you very much.

 

[PeroK]

Ok, so we can rule out number 2 completely, correct? Are you saying that whether 1 or 3 is correct is not fact, and it depends on the person?

One thing to learn from this is that things "are" in mathematics what you define them to be. An imaginary number is what is defined to be an imaginary number. And authors of maths books have a certain flexibility in this. For most things, an author would have to be a real eccentric to define things differently from what is generally accepted. But, for more minor technical points that are of no real significance, often there is no universally agreed standard. This means that if you pick up a maths book, you have to be careful to check and understand the notation, conventions and precise definitions that the author is using.

Whether 0 is imaginary is, in fact, of no real significance. It just means, depending on your definition, you have to include or exclude 0 in certain circumstances.

In fact, if you look back at my post #5, you'll see I said "the most obvious way", not "the only way" to define an imaginary number.

 

[Mark44]

What the hell are you talking about?? $0 = 0+0i = 0i$. This is completely true and not "technically false".

Hallelujah!

 

[Ssnow]

sure it is real ;-)

 

[fresh_42]

sure it is real ;-)

Yep. The complete discussion is nonsense. It would have been much more interesting to discuss whether 0 can be considered natural, although this might be rather philosophic or at best historic.

 

[PeroK]

Yep. The complete discussion is nonsense. It would have been much more interesting to discuss whether 0 can be considered natural, although this might be rather philosophic or at best historic.

... says the self-appointed arbiter of what it is sensible to discuss, insulting all those who made the effort to post on this thread.

Physics Forums is about helping others, not condescending to pass judgement on them. There are plenty of places on line where you can do that.

 

[ehild]

@Isaac0427 said: ↑
Ok, I'm confused. Is this statement true: 0=0i=0+0i?

Yes.

A real number is a complex number whose imaginary part is 0.
An imaginary number is a complex number whose real part is 0.

What does that make 0?

The complex numbers are defined above the real numbers, as ordered pairs of real numbers, written in the form as z=x+yi, with addition and multiplication defined.
"x" is considered the real part of z and "y" is considered the imaginary part.
The addition is defined similarly as in case of two dimensional vectors, z₁+z₂ =(x₁+x₂)+i(y₁+y₂)
There is a neutral (null) element of addition, a complex number denoted by 0_c so as z+0_c=z.
As 0_c is element of the complex numbers, it has to be written in the form of a+ib. So z+0_c=z means that (x+iy)+(a+ib)=(x+a)+ i(y+b) = x+iy which means that 0_c=0+0i.
You can not define real numbers with complex numbers. But you can define an imaginary number as a complex number whose real part is 0. So you can consider 0i an imaginary number.

You can not state that the zero element of the real numbers is identical with the zero element of the complex numbers, as they belong to different sets, as @Number Nine and @fresh_42 and others pointed out. 0 ≠ 0+0i. The left 0 is real, the right side is a complex number.
There are a lot of other sets where you can define addition and zero element. Think about the null-vector or null matrix, or an empty set which is also denoted by ∅. You can not say that they are the same, and 0 is equivalent with the empty set or a null-vector.

Some people, including you, lost his nerves in this thread. You can show that somebody was wrong, even to say that what he wrote had no sense, but you must not hurt other members personally.

 

[jbriggs444]

The complex numbers are defined above the real numbers, as ordered pairs of real numbers, written in the form as z=x+yi, with addition and multiplication defined.

As has been pointed out, this is not correct. There are other foundational approaches that do things differently. The real number zero and the complex number zero may or may not end up being the same entitity, depending on the chosen approach.

The statement

0 ≠ 0+0i

is not justified.

One could argue that the above statement is actually incorrect, but that would open a different can of worms -- the resolution of overloaded notation in mathematical statements.