"Real numbers" versus "Imaginary" numbers, what's the difference?

[NecroCommie]

My father (a higher university graduate in mathematics) brought up "imäginääriluvut" (imaginary numbers?) as an argument in favour of subjective knowledge. The concept is totally strange to me, and from the brief description it seemed like imaginary numbers would be substantives, as opposed to real numbers which sounded like adjectives. He claimed though that both imaginary number 1 and real number 1 are both basically the same concept of number 1, therefore "proving" that 1+1=2 and 1+1=\=2 are both true depending on the context.

I contested this by saying that imaginary number 1 and real number 1 are different things even if they share the same word and symbol. As I explained, real number one is and adjective (one what? 1 cat? 1 raindrop? 1x?) and imaginary number 1 is a substantive (an "abstract object").

But then again I barely finished the short maths in high school, so do enlighten me.

[Devrim]

An imaginary number is one that when squared gives a negative result.

1x1=1
-1x-1=1
but ixi=-1

Devrim

[NecroCommie]

.... ?

So would that fit as a proof of subjective knowledge? That is, if we say that 1+1=\=2 can this happen without altering the definition of number 1?

[NecroCommie]

I am more of a linguist. :D

[Meridian]

I have no idea what this topic does in philosophy, it seems like you are asking a math question yet trying to sprinkle in some grammar.

Real numbers, integers, imaginary numbers.. Aren't these basically different types of counting that mathematicians (and others) use for different purposes?

They are not adjectives or substantives (called nouns in English).

[NecroCommie]

I know they are not, I was merely trying to help you guys understand what I am after. This is in philosophy because math of this theoretical level is borderline philosophy. My father brought up, for example, the definitions of adding and how in different concepts 1+1=x can have varying results.

This is actually quite interesting, how many sciences are near philosophy when at high enough theoretical level. Examples: Quantum physics, the string theory and neurobiology of the consciousness.

[JazzRemington]

If the number and concept are the same thing, then how would 1+1=/=2 be true? Now, it is true that in binary 1+1=/=2 would be true because 1+1=10.

[NecroCommie]

But in binary, doesn't 1 just stand as a symbol for an electronic function instead of being an actual number? Or? That's what I've though... :confused:

[JazzRemington]

But in binary, doesn't 1 just stand as a symbol for an electronic function instead of being an actual number? Or? That's what I've though... :confused:

well, binary can be used for lots of things. Talking about electrical currents is one of them. They can be also the values of a yes or no question, or whether or not there is a problem or error with something. But in terms of using binary in mathematics, it's more abstract and doesn't really "stand" for anything. 1+1=11 just means that one plus one equals one-one (not "eleven").

[Quail]

Binary is just a different way of representing numbers, where each digit represents a power of 2 as opposed to a power of 10.

I'm not sure how real 1 and imaginary 1 fit into proving that 1+1=/=2 or 1+1=2, depending on the context, since 1 is the same number, whether you're looking at real numbers of imaginary numbers. The imaginary numbers are an extension to the real numbers, adding the square root of -1, so the real numbers are a subset of the imaginary numbers, and 1 is a member of both sets. However, I haven't really studied them much at university level (I'm just starting my third year) so as a maths graduate, your dad probably knows something I don't.

[NecroCommie]

Well, you must remember that authority is not an argument so I will not accept that alone. Secondly, your explanation makes more sense. Thridly, he graduated sometime in the 80's, so your knowledge is propably more up to date.

[Meridian]

Successions of numbers are dependant on the rules we have for generating them.

Whether 1 and 1 is 2, or 500 and 500 is 1000, can't tell you anything about "metaphysical truth". It may be true, but that just tells us something about the way we generate numbers.

For the same reason, it does not lead to any conflict about "truth" whether addition using some number system equals X and addition with another system of numbers also equals X. They may both be true.

[black magick hustla]

all math is a game and we made up the rules, including imaginary numbers

[NecroCommie]

So if i take a car, and then take another car I can just make up rules that say I have three cars?

[Meridian]

So if i take a car, and then take another car I can just make up rules that say I have three cars?
No one has said you can make up your own rules.

So, if you claim there are three cars you will be wrong.

[black magick hustla]

So if i take a car, and then take another car I can just make up rules that say I have three cars?

you can but nobody would understand what you are saying

http://en.wikipedia.org/wiki/Formalism_(mathematics)

[Rosa Lichtenstein]

So-called 'imaginary numbers' (which are now known as 'Complex Numbers') were introduced into mathematics when theorists encountered quadratic equations that had no solutions in the number system they already had -- what we might today call the 'Real Numbers'.

http://www.clarku.edu/~djoyce/complex/

This sort of problem has regularly confronted mathematicians. For example, in the middle ages, it became impossible to register debts that merchants and bankers accrued since there were no negative integers. It took a small conceptual innovation to get around this problem and a new system of negative integers was added to the natural numbers (including zero).

Many refused to accept these new numbers since they nowhere exist in the real world -- of course, those who argued this way forgot that the natural numbers nowhere occur in nature, either, although numerals plainly do. And they refused to accept they existed in 'God's mind (where numbers were conceived by many theorists to exist up until then), since they smacked of negativity, and thus of privation, and hence of the 'Devil'.

The so-called 'imaginary numbers' were thus invented to allow mathematics to progress, and those who claim that these numbers are somehow illegitimate since they nowhere exist in reality forget the above point that numbers in general do not exist in reality anyway.

Now, just as negative integers can be represented by extending the real number line leftwards, the Complex Numbers can be represented on an Argand Diagram, moving into two dimensions.

http://en.wikipedia.org/wiki/Complex_plane

Complex numbers are not just a theoretical plaything, and this is not just a game. Many advances in electrodynamics and quantum mechanics depended on them. So, they are just as practical as the negative integers, for example.

This meant that as these new number extensions were added to mathematics, our concept of number changed. So, in mathematics, we no longer mean by 'number' what, for example, the Ancient Greeks did (who, for instance, not only knew nothing of the negative integers, didn't have a zero, and refused to accept One as a number!)

So, mathematicians have to be careful these days to specify which set of numbers they mean, the Natural Numbers (N), the Integers (Z), the Rationals (Q), the Reals (R) or the Complex Numbers (C), among other sets of numbers.

http://en.wikipedia.org/wiki/Number

Problems only arise when mathematicians, scientists and philosophers try to find a correlate in 'reality' answering to these numbers, and out of that attempt arose much of Platonic (and later Hegelian) mysticism.

A much better way of understanding numbers and mathematics in general can be found in Wittgenstein's work, which represented a revolutionary break with tradition, in such a way that if he's right, we can throw away the last 2400 years theorising about the 'ontological status' of numbers and the nature of mathematics -- since he gave the whole of mathematics a social and historical interpretation (thus making it conducive with Marxism -- although, of course, he did not put things this way!).

http://plato.stanford.edu/entries/wittgenstein-mathematics/

http://en.wikipedia.org/wiki/Wittgenstein's_philosophy_of_mathematics