2

[intuitivet]

I'm not sure where to put this seeing as it regards maths, but I'm putting it here as it's generally wondering.
Why is it that 2+2 equates to 4, and yet 2x2 also equates to 4? No other number I can think of does this, why is 2 so special?

 

[walfin]

Not to mention, 2^2=4.

0x0=0 and 0+0=0 too.

The reason is that the operators used are binary.

For unary operators, log1(1)=1 and 1^1=1.

There are no ternary operators I know of in math. Math should use the ?: operator too

 

[intuitivet]

Not to mention, 2^2=4.

0x0=0 and 0+0=0 too.

Ah 0 isn't a proper whole number (I don't think so anyway) but thanks. That's also true, what is it with 2 being special?

 

[BigApplePi]

There is an algebraic formula for that which give that as a solution --

x+x = x^x
x^x-2x = 0
x(x-2) = 0
x = 0; x-2 = 0
Therefore the two solutions: two and zero.

 

[Decaf]

I heard that 2+2=5 for large values of 2

Who am I to question Radiohead?

 

[BigApplePi]

I heard that 2+2=5 for large values of 2

That is absolutely true.

Ripples on a pond. You look to the left. Three ripples. You look to the right three ripples. You add them all checking left and right finding only two on each side. 2+2=6.

Re: Speaking in absolute...

 

[Latro]

This depends on how far back you want to go. The work Principia Mathematica starts with extremely rudimentary axioms and eventually derives (over the course of hundreds of pages!) statements that can, for example, be used to show that 1+1=2 when arithmetic addition is defined. On the other hand, one can also use simpler ideas. Slightly simpler is a definition of numbers as being isomorphic to sets that the ZF axioms say must necessarily exist, and then renaming the operations of ZF on these special sets to match the arithmetic operations. If I remember correctly, under this system 0 is the empty set and 1 is the set whose only element is the empty set. Still simpler is to simply define axioms for arithmetic itself, which is the most common method, with the typical implementation being the Peano axioms.

 

[Hawkeye]

I'm not sure where to put this seeing as it regards maths, but I'm putting it here as it's generally wondering.
Why is it that 2+2 equates to 4, and yet 2x2 also equates to 4? No other number I can think of does this, why is 2 so special?

Multiplication is just stacked adding (and powers are stacked multiplications)

For example:

4 x 3 = 4 + 4 + 4 = 12

3 x 4 = 3 + 3 + 3 + 3 = 12

So I hope now you can see why 2 x 2 is the same as 2 + 2 (and 2^2).


2 is no more special than any other number as the equation in my sig shows




honestly you guys... you are beginning to worry me.

[edit] lol... I just looked at the example wikipedia gives you. 3 x 4 and 4 x 3.

 

[intuitivet]

Thank you knowledgeable people!

 

[Trebuchet]

Don't forget that 2 is the only even prime.

 

[Mary]

Don't forget that 2 is the only even prime.

OOOOO yes.
You have no idea how happy that makes me.
(What about -2? My math teacher never answered that question for me..)

 

[Trebuchet]

Well, I've never seen anyone use negative numbers for prime factorization. Anyway, it would mean a second even prime, and I wouldn't want you to become sad.

 

[Hawkeye]

The prime number patterns were discovered before uses for negative numbers were considered.

Adding the negative number scale buggers up some properties of prime numbers and so, to keep everyones mind at ease... They excluded negative numbers from the prime number theorems.

 

[Mary]

Okie. That would make sense.
YAY~
2 holds its eternal place as most awesome number ever.
(Or is 0 more awesome? Hmmm..)

 

[Hawkeye]

I'd say 3 is more impressive.

It is after all... The Magic Number.

Wikipedias page of the number 3 is also far greater than that of number 2

 

[Trebuchet]

I'd say 3 is more impressive.

It is after all... The Magic Number.

Wikipedias page of the number 3 is also far greater than that of number 2

Well, yeah, but you can't go by how long the Wiki articles are. I mean, the great Dizzy Gillespie's entry is a lot shorter than the entry for the TV show Married...with Children. What does that say about Wikipedia?

I will go along with 3 being more mystical, and a better number of legs for a chair than 2, but 2 is better for dividing things in half, or storing data in magnetic media.

