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Category Archives: counting
do go no to
dodo do go on do no harm do to others go do likewise gogo music go no further goto considered harmful no can do it’s a no go a big nono say no to drugs what to do here or … Continue reading
A new counting system
0 = t__ough 1 = t_rough 2 = th_ough 3 = through So, for example, trough through tough though though. English is so strange.
Computing the Euler totient function, part 4: totient of prime powers
I’ve been on a bit of a hiatus as I’ve been travelling with my family for the past month. So here’s a recap. Our story so far Recall that the Euler totient function, , counts how many numbers from to … Continue reading
Posted in arithmetic, computation, counting
Tagged Euler, multiplicative, totient
Comments Off on Computing the Euler totient function, part 4: totient of prime powers
Computing the Euler totient function, part 3: proving phi is multiplicative
We are trying to show that the Euler totient function , which counts how many numbers from to share no common factors with , is multiplicative, that is, whenever and share no common factors. In my previous post, we looked … Continue reading
Computing the Euler totient function, part 2: seeing phi is multiplicative
In my last post, I claimed that whenever and are relatively prime. (Recall that counts how many numbers from to share no factors in common with .) Let’s get some intuition for this by looking at some Chinese remainder theorem … Continue reading
Computing the Euler totient function, part 1
Recall that Euler’s totient function counts how many of the integers from to are relatively prime to , that is, share no factors in common with . For example, , since only , , , and share no factors with … Continue reading
Games with factorization diagram cards
Since I published a deck of factorization diagram cards last September, a few teachers have picked up copies of the cards and started using them with their students. I’ve started collecting ideas for games you can play using the cards, … Continue reading
Posted in arithmetic, counting, games, pattern, pictures, primes, teaching
Tagged cards, diagrams, factorization, games, SET, war
4 Comments
Factorization diagram cards are here!
It’s been a long process, but factorization diagram cards are finally available for purchase! If you just want to purchase a set right this minute, then click the above link! If you want to learn more, keep reading. History As … Continue reading
Posted in arithmetic, counting, pattern, pictures, primes, teaching
Tagged cards, diagrams, factorization
2 Comments
Factorization diagram card redesign: feedback welcome!
After getting a printed set of factorization diagram cards, I decided there were a few design tweaks I wanted to make. I’ve gone through a few iterations and I think they are definitely better now. Here are some representative samples … Continue reading
Posted in arithmetic, counting, pattern, pictures, primes, teaching
Tagged cards, diagrams, factorization
6 Comments
Factorization diagram cards!
I’ve designed a set of factorization diagram cards and had them actually printed. This is one of the first times in my life when I have caused Actual Physical Objects to be created (other than using a printer I guess) … Continue reading
Posted in arithmetic, counting, pattern, pictures, primes, teaching, video
Tagged cards, diagrams, factorization
5 Comments