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greswd
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How to prove that 3 x 2 = 2 x 3?
This is not an instance of the distributive law but the commutative law for multiplication.greswd said:How to prove that 3 x 2 = 2 x 3?
Erland said:This is not an instance of the distributive law but the commutative law for multiplication.
The distributive law says that a x (b + c) = a x b + a x c.
The proofs of these laws will be different depending upon which number system we talk about: natural numbers, integers, rational numbers, real numbers, complex numbers, and even others can occur, and also upon which axioms or constructions we use for these systems.
micromass said:You will have to be content with a reference then. Get Rudin's "Principles of Mathematical Analysis". The commutative law for real numbers is proven in the appendix of chapter 1.
greswd said:alright, I'll check it out. Is it complicated?
42Physics said:See, in mathematics distributing different numbers/factors, EVENLY, doesn't change anything. Eg.
1(100+10)=110 is the same exact as 2(100+10)=220. We distributed evenly throughout the equation.
In my own theory, we can do this for all mathematical equations, even e=mc^2. If we added a square to "e" it would be e^2=mc^2^2 since you have to add the square to both sides to balance the equation. If you were to square root it, you would return to nice ol' e=mc^2!
Hope this helped.
Then it depends on exactly how you define "real numbers"- and there are several different ways to do that.greswd said:Let's say in the case of real numbers.
HallsofIvy said:Then it depends on exactly how you define "real numbers"- and there are several different ways to do that.
micromass said:Yes, it is quite complicated. There are actually various layers that you need to go through and Rudin only proves the final layer (which is the most complicated one). The first layer is the construction of the natural numbers and proving the distributive law for that. You can find this in many standard set theory books such as Hrbacek and Jech. The next layer is the construction of the integer and proving the distributive law there. Then you construct the rationals and prove the distributive law there. These two things are rather straightforward.
The most complicated layer is constructing the real numbers from the rational numbers and proving the distributive law for that. That is done in Rudin.
micromass said:This is however just one approach to the real numbers. Many people also prefer to accept the real numbers axiomatically. The axioms that govern the real numbers are called the field axioms. One of the field axioms is the distributive law. In that case, the distributive law becomes an axiom and doesn't need a proof.
The problem with an axiomatic approach is that there doesn't need to be anything that actually satisfies the axiom. To prove that the real numbers actually exist, you really do need to construct them. But many books don't really bother with that.
greswd said:Lol, no mention of commutativity?
I see. But real numbers only exist in our minds, don't they?
[IMG codes don't seem to be working]
greswd said:Lol, no mention of commutativity?
I see. But real numbers only exist in our minds, don't they?
micromass said:But that does not mean that they don't have a precise definition and construction!
HallsofIvy said:Yes, but you still have to prove that their properties are consistent.
I mentioned before that there are several different ways to define the real numbers.
One is "Dedekind cuts"- a real number is a set of rational numbers such that:
The set is not empty.
If x is in the set and y< x then x is also in the set.
There exist at least one rational number not in the set.
There is no "largest member" of the set.
Another way is as an equivalence relation on the set of all increasing sequences of rational numbers having an upper bound. We say that two such sequences [itex]\{a_n\}[/itex] and [itex]\{b_n\}[/itex] are "equivalent" if and ony if the sequence [itex]\{a_n- b_n\}[/itex] converges to 0. Then the "real numbers" are equivalence classes defined by that equivalence relation.
A third way is to define rational numbers as that same equivalence relation defined on the set of all Cauchy sequences of rational numbers.
We can then prove that xy= yx for rational numbers by using that property for rational numbers.
Of course, all of those depend upon the rational numbers. We can define rational numbers as equivalence classes using an equivalence relation on the set of pairs (x, y) where x is an integer and y is a positive integer, defining (x, y) to be equivalent to (x', y') if and only if xy'= x'y.
We can prove that xy= yx for rational numbers by using the same property for integers.
And we define integers similarly: integers are equivalence classes using an equivalence relation on the natural numbers (positive integers) where (x, y) is equivalent to (x', y') if and only if x+y'= x'+y.
We can prove that xy= yx for integers by using the same property for natural numbers.
Finally, we can define the natural numbers using the "Peano axioms":
The natural numbers consist of a set of objects, N, called "numbers", and a function, s(x), called the "succesor" function such that:
There exist a unique number, "1", such that s(x) maps N one to one and onto N-{1}.
If a set X, of numbers, contains 1 and, whenever it contains x it also contains s(x), then X= N.
What would be an "actual proof"?greswd said:After reading through, it seems like there's no actual proof of commutativity.
I don't know. Maybe something went wrong when the forum changed to a new style.greswd said:By the way, why are IMG codes disabled?
Fredrik said:What would be an "actual proof"?
greswd said:How to prove that 3 x 2 = 2 x 3?
lavinia said:a 2 by 3 rectangle of six squares groups the squares in two ways - 3 groups of 2 and 2 groups of three.
The distributive law can be one with cubes.
greswd said:yeah, the intuitive geometric proof.
but it could be that we describe geometry in commutative terms, and not the other way round.
HallsofIvy said:Well, that's the problem, isn't it? You asked a question about the fundamentals of the number system without knowing much about the fundamentals.
I think post #17 does that. To fully understand this stuff, you need to study the basics of set theory and the basics of abstract algebra. Then you can use a book on set theory (e.g. Goldrei or Hrbacek and Jech) to find out how to use set theory to define natural numbers, integers, rational numbers and finally real numbers (a Dedekind-complete ordered field).greswd said:I was hoping someone could simplify it. Or at least you could set me on the right track.
Not sure you will need anything to prep for that, but you may find some of the stuff recommended in this thread useful, in particular the book linked to in post #2 and the 10-page pdf linked to in #5.greswd said:What should I read as prep before Goldrei?
Fredrik said:Not sure you will need anything to prep for that, but you may find some of the stuff recommended in this thread useful, in particular the book linked to in post #2 and the 10-page pdf linked to in #5.
I've been discussing similar things with a guy in this thread, and he seems to find both of those useful.
Chapter 5? Problem 5? You may need to be more specific.greswd said:I'm struggling with No.5 in Book of Proof.
The distributive property of multiplication states that when multiplying a number by a sum, you can first multiply each addend by the number and then add the products together to get the same result as multiplying the number by the sum.
To use the distributive property to solve equations, we can break down the equation into smaller parts and apply the property to simplify it. This is particularly useful when dealing with equations involving variables.
Sure, for example, if we have the equation 2(x + 3), we can use the distributive property to rewrite it as 2x + 6. This is because we first multiply 2 by x and then 2 by 3, and then add the two products together.
The distributive property can be proven using algebraic manipulation, where we show that the left side of the equation is equal to the right side. We can also use visual representations, such as area models or arrays, to demonstrate how the property works.
The distributive property is important in mathematics because it allows us to simplify complex equations and expressions, making them easier to solve. It is also a fundamental concept in algebra and is used in many mathematical operations, such as factoring and expanding expressions.