 

[Mary]

Well, yeah, but you can't go by how long the Wiki articles are. I mean, the great Dizzy Gillespie's entry is a lot shorter than the entry for the TV show Married...with Children. What does that say about Wikipedia?

I will go along with 3 being more mystical, and a better number of legs for a chair than 2, but 2 is better for dividing things in half, or storing data in magnetic media.

Yes. 2 is logic, 3 is fluffy.

Without 2, there would be no half empty things. Or half full. And ordinary people wouldn't know how to define optimism!
:O

 

[Hawkeye]

haha, yeah. I don't like wikipedia myself I was just using it as an example as it shows how the number 3 is used a lot in philosophy and religions.

2 is better for dividing things in half

This made me laugh for some reason.

 

[BigApplePi]

Well, I've never seen anyone use negative numbers for prime factorization. Anyway, it would mean a second even prime, and I wouldn't want you to become sad.

Even if minus two is a prime, that is ,ugh, negative thing to say.

 

[BigApplePi]

Oh yeah? 12 is the only number that is exactly one dozen!

 

[Hawkeye]

Oh yeah? 12 is the only number that is exactly one dozen!

Although a dozen can be used to mean a number other than 12.

 

[Mary]

What about a dozen from the POV of a baker, HMMMMM?
That's not 12.

 

[Trebuchet]

haha, yeah. I don't like wikipedia myself I was just using it as an example as it shows how the number 3 is used a lot in philosophy and religions.

Actually, I do like Wikipedia. I wouldn't use it for serious research, but for this discussion it seems about right. Your use of it to advance the 3 cause was really funny.

Oh, and to those advocating how great a dozen is, 12 dozen is just gross.

 

[BigApplePi]

Here's a tougher one -- 1,729.

It is the only number that is the sum of two cubes two different ways. I'll bet you can't tell why.

8, for example, is a cube because 2x2x2 = 8.

 

[Hawkeye]

Here's a tougher one -- 1,729.

It is the only number that is the sum of two cubes two different ways. I'll bet you can't tell why.

8, for example, is a cube because 2x2x2 = 8.

Mmmm Taxicab numbers


This is quite interesting

[SIZE=+1][SIZE=+3]8586[/SIZE] is a house number.[/SIZE]

 

[Agent Intellect]

YouTube- One is the Loneliest Number

 

[bananaphallus]

9 was always the one that did it for me.

Take this for example:

[multiples of nine]

...18, 27, 36, 45, 54, 63...THE DIGITS ALL ADD UP TO NINE, FOR F*CK'S SAKE!

*goes absolutely nuts*

 

[Mary]

Hehehe..
You have fun with that..
I'LL JUST WORSHIP MY 2 OVER HERE.
Not 9 nutcakes allowed.
And 0, 0 is good. 0 makes me happy. It is nothing.
WHEEEE~
Emptiness is good.
As in, the glass is 1/2 empty
Hehehee..

 

[RubberDucky451]

9 was always the one that did it for me.

Take this for example:

[multiples of nine]

...18, 27, 36, 45, 54, 63...THE DIGITS ALL ADD UP TO NINE, FOR F*CK'S SAKE!

*goes absolutely nuts*

Is that the reason you can multiply 9's with your fingers?

I can recall times where I added and subtracted numbers to try and find patterns in them.

 

[Geminii]

...18, 27, 36, 45, 54, 63...THE DIGITS ALL ADD UP TO NINE, FOR F*CK'S SAKE!

An artifact of a base-10 numbering system. And it falls over at 99.

Non-base-10 systems have similar things show up for their respective base-minus-one numbers. Nine itself isn't anything special in that regard.

 

[bananaphallus]

I didn't mean to imply that it's the 'rule', that the digits of all multiples of nine taken individually add up to nine, I'm sorry...frankly I'm ashamed of myself, but would it be fair to say that within the context of a base-10 numeral system/my heart, nine is something special?

 

[Mary]

No, it still works.
108, for example.
1+0+8=9
And 9*23=207
2+0+7=9
9*21=189
1+8+9=18
1+8=9

 

[BigApplePi]

The Moldy Peaches - Lucky Number Nine
YouTube- The Moldy Peaches - Lucky Number Nine

 

[Trebuchet]

I learned casting out nines and the rules of factorization back in 5th grade, and never forgot them. In honor of Carl Jacober, a great math teacher, I shall post them here.

