hatty
Banned
do they ever get u 2 prove something in 4 unit? \
like theorems of formulas etc...
like theorems of formulas etc...
-
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proofs in 4 unit... (1 Viewer)
- Thread starter hatty
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hatty
Banned
dmt = de moivres theorem?
From what I can remember, you don't actually prove anything as such.
Probably the biggest thing you 'prove' is DeMoivre's theorem. It is easy to prove when n is an integer (induction), but much harder if n is not an integer....first prove it when n is rational, then when n is irrational....then you might run into problems like what irrational means and how to properly define irrational in a useful way (not really useful to say that irrational is not rational, or non-repeating decimal)....I don't think they get you to do this. They just tell you it (DeMoivre's theorem) is true (for any real n).
Some other things to think about:
1. fundamental theorem of calculus (integration is the 'opposite' operation to differentiation)
2. the 'formula' for integration by parts (easy)
3. the derivative of trigonometric functions (go back to first principles here)
4. prove the quadratic formula (easy)
5. show that there are infinitely many primes
6. show that an integer, whose remainder after dividing by 4 is 3, can never be written as a sum of two squares.
7. show that sqrt(2) is irrational (easy)
8. what can you say about a^b where both a and b are both irrational?
9. Formulate a 6-day week with 168 hours and live it out (you get to sleep longer and wake up at weird hours of the day...).
10. When putting on pants/jeans, stop and put the other leg in first.
OK it's starting to get ridiculous.
1 through to 8 are legit questions, though their relevance to your ability to do 4 unit math questions is questionable.
But most things in the 4 unit course, you just accept and plod along doing questions...there are those big questions that tell you to prove the irrationality of e and pi (these are big theorems!), but set it such a way that you are guided through them, making them do-able.....
/end rant.
Probably the biggest thing you 'prove' is DeMoivre's theorem. It is easy to prove when n is an integer (induction), but much harder if n is not an integer....first prove it when n is rational, then when n is irrational....then you might run into problems like what irrational means and how to properly define irrational in a useful way (not really useful to say that irrational is not rational, or non-repeating decimal)....I don't think they get you to do this. They just tell you it (DeMoivre's theorem) is true (for any real n).
Some other things to think about:
1. fundamental theorem of calculus (integration is the 'opposite' operation to differentiation)
2. the 'formula' for integration by parts (easy)
3. the derivative of trigonometric functions (go back to first principles here)
4. prove the quadratic formula (easy)
5. show that there are infinitely many primes
6. show that an integer, whose remainder after dividing by 4 is 3, can never be written as a sum of two squares.
7. show that sqrt(2) is irrational (easy)
8. what can you say about a^b where both a and b are both irrational?
9. Formulate a 6-day week with 168 hours and live it out (you get to sleep longer and wake up at weird hours of the day...).
10. When putting on pants/jeans, stop and put the other leg in first.
OK it's starting to get ridiculous.
1 through to 8 are legit questions, though their relevance to your ability to do 4 unit math questions is questionable.
But most things in the 4 unit course, you just accept and plod along doing questions...there are those big questions that tell you to prove the irrationality of e and pi (these are big theorems!), but set it such a way that you are guided through them, making them do-able.....
/end rant.
hatty
Banned
you my good friend
are a legend
are a legend
most primes are odd, 2 being the only even prime. Note that 1 is not considered a prime.
If an odd number were a sum of two primes, one would have to be 2, making the other number 2 less. The 2-less number is not always prime (eg. 11=2+9). This should make you change your conjecture.
On the other hand, looking at even numbers, it could possibly be that every even number bigger than 4 is the sum of two primes. Here is the start of the list:
4=2+2
6=3+3
8=5+3
10=5+5
12=7+5
14=7+7
16=11+5=13+3
18=13+5=7+11
20=3+17=7+13
22=3+19=11+11
The list can, and does, go on. But it doesn't prove anything.
For more information, though, punch into google "Goldbach's Conjecture". It is the name of this open problem.
As far as I know, a counter-example has not been found yet...
If an odd number were a sum of two primes, one would have to be 2, making the other number 2 less. The 2-less number is not always prime (eg. 11=2+9). This should make you change your conjecture.
On the other hand, looking at even numbers, it could possibly be that every even number bigger than 4 is the sum of two primes. Here is the start of the list:
4=2+2
6=3+3
8=5+3
10=5+5
12=7+5
14=7+7
16=11+5=13+3
18=13+5=7+11
20=3+17=7+13
22=3+19=11+11
The list can, and does, go on. But it doesn't prove anything.
For more information, though, punch into google "Goldbach's Conjecture". It is the name of this open problem.
As far as I know, a counter-example has not been found yet...
Grey Council
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lol, what a useless piece of trivia. 
and what did rumanujan actually do? why is he famous?
and what did rumanujan actually do? why is he famous?
Grey Council
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ah k.