A well-known fallacy committed by students is the so-called “Law of Universal Linearity” (the link is to a discussion of this phenomenon on Mathematics Stack Exchange). The most famous example of this is the statement
$$\left(x+y\right)^n = x^n + y^n,$$
known as the Freshman’s dream.
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I ask that anyone who has had multivariable calculus please stop right now and try to work out this problem.
$$\int_0^1\int_0^1\frac{dx\,dy}{1-xy}.$$
Hint: The integrand looks familiar!
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I have just finished moving all of my posts from the old location of this blog from a free webhost. I hope to be more productive from now on!
So as I study for my comprehensive exams, I will post some of my favorite problems from my studies.
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There are many rules for determining whether a number is divisible by another. For example, we know that a number is even (divisible by 2) whenever the one’s digit is even. We also know that a number is divisible by 3 if the sum of its digits is divisible by 3. A less known one is the rule for divisibility by 11, which says that a number is divisible by 11 when the number obtained by starting with the one’s digit, subtracting the 10’s digit, adding the 100’s digit, and so on, alternating between adding and subtracting, until you’ve used all digits, is itself divisible by 11. For example, we know that 628474 is divisible by 11 because $4-7+4-8+2-6=-11$, which is divisible by 11. Indeed, the quotient of 628474 by 11 is 57134.
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When I was in grade school learning about primes, I would ask myself: How many primes are there? If I pick a number at random, will it be prime?
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