Friday, January 31, 2020

You Get What You Pay For

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

And keep in mind that this comic is free ...

No, I didn't need to do this on an index card. It was just a thing I did a few times in the past. Imitation, flattery, all that.

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Wednesday, January 29, 2020

You Load 60 Sheets

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

I'm going to ream the first person who complains ...

While this parodies the Tennessee Ernie Ford song of yore, my first exposure to this song (that I remember) was actually Bowzer and Sha Na Na ... of yore.

Update: Actually, I've been reminded by family that my brother (and my father?) used to sing that song a lot when I was a kid, which is probably why it sticks out in my memory the time Bowser sang it.

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Monday, January 27, 2020

Off on a Tangent

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

I guess a secant line could bring him back.

I was going to label what the actual conversation was ... but I forget it after I started reading stuff online.

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Saturday, January 25, 2020

Blog: Difference of Squares, Cubes and Quarts

I wanted to do this before the Algebra Regents exams when it might have been more helpful to some students, but the fact is, they don't actually follow me. They just find the posts with the questions and answers on them through search engines.

The Difference of Squares is a common topic for Algebra 1, but what about other powers, such as cubes and quarts? (Quarts are is "quartic", fourth power.)

Squares and Quarts

In a binomial that looks like x2 - n2, where n is any number, which will be squared, we can factor the two terms into conjugates:

(x + n)(x - n)

All nice and neat. When you multiply the two conjugates (which have the same terms, but one has a plus and one has a minus), the two x terms -- (nx) and (-nx) -- are additive inverses and will sum to zero.

Every now and then, state exams like to throw a curve ball at you, and use a higher power, usually 4.

What if the problem said:

Factor completely x4 - 16

Since 4 is an even power, x4 is a perfect square. And, of course, 16 is a perfect square. So the regular rule applies:
x4 - 16 = (x2 + 4)(x2 - 4)

Ah, but have we factored "completely" as the question asked? We can't do anything with the first factor, (x2 + 4), which has no real roots, but what about the other factor? It's another Difference of Squares, so we can apply the rule again:
x4 - 16 = (x2 + 4)(x2 - 4) = (x2 + 4)(x + 2)(x - 2)

So, if they really wanted to be mean, er, I mean "challenging", they could go another step further and ask:

Factor completely x8 - 256

Sum of Cubes

I know that I started by saying "difference of ...", but that was to keep everything flowing.

Unlike squares (and other even powers), you can factor the sum of two cubes. Also unlike squares, a perfect cube can be a negative number, so it also could be written as a difference of cubes. So I'm not wrong -- I just renamed it for no reason.

There's a simple format for the Sum of Two Cubes:

(x3 + n3) = (x + n)(x2 - nx + n2)

First thing you should see is that (x + n) means that there is a root at x = -n.

The second thing you should see is that that is the only real root. If you check the discriminant, b2 - 4ac, you get the following:
(n)2 - 4(1)(n2).

As long as n has a nonzero value, the discriminant will always be less than zero. This example had a leading coefficient of 1, but it is still true if there is a different number in front the x3 term.

Final Example

If we wanted to put this together, we could try to factor x6 - 64.

Or really put it all together and got with an x12 term. But who wants to do that?

Wednesday, January 22, 2020

The Mathemagic Goes Away

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

It's kind of a cumulative effect.

With any magic trick, if you look into it, you gain more understanding. However, if you do this, you lose some of the magic.

One of the simplest, most beautiful equations in math is 1 + 2 = 3.

The sum of two consecutive whole numbers add up to the third. You can't have three consecutive numbers add up to a fourth, but you can have it add up to a fourth and a fifth. If I didn't know what those numbers were, I could solve it by writing an equation:

n + n + 1 + n + 2 = n + 3 + n + 4

And when I solve 3n + 3 = 2n + 7, I'd get n = 4, so 4 + 5 + 6 = 7 + 8.

However, as the table gets filled in, it actually gets less exciting, even though there are exciting things happening in it.

1 + 2 = 3
4 + 5 + 6 = 7 + 8
9 + 10 + 11 + 12 = 13 + 14 + 15
16 + 17 + 18 + 19 + 20 = 21 + 22 + 23 + 24
25 + 26 + 27 + 28 + 29 + 30 = 31 + 32 + 33 + 34 + 35
... = ...