2 is a factor if the last digit is 0,2,4,6, or 8
3 is a factor if the sum of the digits is a multiple of 3
4 is a factor if the last to digits are a multiple of 4
5 is a factor if the last digit is 0 or 5
6 is a factor if 2 and 3 are factors
7 is a factor if you can find a rule for it (he never did adopt my solution of converting to base 8 and casting out 7’s)
8 is a factor if the last 3 digits are a multiple of 8
9 is a factor if the sum of the digits is 9 (as Mary pointed out, you might have to repeat a few times)
10 is a factor if the last digit is 0

May he rest in peace.

 

[Geminii]

Seven's not an easy one to eyeball, especially on larger numbers. One solution is to mentally add the appropriate number of 7s until the number is divisible by 10, then take off the zero and repeat.

So for 5922945, you'd add 35 to get 5922980, remove the zero to get 592298.
592298 plus 42 is 592340 -> 59234
59234 plus 56 is 59290 -> 5929
5929 plus 21 is 5950 -> 595
595 plus 35 is 630 -> 63
63 plus 7 is 70 -> 7
7 is divisible by 7, so therefore 5922945 is divisible by 7.

It's slightly mentally easier if you also allow subtracting an appropriate multiple of 7 as well as adding it, because you're less likely to have to break a three-digit boundary.

So in the above case:

5922945 - 35 = 5922910
592291 - 21 = 592270
59227 - 7 = 59220
5922 +28 = 5950
595 - 35 = 560
56 is divisible by 7.

Another method is to repeatedly subtract obvious visual multiples of 7. It helps to remember that 105, 203, and 301 are multiples of 7 (differing by 98 - you can see the pattern).

+5,922,945
-5,621,742 (56, 21, 7, and 42)
=0,301,203
Which is divisible by 7.

 

[Trebuchet]

Those tricks for testing if a number is a multiple of 7 are just lovely! While they aren't really a factorization rule like the others, they are very neat tricks I hadn't come across before, and I shall use them with pleasure. Thanks.

 

[BigApplePi]

Re: 7. Have you tried comparing that technique with basic long division?

 

[BigApplePi]

Here's a tougher one -- 1,729.

It is the only number that is the sum of two cubes two different ways. I'll bet you can't tell why.

8, for example, is a cube because 2x2x2 = 8.

No one solved this puzzle yet? What about just one answer?

 

[Trebuchet]

No one solved this puzzle yet? What about just one answer?

I think a lot of people have solved it, given the references to taxicab numbers. I expect no one wanted to spoil the answer for others, and it is easy enough to look up, given how famous the number is. That is why I didn't post an answer, anyway.

 

[Luzian]

I'm not sure where to put this seeing as it regards maths, but I'm putting it here as it's generally wondering.
Why is it that 2+2 equates to 4, and yet 2x2 also equates to 4? No other number I can think of does this, why is 2 so special?

This is a stupid and pointless question
Also, have you realized there are two arguments for the addition function? There's a hint.

 

[BigApplePi]

This is a stupid and pointless question
Also, have you realized there are two arguments for the addition function? There's a hint.

Someone once told me, "There is no such thing as a stupid and pointless Q."
How do you define "stupid" and "pointless"? Consider these two cases:

(1) That is a good question
(2) That is a stupid and pointless question

I eagerly await the answer because .... well I don't know why.

 

[BigApplePi]

I think a lot of people have solved it, given the references to taxicab numbers. I expect no one wanted to spoil the answer for others, and it is easy enough to look up, given how famous the number is. That is why I didn't post an answer, anyway.

Trebuchet. True.

I think it's more fun to solve it oneself wo looking it up. I suppose I could have said, "Don't look it up", but that could be an invitation to do just that, lol.

Want a harder one? Solve this for positive whole numbers:
x^x + 2 = y^y^y. Clue: both x and y are small. I haven't tried to look it up. One solution is easier but are there any other solutions?

 

[Trebuchet]

Trebuchet. True.

I think it's more fun to solve it oneself wo looking it up. I suppose I could have said, "Don't look it up", but that could be an invitation to do just that, lol.

