1x1=2

[Chriliman]

You are still wrong. Might be because your math skills are a little dull... but it is quite easy to sharpen them.

Multiplication is bascially a repeated addition. 3x3 does not mean you "multiply it 3 times"... that would be "3x3x3" (27).
As you correctly identified, it is "3+3+3"... you add it three times.

This is also why it is possible (oh wonder!) to multiply different numbers. Say 3x4... you take the "3", add another "3", and another "3" and a fourth "3"... and add all these.

To go one step further, when you do this, you can also take "4", and add it just three times. It will get the same result.
In math-speak, it is said that multiplication is commutative. Addition, too, is commutative... you can do it in any order you want.

And so we can see why 1x1 isn't 2. You take 1. Just once. You do not add anything else. Just this single one.

Right, that's what I said

 

[Freodin]

Right, that's what I said

Not quite what you said, but close enough.

 

[Sultan Of Swing]

Or you could actually think about it yourself in an objective way and see what you discover.

Let's say you have a basket with one ball in it.

If you have two baskets with balls in them, you have 2x1, 2 lots of 1 ball, which equals 2.

If you have one basket, this is 1x1, one lot of 1, which is 1.

 

[Chriliman]

Let's say you have a basket with one ball in it.

If you have two baskets with balls in them, you have 2x1, 2 lots of 1 ball, which equals 2.

If you have one basket, this is 1x1, one lot of 1, which is 1.

Got it! Now explain that to Howard

 

[Sultan Of Swing]

Now 1x1 may equal 1.

But did ya know 1+1 also equals 1?

Let's say

a = 1
b = 1

a = b
a^2 = b^2
a^2 - b^2 = 0
(a-b)(a+b) = 0
(a-b)(a+b)/(a-b) = 0/(a-b)
1(a+b) = 0
(a+b) = 0
1 + 1 = 0
2 = 0
1 = 0
1 + 1 = 1

Goodbye Maths. It's been nice knowing you.

 

[Chriliman]

Now 1x1 may equal 1.

But did ya know 1+1 also equals 1?

Let's say

a = 1
b = 1

Contradiction! Therefore the rest is illogical.

a = b
a^2 = b^2
a^2 - b^2 = 0
(a-b)(a+b) = 0
(a-b)(a+b)/(a-b) = 0/(a-b)
1(a+b) = 0
(a+b) = 0
1 + 1 = 0
2 = 0
1 = 0
1 + 1 = 1

Goodbye Maths. It's been nice knowing you.

 

[Sultan Of Swing]

Contradiction! Therefore the rest is illogical.

Where's the contradiction? There's nothing wrong with defining both a and b as 1.

 

[Chriliman]

Where's the contradiction?

If we're saying 'a' is different than 'b' then a and b cannot both equal 1.

If we're saying 'a' is the same as 'b' then we should just use one or the other, not both.

So one logical equation is a=1

Another logical equation is b=1

But it's illogical to say a=1 and b=1 in the same equation because it's assumed that a and b are different variables.

 

[Freodin]

Now 1x1 may equal 1.

But did ya know 1+1 also equals 1?

Let's say

a = 1
b = 1

a = b
a^2 = b^2
a^2 - b^2 = 0
(a-b)(a+b) = 0
(a-b)(a+b)/(a-b) = 0/(a-b)
1(a+b) = 0
(a+b) = 0
1 + 1 = 0
2 = 0
1 = 0
1 + 1 = 1

Goodbye Maths. It's been nice knowing you.

I'd say you know where this fails, don't you?

(a-b)(a+b)/(a-b) = 0/(a-b)
1(a+b) = 0

Aye, there's the rub that makes mathematical paradoxa of so long life.

You took (a-b)/(a-b) to be 1. This usually works perfectly, for every possible number.... except for 0. Any number divided by 0 is undefined in common arithmetics, even if it is 0 divided by 0.

And as a=b, (a-b) is 0.

Thus the rest of the expression is also undefined.


Nice try though. Hurray Maths!

 

[Freodin]

If we're saying 'a' is different than 'b' then a and b cannot both equal 1.

If we're saying 'a' is the same as 'b' then we should just use one or the other, not both.

So one logical equation is a=1

Another logical equation is b=1

But it's illogical to say a=1 and b=1 in the same equation because it's assumed that a and b are different variables.

Ouch, Chriliman, your mathskills really ARE a little dulled.

a=b is a completely and perfectly valid expression in mathematics. There is nothing illogical in it. Two variables can indeed hold the same value.

Yes, it means we can interchange them. In fact, this is the trick upon which this (faulty) expression rests. It makes it appear as if (a-b) is just an ordinary mathematical expression. But it isn't. If we were to exchange a for b, we would see that the term always comes down to (a-a)... which is of course 0.

 

[Chriliman]

Ouch, Chriliman, your mathskills really ARE a little dulled.

a=b is a completely and perfectly valid expression in mathematics. There is nothing illogical in it. Two variables can indeed hold the same value.

If two variables hold the same value, then they should be considered the same variable, otherwise their value is meaningless.

 

[Freodin]

If two variables hold the same value, then they should be considered the same variable, otherwise their value is meaningless.

Err, no, not quite. A value is not in any way influenced by how many variables hold them.

If two variables hold the same value, they are considered equal. That is the whole point.

Usually, this is the endpoint of a calculation, not something to start with... but it is completely valid. Nothing illogical, nothing contradicting here.

 

[Loudmouth]

Anyone care to explain why Howard is wrong?

1 group of 1 is . . . 1.

5 groups of 2 is 10. 2 groups of 5 are 10. 1 group of 5 is 5. 1 group of 1 is 1, not 2.

If you want to change this to binary, you might get some interesting answers.

 

[Chriliman]

Err, no, not quite. A value is not in any way influenced by how many variables hold them.

If two variables hold the same value, they are considered equal. That is the whole point.

Usually, this is the endpoint of a calculation, not something to start with... but it is completely valid. Nothing illogical, nothing contradicting here.

In this case the value was influenced by the variable because instead of 1+1=2 we get 1+1=1. Or a+b=1.

Whereas logically it should read a+a=2 or b+b=2 or 1+1=2 so the variable does not influence value.

 

[keith99]

Are you pulling our chains?

You have a Prince Albert?

 

[Freodin]

In this case the value was influenced by the variable because instead of 1+1=2 we get 1+1=1. Or a+b=1.

Whereas logically it should read a+a=2 or b+b=2 or 1+1=2 so the variable does not influence value.

Huh?

How do you ever get to the 1+1=1 part?

You are correct a+a=2. b+b also equals 2. (Assuming a and b are 1). But so would a+b equal 2.

 

[Sultan Of Swing]

In this case the value was influenced by the variable because instead of 1+1=2 we get 1+1=1. Or a+b=1.

Whereas logically it should read a+a=2 or b+b=2 or 1+1=2 so the variable does not influence value.

@Freodin correctly said where the fault was, it's not to do with a=b which is perfectly valid.

If a=1 and b=1, a+b=2. This doesn't change anything.

 

[Warden_of_the_Storm]

This last page just made my head hurt...

 

[Freodin]

This last page just made my head hurt...

Mine too. But for the reason that I try to figure out how to display real mathematical equations on this forum.

Does anyone here know how MathJax works?

 

[Warden_of_the_Storm]

Mine too. But for the reason that I try to figure out how to display real mathematical equations on this forum.

Does anyone here know how MathJax works?

I still love how that's a distinction that exists: real maths and real maths.