Morning Mathematical Monsters & Maniacs!
(Today’s post is sponsored by the letter “M”)
Over the past 600+ episodes, The Simpsons has taken us on an amazing mathematical journey involving fractions, probability, Fermat’s last theorem, and hundreds of other aspects from the wonderful world off mathematics.
And what better way to start your week, then by discussing math Monday morning?
With season 30 officially in our rearview mirror after last nights season premiere (shout out to the use of the Pythagorean theorem in last nights episode), we continue our journey today on September 30th, by looking at some properties of the number 30 – Square Pyramidal and Harshad numbers.
In Bart’s Girlfriend (Season 06, Episode 07), Bart develops a crush on Reverend Lovejoy’s daughter Jessica. When Jessica invites Bart over for dinner, Reverend Lovejoy test Bart’s mathematical aptitude.
Helen Lovejoy: “So Bart, how’s school going? Jessica always gets Straight A’s.”
Bart: “Well, in my family, grades aren’t that important. It’s what you learn that counts.”
Reverend Lovejoy: “Six times five! What is it?”
Bart: “Actually, numbers don’t have much use in my future career: Olympic gold-medal rocket-sled champ!”
Helen Lovejoy: “I didn’t know the rocket sled was an Olympic event.”
Bart: “Well no offense, lady, but what you don’t know could fill a warehouse.”
Now for those of you who do make it as Olympic rocket-sled athletes, you may not need to know what six times five is, but for the rest of us:
6 x 5 = 30
So let’s look at a couple properties of the number 30:
Square Pyramidal Number
A square pyramidal number is a number that represents the number of stacked spheres in a pyramid with a square base. So the first four square pyramidal numbers are:
12 = 1
12 + 22 = 1 + 4 = 5
12 + 22 + 32 = 1 + 4 + 9 = 14
12 + 22 + 32 + 42 = 1 + 4 + 9 + 16 = 30
Graphically, 30 as a square pyramidal number can be depicted as follows:
Formulas for calculating square pyramidal numbers was first documented by 12th century Italian mathematician Fibonacci and later refined by 16th century German mathematician Johann Faulhaber.
20th century recreational mathematician Dattatreya Ramchandra (D.R.) Kaprekar was an Indian school teacher with no formal postgraduate training. In his spare time, he enjoyed dabbling in numbers and discovered many interesting properties of various numbers; one of which were harshad numbers.
Dattatreya Ramchandra Kaprekar
The team harshad comes from the Sanskrit meaning ‘joy-giver’. D.R. Kaprekar found great joy in the number 30, as he found this number was divisible by the sum of its digits.
30 ÷ (3 + 0) = 30 ÷ 3 = 10 with no remainder
If we go back to our first three Math Mayhems from January, we can see if any ot the three numbers we looked at were harshad numbers:
8,191 ÷ (8 + 1 + 9 + 1) = 8,191 ÷ 19 = 431 with remainder 2
8,128 ÷ (8 + 1 + 2 + 8) = 8,128 ÷ 19 = 427 with remainder 15
8,208 ÷ (8 + 2 + 0 + 8) = 8,208 ÷ 18 = 456 with no remainder
So only 8,208 is a harshad number. How narcissistic is that?
20th century Canadian mathematician Ivan Niven, who served as professor of mathematics at University of Oregon for many years as well as president of the Mathematical Association of America, extended harshad numbers beyond base 10 to any n-base in 1977, resulting in Niven numbers. Niven numbers are essentially harshad numbers, but in any base. So 30 in base 10 is a Niven number, in addition to being a harshad number.
Now that we’ve completed a look at some properties of the number 30, be sure to keep them in mind when you watch Treehouse of Horror 30 in a couple weeks. Were you able to quickly calculate six times five the first time you saw Reverend Lovejoy pose the question to Bart? Were you familiar with square pyramidal numbers? What about harshad numbers or Niven numbers? Do you know any other numbers that are both square pyramidal and harshad? Sound off in the comments below. You know we love hearing from you.