Proof - Eddie Woo never fails to excel while teaching Setup: By definition, we let x be the number of dislikes on average for Eddie Woo's videos. Assume that x is bigger than Graham's number. By averaging out the dislikes, it is obvious the difference between x and 0 is infinitesimally small. Therefore, our original assumption is false. This implies the premise of our proof.
Part (a) is easy to explain by contrapositive. Integers which don't have this form are either a multiple of 4 or a two less than a multiple of 4 in which case such integers would be even. Done.
Alternative for a) n = 4q + r: r is the remainder 1) r = 1 => n = 4q + 1 2) r= 3 => n = 4(q+1) - 1 r can't be even because n is odd And so we are done!
b), can I say that since 4n+1 is odd for any n (as proven in part a), then since the product of two odd numbers is always odd, then the product can always be expressed in the form 4n+1?
You can’t, but you’re pretty close to a proof using modular arithmetic. 4n + 1 is congruent to 1 mod 4. Basically, 1 is the remained when dividing by 4. The multiplying two numbers, the residues (remainders) when dividing by some number (say 4) can also be multiplied. So if one number has a remainder of 1 when divided by 4 and you multiply that by anything, the remainder of the product (when divided by 4) will be equal to the remainder of the number you are multiplying by. In this case BOTH have remainders of 1, the the product must have a remainder of 1 (when divided by 4). Put simply: a (4n + 1) number times (4n + r) [r is 0,1,2,3] will be a (4n + r) number.
How can you prove geometrically, visually or graphically, that the determinant when it is worth zero is different from steel, the planes intersect or are parallel and really waht is the range.
Proof - Eddie Woo never fails to excel while teaching
Setup: By definition, we let x be the number of dislikes on average for Eddie Woo's videos. Assume that x is bigger than Graham's number.
By averaging out the dislikes, it is obvious the difference between x and 0 is infinitesimally small. Therefore, our original assumption is false. This implies the premise of our proof.
lmao
Part (a) is easy to explain by contrapositive. Integers which don't have this form are either a multiple of 4 or a two less than a multiple of 4 in which case such integers would be even. Done.
Alternative for a)
n = 4q + r: r is the remainder
1) r = 1 => n = 4q + 1
2) r= 3 => n = 4(q+1) - 1
r can't be even because n is odd
And so we are done!
series of numbers in general terms, pretty cool. Simple way to do it. The first example is an increasing sequence of numbers.
Alternative for b)
(4a+1)(4b+1) = 4(4ab+a+b) +1
And so we are done!
Have you ever been to your home country Malaysia? If so, have you caught the MRT (Mass Rapid Transit) in Malaysia, yes or no? BTW I like trains.
Sir, You ou are great.
From - India
Great stuff!
I have a question sir ... We can also prove 2 irrational by contradiction method .......plz
b), can I say that since 4n+1 is odd for any n (as proven in part a), then since the product of two odd numbers is always odd, then the product can always be expressed in the form 4n+1?
Not every odd number can be expressed in the form 4n+1.Take 3 for instance.
@@damianflett6360 Thanks, I'd better go over that again.
You can’t, but you’re pretty close to a proof using modular arithmetic. 4n + 1 is congruent to 1 mod 4. Basically, 1 is the remained when dividing by 4.
The multiplying two numbers, the residues (remainders) when dividing by some number (say 4) can also be multiplied.
So if one number has a remainder of 1 when divided by 4 and you multiply that by anything, the remainder of the product (when divided by 4) will be equal to the remainder of the number you are multiplying by.
In this case BOTH have remainders of 1, the the product must have a remainder of 1 (when divided by 4).
Put simply: a (4n + 1) number times (4n + r) [r is 0,1,2,3] will be a (4n + r) number.
But why did you think about it as a series? What made you think it is a series?
How can you prove geometrically, visually or graphically, that the determinant when it is worth zero is different from steel, the planes intersect or are parallel and really waht is the range.
How about 3 and 5?
Every odd number is one less than a multiple of 2
Sir, which app are you using for teaching ?
Notability and digital pen
@@aashsyed1277 ok thank you.
Has Eddie Woo met Terence Chi-Shen Zhenxuan Tao?
Who is that?
@@aashsyed1277 Bruh Terence Tao, Tao-Green conjecture, twin primes conjecture, solved Erdos discrepancy problem.
Does anyone know?
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