bah! Induction

[69dodge]

2^2 = 2*2
3^2 = 3*3
4^3 = 4*4*4
5^2 = 5*5
2^3 = 2*2*2
2^4 = 2*2*2*2

You get the idea.

In those equations...what is that funny star thing mean?

Silly computer programmers, making up new symbols for everything...

(Ok seriously, Iamme, the asterisk means multiplication.)

 

[gnome]

What kind of equations do you call these things, anyway? And what good do they do a person? Just curious.

If you're asking about mathematical induction, it is particularly useful in proving something to be true for ALL numbers. It metaphorically works like setting up dominoes... once you know the first domino knocks over the next, you know the whole lot will fall into place.

 

[Beausoleil]

I'm guesing e but I really can't be arsed to check right now but I don't see it being anything else.

Here's my stab

Set y = x^a - a^x
y = 0 at x=a, obviously
Want this to be a minimum so that x^a => a^x for all x
Differentiate, set a=x and dy/dx = 0
Get ln(x) = 1, so x = e.

 

[bruto]

In those equations...what is that upside down v thing mean?

Because standard computer keyboards and fonts do not contain the mathematical symbols, most plain text renditions of equations and inequalities use a different set of symbols from what you may have learned in school; these are more or less the same as the ones found in computer languages such as fortran and basic, which were designed to be written in plain text.

The caret is "raise to the power of," "sqr" is the square root, and other roots can be represented by fractional powers (a cube root is equivalent to ^1/3, for example). Because there is no division sign, the slash is used for both division and fractions, since they are functionally equivalent, and the asterisk is used for multiplication. Inequalities are represented by the greater-than and less-than symbols, as always, but because there is no combination symbol, "less than or equal to" is represented by the two symbols in succession, "greater than or equal to" similarly, and "not equal to" is represented by the two inequality symbols back-to-back, thus: <>.

 

[RandFan]

Because standard computer keyboards and fonts do not contain the mathematical symbols, most plain text renditions of equations and inequalities use a different set of symbols from what you may have learned in school; these are more or less the same as the ones found in computer languages such as fortran and basic, which were designed to be written in plain text.

The caret is "raise to the power of," "sqr" is the square root, and other roots can be represented by fractional powers (a cube root is equivalent to ^1/3, for example). Because there is no division sign, the slash is used for both division and fractions, since they are functionally equivalent, and the asterisk is used for multiplication. Inequalities are represented by the greater-than and less-than symbols, as always, but because there is no combination symbol, "less than or equal to" is represented by the two symbols in succession, "greater than or equal to" similarly, and "not equal to" is represented by the two inequality symbols back-to-back, thus: <>.

When I first started lurking on USENET and chat rooms that dealt with science issues, I noticed that many simply ignored such questions. I always felt intimdated.

Since I have come to JREF I've had some really great questions answered. Some I have asked and some just by lurking. Things have probably changed for the better on many similiar forum but I just wanted to say JREF is cool. And though I give Iamme a hard time I have to give him credit for being willing to ask questions. I know special charachters for logic. What is the tilde in math?

 

[DevilsAdvocate]

What is the tilde in math?

Tilde means "is proportional to or similar to". Sometimes in common usage it will be used for "is approximately eaqual to" because the correct double tilde ≈ is not availbe in the ASCII character set.

 

[Art Vandelay]

What kind of equations do you call these things, anyway? And what good do they do a person? Just curious.

What do you mean?

If you're asking about mathematical induction, it is particularly useful in proving something to be true for ALL numbers.

Or at least, the integers.

Here's my stab

Set y = x^a - a^x
y = 0 at x=a, obviously
Want this to be a minimum so that x^a => a^x for all x
Differentiate, set a=x and dy/dx = 0
Get ln(x) = 1, so x = e.

That's a start, but there are still some dots to connect.

Inequalities are represented by the greater-than and less-than symbols, as always, but because there is no combination symbol, "less than or equal to" is represented by the two symbols in succession, "greater than or equal to" similarly, and "not equal to" is represented by the two inequality symbols back-to-back, thus: <>.

For great than or equal to, I prefer a >= b rather than a => b, since the latter can also mean "b logically follows from a".

 

[RandFan]

Tilde means "is proportional to or similar to". Sometimes in common usage it will be used for "is approximately eaqual to" because the correct double tilde ≈ is not availbe in the ASCII character set.

Thank you. BTW, all charachters are available and since many forums allow for them if you want you can use them.

÷ ≤ ± → ⊂ ⊆ ⊃ ⊇ ≠

But then ASCII is so much simpler.

 

[LW]

I didn't realise induction was so tricky to get the hang of.

But the basic principle is very nice and clean. I finally understood what induction was about by using the following example.

You have the problem that you want to prove that something holds for all natural numbers. In effect you have an infinite number of smaller claims: "0 has the property", "1 has the property", "2 has the property", and so on. Let's use the syntax P(x) to mean "the natural number x has the property". So, in logic, you would now want to prove the claim "for all x : P(X) holds".

When you use induction, you prove the original claim by proving two easier claims:

• that P(0) is true; and
• that for all x : if P(x) is true, then P(x+1) is necessarily true.

These two claims together imply the original claim. [Argh, I posted this by mistake before it was complete] For example, if you want to be convinced that P(3) is true, you can start from your proof for P(0) and then note that by substituting 0 for x in the second claim, you find that P(1) is true, and by substituting 1 for x you get that P(2) is true, and so on up to P(3) or whatever number you are interested on.

In the actual induction proof you prove the two claims in three steps:

• base case where you prove P(0);
• you take the induction hypothesis where you suppose that P(x) is true for some x; and
• you complete the induction step and show that if P(x) is true, then P(x+1) is also true.

 

[LW]

Tilde means "is proportional to or similar to".

In formal logic it occasionally means complementation: '~a' is read as "not a". This is usually used only if using the standard negation symbol '¬' would be confusing for a reason or another. Two most common occurrences are when the semantics of the negation differs from the classical negation (for example, when we use "default negation" where "not a" means "we can't prove that 'a' is true) or when we are taking complements of literals (formulas of the form 'p(t)' and '¬p(t)') and and don't want to worry about double negations.

 

[LW]

Or at least, the integers.

With a small extension induction works also for transfinite numbers.

Though, transfinite stuff can be a bit tricky. When I first TAd an exercise session having a transfinite induction in it, it took me 40 minutes to understand the model solution.

And then it took me four years to find the error in it ...