First, the numbers are consecutive. No number is skipped.

Second, if we call the first row, Row 1, then Row 2, etc., we see that for Row N, there are N + 1 numbers on the left side and N numbers on the right side of the equation.

Third, the first number in Row N is N2. But this isn't surprising because the first number is 1, and then we add 3 more numbers, and then add 5 more, and then add 7 more. From row to row, we are counting up by consecutive odd numbers, which means we are moving from one perfect square to the next.

In every Row N, the last of the N + 1 numbers on the left side will be N2 + N. This makes the N numbers on the right side N2 + N + 1 through N2 + N + N.

That last term on the left can be written as N(N + 1), and there are N terms before it on that side of the equation. This means that you can add (N + 1) to each of those N terms.

If you do that, you get N2 + (N + 1), N2 + 1 + (N + 1) through N2 + (N - 1) + (N + 1), which are the exact terms on the right side. Thus, it isn't really a mystery at all why the two sides are equal.

So, yes, I looked at it too closely, and the "mathemagic" went away.

Final note: the title is a take on Larry Niven's "The Magic Goes Away".

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Monday, January 20, 2020

Power Surge

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

Nothing kills a ''quick'' joke like pixelation of older images. Sigh.
Originally, I was just going to substitute the other AnthroNumeric(tm) characters, but then I decided I should match the colors and faces. Ooooh, bad decision.

If you take numbers ending in 1 to 9 and raise them to the second power, they will end in 1, 4, 9, 6, 5, 6, 9, 4, 1.
If you raise them to the third power, they will end in 1, 8, 7, 4, 5, 6, 3, 2, 9.
If you raise them to the fourth power, they will end in 1, 6, 1, 6, 5, 6, 1, 6, 1.
And, finally, if you raise them to the fifth power, you return to 1, 2, 3, 4, 5, 6, 7, 8, 9, and the cycle will start over again.

The interesting things here:
In 1, 5, 6, and 0, if included, never change.
Row 2 is symmetrical (if you leave out 0). Also, no perfect square ends in 2, 3, 7 or 8.
Row 3 uses all nine numbers from 1 to 9.
Row 4 is also symmetrical and only contains 3 numbers. (That pesky 5!)

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Thursday, January 16, 2020

Locus of Points

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

There's a step ladder in the garage ... pity that it isn't closer.

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Tuesday, January 14, 2020

Unit Rates

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

If you know what I Mina. Or, mean-a?

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Friday, January 10, 2020

Blog: Zeroth Power and Fractional Exponents

I'm starting a new, old math book (report to come), and I came across something interesting in the first chapter about numbers: another way to look at why N0 = 1.

The rules for exponents are simple:

Na * Nb = N(a + b)
and Na / Nb = N(a - b)

In the case of multiplying repeatedly by the same number:

Na * N = N(a + 1)

So, say, 24 * 2 = 24 + 1 = 25
and 25 * 2 = 25 + 1 = 26

Likewise, when dividing, repeatedly, by the same factor, the exponent would be decremented:

Thus, 23 / 2 = 23 - 1 = 22
and 22 / 2 = 22 - 1 = 21
and, finally, 21 / 2 = 21 - 1 = 20.

So what is 20?
If we evaluate the other expressions, we have the following:
8 / 2 = 4; 4 / 2 = 2; 2 / 2 = 1. So 20 = 1.

Note that we could continue the progression into negative exponents if we keep dividing. This will lead to fractions. Perhaps on another day...

Fractional exponents

What does it mean to have a fraction as an exponent?

Keep in mind that negative exponents have nothing to do with negative numbers. They create fractions. So fractional exponents won't create fractions.

Let's review one more rule about exponents:

(Na)b = N(ab)

So (23)4 = 2(3*4) = 212 = 4096.

What if there is an exponent of 2/3 or 3/5? First, consider that 2/3 = (2)(1/3) and 3/5 = (3)(1/5). The 2 and 3 still mean the second and third powers, but what about the unit fractions of 1/3 or 1/5?

If I take the positive square root, which I'll abbreviate SQRT(), of N2, I will get N, because N * N = N2.
If I take the positive square root of N4, I will get N2, because N2 * N2 = N4, etc.

So SQRT(Na) will give (N(1/2)a).
However, our rule tells us that (N(1/2)a) = (Na)1/2.
So (Na)1/2 is another way to write SQRT(Na), and N1/2 is another way to write SQRT(N).

