Factor Song

(Click on the comic if you can't see the full image.)

(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

A true artist is never appreciated in his own time. Or class.

Folks who follow the math problem calender on Twitter might've gotten a sneak peek last week at this little ditty. The plan was to use it here as well, but I wasn't exactly sure how I was going to use it. Or when, since the new school year keeps me busy until I get things settled into a routine.

The problem written in the top panel is how it appeared on the calendar. In the bottom panel, I rewrote it in a way more commonly seen in math class when covering combinations.

I also write Fiction! You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below. Preorder the softcover or ebook at Amazon.

Come back often for more funny math and geeky comics.

(x, why?) Mini: Canceling

(Click on the comic if you can't see the full image.)

(C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Eliminate. Terminate. Totally, completely Annihilate.

Someone obviously used the Difference of Square rule in this example.




Come back often for more funny math and geeky comics.

Weird

(Click on the comic if you can't see the full image.)
-->
(C)Copyright 2019, C. Burke.

So, yes, you're weird!

Weird numbers are a subset of Abundant numbers.
In brief:
A perfect number is one where the sum of the number's factors, excluding the number itself, equal the number. Ex: 1+2+3 = 6.
An abundant number is one where the sum of the number's factors, excluding the number itself, is greater than the number. Ex: 1+2+3+4+6=16 > 12.
A semiperfect number is one where a subset of the number's factors have a sum equal to the number. Ex: 1+2+3+6 = 12.
A perfect number is also considered to be semiperfect, unlike my wife who is perfect and I would never consider to be semiperfect.
A weird number is abundant but not semiperfect: there is no subset of factors that add up to the number.
Ex: no combination of 1, 2, 5, 7, 10, 14, and 35 add up to 70, but the sum of the factors is 74.

I was familiar with semiperfect, but not the "weird" term until I was looking up what the prefixes for "abundant" numbers were.




Come back often for more funny math and geeky comics.

Abundant

(Click on the comic if you can't see the full image.)
-->
(C)Copyright 2019, C. Burke.

Good answer! Good answer! Incorrect, but good answer!

Unfortunately, 12 is not only abundant but superabundant. You could even go as far as to say colossally abundant, but I wouldn't go and say that, if I were you.




Come back often for more funny math and geeky comics.

Prime Factorization, Perfect Squares and Irrational Numbers

Moving on from yesterday's discussion about Rational Numbers, what about irrational numbers, numbers which cannot be written as a ratio of two integers?

Most whole numbers have square roots which are irrational numbers, but not everything with a radical is irrational. The square root of a perfect square is perfectly rational. So how can you tell if a number is a perfect square without a calculator?

One way is through prime factorization. (Remember those factor trees from a long time ago. C'mon, they were fun to do -- and you can do them again.... just not when you're typing in a blog. Then, they're kind of a pain, but I'll try.)

Take a number such as 60. It's prime factorization is 2 X 2 X 3 X 5, or 2² X 3 X 5.
If we were to square 60, we'd multiply 60 X 60, but we could also multiply 2² X 3 X 5 X 2² X 3 X 5.
That number (3600) would have a prime factorization of 2⁴ X 3² X 5².

Notice what happened to the exponents. They've all doubled from 1 to 2 or 2 to 4. Every time you square a number, the exponents of its prime factors double. So if a number has been square, then all of the exponents of its prime factors will be even numbers because they are multiples of two.

Going back to our original number, is the square root of 60 a rational number?

It could only be a rational number if 60 were a perfect square, and it can only be a perfect square if all the exponents of its prime factors are even. However, the prime factorization is 2² X 3 X 5. Only one factor is even, so it is not a perfect square and the square root of 60 is irrational.

But wait! There's more!

As long as we've done the legwork, there is one more thing that we can do. Radicals that are irrational can be simplified. This is done by factoring out the largest perfect square. If we look back at the prime factorization, 2² X 3 X 5, we can see that there are two factors of 2.

So the square root of 60 is the same as (the square root of 2²) X (the square root of 3 X 5). The square root of 2² is just 2.
That means that the square root of 60 is (2) times (the square root of 3 X 5), or 2(radical 15).