Want a harder one? Solve this for positive whole numbers:
x^x + 2 = y^y^y. Clue: both x and y are small. I haven't tried to look it up. One solution is easier but are there any other solutions?

Spoiler

I get x = -1 and y = 1
Oh, never mind, you said positive whole numbers.

Richard Wiseman posted a cool one recently, on his blog.

Imagine you have to make the number 8 from the numbers 4, 7, 6, and 3. You can add, subtract, multiply, or divide to make the number 8. (No exponents, logs, or other operations.)

4 + 7 - 6 + 3 = 8

So, can you make the number 24 with 5, 5, 5, and 1?

 

[BigApplePi]

Spoiler

I get x = -1 and y = 1
Oh, never mind, you said positive whole numbers.

Richard Wiseman posted a cool one recently, on his blog.

Imagine you have to make the number 8 from the numbers 4, 7, 6, and 3. You can add, subtract, multiply, or divide to make the number 8. (No exponents, logs, or other operations.)

4 + 7 - 6 + 3 = 8

So, can you make the number 24 with 5, 5, 5, and 1?

Just to make sure the symbols are understood, that was 2 squared + 2 = 3 cubed. So even the negative "-1" doesn't work. I think the answer easier than the one you proposed.

For the 24 with 5,5,5,1 I get "no" for the answer so far. Must be something I missed. I have a solution if I use just two of the fives, not all three.

 

[LucasM]

Even if minus two is a prime, that is ,ugh, negative thing to say.

that made me wonder...
If we use negative numbers
8 = 2x2x2 = -2x-2x2 = 2i x 2i x -2
Why do factors have to be positive whole numbers?

 

[BigApplePi]

that made me wonder...
If we use negative numbers
8 = 2x2x2 = -2x-2x2 = 2i x 2i x -2
Why do factors have to be positive whole numbers?

LucasM. Good one. Or rather good "2." I guess they don't have to be.

 

[Trebuchet]

Just to make sure the symbols are understood, that was 2 squared + 2 = 3 cubed. So even the negative "-1" doesn't work. I think the answer easier than the one you proposed.

For the 24 with 5,5,5,1 I get "no" for the answer so far. Must be something I missed. I have a solution if I use just two of the fives, not all three.

Ahhhhh! Thanks for clarifying. I read it as 2 to the power of 2 + 2 = 3 to the power of 3 to the power of 3, which is 19,683. The -1 solution does work for that, but it isn't what you asked. Now I have an answer.

Spoiler

5 squared plus 2 equals three cubed.

And there is a solution to the one I posed. Want a hint? I did not put the answer.

Spoiler

Not all the intermediate steps are whole numbers.

 

[BigApplePi]

Ahhhhh! Thanks for clarifying. I read it as 2 to the power of 2 + 2 = 3 to the power of 3 to the power of 3, which is 19,683. The -1 solution does work for that, but it isn't what you asked. Now I have an answer.

Spoiler

5 squared plus 2 equals three cubed.

And there is a solution to the one I posed. Want a hint? I did not put the answer.

Spoiler

Not all the intermediate steps are whole numbers.

Your spoiler is correct for x squared plus two equal three cubed, but here is the puzzle. That is the ONLY solution. Apparently there is no easy way to prove that. It's advanced number theory and I don't know how.

For the 5,5,5,1 I have only 5*5-1 = 24 or 5*5-(1 to the 5th) = 24. Don't know any fractional solution.

 

[Latro]

\o/ Fermat's Last Theorem is awesome stuff. (Somewhat related to what you guys are talking about.) The result, btw:
If a, b, c, and n are positive integers and n>2, then:
a^n+b^n != c^n
This was one of the hardest results to prove in the history of mathematics.

 

[BigApplePi]

Fermat's last theorem is a good one because it's so easy to understand. It's a generalization of the Pythagorean theorem which may be the most famous math'l theorem.

The latest hard problem is called the Riemann Hypothesis. It's something about 1 + 1/2 + 1/3 + 1/4 + ... (or maybe the squares of those, I forget) and the solution is supposed to be very important, but no INTP has been able to solve it yet. (I'm assuming it would take an INTP, lol.)