Beautiful Mathematics

A

Ambrose Ackroyd

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1 x 8 + 1 = 9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 9876543
12345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321

1 x 9 + 2 = 11
12 x 9 + 3 = 111
123 x 9 + 4 = 1111
1234 x 9 + 5 = 11111
12345 x 9 + 6 = 111111
123456 x 9 + 7 = 1111111
1234567 x 9 + 8 = 11111111
12345678 x 9 + 9 = 111111111
123456789 x 9 +10= 1111111111

9 x 9 + 7 = 88
98 x 9 + 6 = 888
987 x 9 + 5 = 8888
9876 x 9 + 4 = 88888
98765 x 9 + 3 = 888888
987654 x 9 + 2 = 8888888
9876543 x 9 + 1 = 88888888
98765432 x 9 + 0 = 888888888

Brilliant, isn't it?
And finally, take a look at this symmetry:

1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111=12345678987654321
 
It seems to me now that mathematics is capable of an artistic excellence as great as that of any music, perhaps greater; not because the pleasure it gives (although very pure) is comparable, either in intensity or in the number of people who feel it, to that of music, but because it gives in absolute perfection that combination, characteristic of great art, of godlike freedom, with the sense of inevitable destiny; because, in fact, it constructs an ideal world where everything is perfect but true


Bertrand Russell (1872-1970), Autobiography, George Allen and Unwin Ltd, 1967, v1, p158
 
Yes it is, and you can also see why numbers have turned some people insane over the years.

There's one equation, I don't know what it is, where you always feel you're getting to the end but never do, but you never know you're not getting to the end, and that's what gets you.

Like seeing a door where the prize might be on the other side, but when you open it, there's another door, and so on and so on........
 
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/thread
 
you are assuming that 1 + 1 = 2 and that 1x0 is the same answer as 2 x 0:)

"What does '1' mean?"
"What does '2' mean?"
"What does '+' mean?"
"What does '=' mean?"
This is traditionally done with the Peano axioms:
1) 1 is a natural number. *
2) Every natural number has a successor. *
3) No natural number has successor 1 (or 1 has no predecessor) *
4) Every natural number has a predecessor except for 1. *
5) You don't need to know this one right now. *
We then define addition a+b for a, b natural numbers, as follows:
If a is 1, then a + b is b's successor. *
Otherwise, a has a predecessor, denoted a'. Then, a+b equals the successor of a' + b. * (Axiom 4)
For example, 3 + 2 is the successor of 2 + 2 which is the successor of the successor of 1 + 2 which is the successor of the successor of the successor of 2 which is the successor of the successor of 3 which is the successor of 4 which is 5.
We can also define equality as follows (a = b?)
If a is 1, and b is 1, then a = b is true.
If either a or b is 1, but the other is not, then a = b is false.
Otherwise, there exist a' and b', predecessors of a and b, respectively (Axiom 4). Then, if a' = b' is true, a = b is true, otherwise a = b is false.

The decimal number system and Arabic numerals are not part of this theory - they simply represent natural numbers. In any case, the successor of "1" is denoted "2", the successor of "2" is denoted "3", and so on. *

So given these definitions, it is clear that 1 + 1 = 2.

Furthermore, once a mathematical fact has been proven true, it is never possible to prove it false, unless you change some of the underlying axioms (above, the axioms are starred; only the first 5 are "real" axioms). The exception is when the axioms contradict each other, but these don't.

So if you redefine 2 to mean the successor of the successor of 1, then obviously 1 + 1 = 2 is false, because then you are asserting that one plus one is three. Or if you redefine the '+' sign to mean multiplication, again 1 + 1 = 2 is false. Or if you redefine the '=' sign to mean "not equal", again 1 + 1 = 2 is false.

Assuming that we don't modify any of these axioms, though, and add more self-consistent axioms, you can't prove 1 + 1 = 2, because a false statement can never be proven true.

There are many erroneous proofs that 1 + 1 = 2, but they all involve something that you're not allowed to do, such as dividing by zero, or using certain exponent rules with complex numbers.

Some of these are given by other answerers. I see that they have thumbs-down, but it's undoubtedly either a troll or an idiot who did that - I stand by their answers one hundred percent.
 
It's a language and like any language, it is capable of expressing great beauty and insight. You just have to learn to understand it.

Why art is 'art' and science is 'science' is beyond me. The polarisation has been applied by the poorly-educated and that unfortunately happens to be the majority of human beings.
 
An interesting proof:

Assume two unknowns - a and b.

Let a = b. Then,

a^2 = ab,
a^2 + a^2 = ab + a^2,
2a^2 = ab + a^2,
2a^2 - 2ab = a^2 + ab - 2ab
2a^2 - 2ab = a^2 - ab.
Factoring gives,
2(a^2 - ab) = a^2 - ab.
Dividing both sides by (a^2 - ab) gives,
2 = 1.

:D
 
amit1986 said:
An interesting proof:

Assume two unknowns - a and b.

Let a = b. Then,

a^2 = ab,
a^2 + a^2 = ab + a^2,
2a^2 = ab + a^2,
2a^2 - 2ab = a^2 + ab - 2ab
2a^2 - 2ab = a^2 - ab.
Factoring gives,
2(a^2 - ab) = a^2 - ab.
Dividing both sides by (a^2 - ab) gives,
2 = 1.

:D
Click to expand...

Is the flaw in the dividing at the end? ie we lose a solution. If 2 = 1 and a = b then a^2 = a^1 = ab so a = b = 1 and so we are dividing by 0 which is a sin
 
Scotty2Cues said:
Is the flaw in the dividing at the end? ie we lose a solution. If 2 = 1 and a = b then a^2 = a^1 = ab so a = b = 1 and so we are dividing by 0 which is a sin
Click to expand...

No, the division at the end is fine. The flaw lies at the beginning of the proof in assuming that a = b.
 
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