# Category Archives: number theory

## A Lesson on Converting Between Different Bases

We’re in the time of COVID-19, and that has meant taking far more direct responsibility for my children’s learning than I ever have before.  It’s been a lot of work, but it’s also been fun.  In fact, I’ve been surprised … Continue reading

## Strong Induction Wasn’t Needed After All

Lately when I’ve taught the second principle of mathematical induction – also called “strong induction” – I’ve used the following example to illustrate why we need it. Prove that you can make any amount of postage of 12 cents or … Continue reading

## Arguments for 0.9999… Being Equal to 1

Recently I tried to explain to my 11-year-old son why 0.9999… equals 1.  The standard arguments for (at least the ones I’ve seen) assume more math background than he has.  So I tried another couple of arguments, and they seemed … Continue reading

## A Quasiperfect Number Must Be an Odd Perfect Square

Let be the sum of divisors function.  For example, , as the divisors of 5 are 1 and 5.  Similarly, , as the divisors of 6 are 1, 2, 3, and 6. A perfect number is one that is equal … Continue reading

## The Validity of Mathematical Induction

Suppose you have some statement .  Mathematical induction says that the following is sufficient to prove that is true for all natural numbers k. is true. For any natural number k, if is true, then is true. The idea is that the first … Continue reading