Maths Test! 2(1+2)2/(1+2)

[Takhisis]

Isn't ≅ for congruence?

 

[Terxpahseyton]

Which means the properties are exactly the same, in a mathematical sense.

Doesn't = already express that?

So in your arithmetic modulo 4, couldn't one simply say 1 = 5 (under the conditions of the modulo).

 

[Earthling]

So, do you think it is 9 or 1? I think it is 1.

And you would be correct. Implicit multiplication takes precedence according to the standards established by journals, mathematics societies and taught in textbooks. Also, it's worth noting an expression written like that may not even be computable by some calculators/programs due to the lack of an operation symbol - sometimes it could yield an error and calculators can just be programmed differently.

 

[cardgame]

6/2(1+2)
6/2+4
3+4
7
It's not difficult

You pulled that four from thin air.

 

[gangleri2001]

Congruent

No, this is congruent: ≅

And this is is equal by: ≡ (it may be used as an alternate symbol of congruence according to the goddess wiki)

 

[ParadigmShifter]

Doesn't = already express that?

So in your arithmetic modulo 4, couldn't one simply say 1 = 5 (under the conditions of the modulo).

Nah, 1 is not 5 But they are congruent modulo 4.

If you have an equivalence relation (such as modulo operation, fractions), it partitions everything into disjoint subsets. The members of each disjoint subset are congruent to each other. They are not equal though.

 

[ParadigmShifter]

No, this is congruent: ≅

And this is is equal by: ≡ (it may be used as an alternate symbol of congruence according to the goddess wiki)

I thought the first symbol means "is isomorphic to". I am drunk tho;

EDIT: Mathworld agrees, with a note about the unfortunate clash with congruence
http://mathworld.wolfram.com/Isomorphism.html

 

[CKS]

The 3 line symbol is often used for "is defined as" rather than "happens to be equal to."

 

[Leoreth]

You pulled that four from thin air.

He didn't. He just started by 2(1+2) = 2+4. I like this approach.

It shows a truly skeptical and creative mind that's neither slowed down by the mathematicians' community's close-mindedness nor the Big Calculator lobby.

 

[Terxpahseyton]

If you have an equivalence relation (such as modulo operation, fractions), it partitions everything into disjoint subsets. The members of each disjoint subset are congruent to each other. They are not equal though.

I don't understand If they are congruent, they are interchangeable, right? If they are interchangeable, they are not also automatically equal?

I mean sure 1 = 5 is false, but that's why I added "under the conditions of the modulo". So in the context of this particular mathematical relation. Just like 1 does not equal 2 but under the conditions of the function x = 2y and 1 being an element of x and 2 of y, 1 = 2.

 

[ParadigmShifter]

I think it just helps avoid ambiguity.

Congruence, isomorphism, they are effectively the same objects, but we need to prevent confusion between them.

 

[Terxpahseyton]

So it is more about pointing people in the right direction than an actual mathematical distinction? Well - it just confused me!

 

[azzaman333]

Because it uses the divided sign, rather than /, there is less ambiguity where the 2 belongs. So, in this case, definitely 9.

Also, the thread title makes no sense.

 

[Leoreth]

So it is more about pointing people in the right direction than an actual mathematical distinction? Well - it just confused me!

I don't think so. It's to make clear that you're using two different equivalency relations. The equivalency relation on the set of natural numbers is "=". If you take a subset of the natural numbers modulo 4, you have to define an equivalency relation for it, too, and that's not the same as "=", so you need a different symbol. The definition could look like this:

x == y <=> x mod 4 = y mod 4

It would make no sense if you'd use "=" on the left side as well.

 

[Integral]

Doesn't = already express that?

Nope!

Consider two equations.

F(x) = 0. This says that F(x) can take on many values, but there are certain x's for which F(x) equals zero.

F(x) == 0. This says that for all x, F(x) evaluates to zero.

At least, that's how I often see it used.

 

[Terxpahseyton]

Ah thanks that I understand. So congruent removes any ambiguity a particular equivalence may possess when using equal.

 

[ParadigmShifter]

Well, the number zero and the zero function are completely different. I guess that's why we distinguish between scalars and vectors (since the zero function is the additive identity in the vector space of continuous functions).

 

[NickyJ]

He didn't. He just started by 2(1+2) = 2+4. I like this approach.

But that isn't the way it's done. 1+2=3, not 4.

 

[Takhisis]

2(1+2)=2×1 + 2×2.

Where did he say that 1+2=4?

 

[NickyJ]

2(1+2)=2×1 + 2×2.

Where did he say that 1+2=4?

6/2(1+2)
6/2+4
3+4
7
It's not difficult

That's where he said it.

Also, where do you get 2x1 + 2x2 from? When following the order of operations, you do the parenthesis first, which gives you (3). Then, you go from left to right, so you do 6/2, which equals 3. 3 times (3) equals 9.