0² = 0
1² = 1
Does that pattern hold for all natural numbers? Could we claim that n² = n?
Yes, we can, and I’ve written a proof to prove it! The proof uses a valuable concept in mathematics called induction. I remember being introduced to proofs by induction when I was in Junior High. Nowadays, if it is not part of Common Core, it wouldn’t be taught much anymore. Nevertheless, I will use it here to prove that n² = n.
Using a similar proof, we can also prove that n³ = n, n⁴ = n, n⁵ = n, n⁶ = n, and so forth!
Today is the perfect day to review how to use proof by induction so try your hand at proving at least one of those mathematical statements on your own. Use the same steps in my example: prove true for n=1, assume true for n = k, prove true for k + 1, write your conclusion. then have a very Happy April Fools’ Day, Everyone!
Today is also a very good day to review that (x + y)² = x² +2xy + y² and NOT x² + y², a very common error students make. Confession: I remember making that exact error in high school when I definitely should have known better. Using induction to prove something in mathematics is a valid technique, but if you use invalid equations like
(x + y)³ = x³ + y³, you will make invalid conclusions. Thus, today might also be a good day to review the binomial theorem and Pascal’s triangle. (Pascal’s triangle has numbers in its interior, not just 1’s going down the sides, after all.)
My post today was inspired by a post written by Sara Van Der Werf titled Why I’ve Started Teaching the FOIL Method Again. In her post, she not only plays a great April Fools’ joke on her readers, but she explains a tried and true way to multiply binomials and other polynomials.
I read her post exactly one year ago today, and since then, I have been waiting for April Fools’ Day to roll around again so that I could share this post with you. It is my hope that you will enjoy my little prank and learn a little mathematics from it as well.
Now I’ll write a little bit about the number 1371:
- 1371 is a composite number.
- Prime factorization: 1371 = 3 × 457
- 1371 has no exponents greater than 1 in its prime factorization, so √1371 cannot be simplified.
- The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1371 has exactly 4 factors.
- The factors of 1371 are outlined with their factor pair partners in the graphic below.
1371 is the hypotenuse of a Pythagorean triple:
504-1275-1371 which is 3 times (168-425-457)
OEIS.org informs us that 1² + 37² + 1² = 1371, and there’s no April Fooling about that!