Fractions and Zero

Consider the following progression:

SQRT(16) = 4
SQRT( SQRT(16)) = 2
SQRT( SQRT( SQRT(16))) = 1.414...
SQRT( SQRT( SQRT( SQRT(16)))) = 1.189...
SQRT( SQRT( SQRT( SQRT( SQRT(16))))) = 1.090...

As you repeatedly take the square root, the answer will get closer and closer to 1.
If we rewrite that using exponents, we get the following:

16(1/2) = 4
(16(1/2))(1/2) = 16(1/4) = 2
((16(1/2))(1/2))(1/2) = 16(1/8) = 1.414...
(((16(1/2))(1/2))(1/2))(1/2) = 16(1/16) = 1.189...
((((16(1/2))(1/2))(1/2))(1/2))(1/2) = 16(1/32) = 1.090...

As the denominator gets larger, the fraction gets smaller. As the denominator goes toward infinity, the fraction goes toward zero. And the value on the right side of the equal sign goes toward 1.

Wednesday, January 08, 2020

Non-abelian Group

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

For years, I have resisted making abelian group jokes because I don't want to explain them. I still don't.

Trust me, there are a handful of people laughing hysterically (or chuckling politely) at this, and I hope that they will share it with others who do not require explanation.

However, for those who would like more information, allow me to point to this tweet from the account Great Women of Mathematics.

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Monday, January 06, 2020

I vs i

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

Know what I talkin' bout?

A long time ago, I did a "Pop i" that had big sailor forearms, but then the Roman I would've needed arms, too, but not a Gladius

I did a Pi vs. Pi comic years ago. I never did draw the followup.

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Thursday, January 02, 2020

End of Year / End of Decade Report

I wanted to do this before the end of the year, but I was a little busy, and that was a good thing. I ended the year -- and decade -- on a happy note.

This made me happy because for the past month I've read others' account of the achievements and milestones of the past ten years. And that's where I ran into problems.

Thinking of achievements was difficult when they were weighed down by loss. I'm at an age when most of my achievements are behind me while the next generation steps out and marks their own milestones. On the other end of the spectrum, the older generation ... well, we know what eventually happens when you get older. You, one day, stop getting older. There was a lot of that in the past decade, and -- possibly a statistical anomaly, or just the edge of normal -- many of them fell within an eight-month period. That wasn't a great year.

Professionally, I was excessed from a position I held for more that ten years. Then I made it back, and I was let go again. I've bounced around the system since, including what I thought was a new permanent position, only to be let go, again, at the end of the term. Nothing that I did -- they loved me -- but they all play budget games. They love me when they're desperate, but when they catch their breath -- and I've done the hard part -- they'll look for someone with under two years experience who they can push around more and pay less.

But, yes, there were some positive things that I can focus on:

For starters, I'm still writing this blog, and still creating comics. These both started the decade before last, but the fact that they continue shows a great commitment (even if Wikipedia still won't list my twelve-year-old webcomic).

For another, nine of those comics made it into a Logic textbook at the University of Sweden. It's not the "sexiest" thing I've been published in, but I'd like to think college students are getting a chuckle (or a groan) from some of Mr. Keegan's exploits.

Also professionally related, there was a book proposal called Fueled By Coffee & Love that was to be filled with teacher stories. I honestly didn't know what that meant, and the guidelines were so open, I wasn't sure what, or how much, to write. After the first book came out and a sequel was proposed, I was probably one of the first to submit, and "My Teaching Journey" was accepted for publication. (Side note: it reminded me how much I had to owe Tracy S. for me still being a teacher.)

But fiction writing? That has been on the back burner for the past 20-plus years. Many excuses have been made. And the few times I sat down to type and finished something, nothing came of it. That changed with a chance meeting with Danielle Ackley-McPhail, a writer, editor, and owner or eSpec Books. We're just friends, acquaintances, really, who see each other a couple times per year at sci-fi conventions.

I started reading the company blog, and I saw their first flash fiction contest. I submitted something under a thousand words long (it could have been up to twice as long) and it won! Well, it co-won. Danielle told me she liked my story because it had the most "catty-ness" of all the entries (and that was the theme).

After that, I tried to enter as often as I could, usually writing in the final days of the month. Sadly, I discovered that a story that she liked was the only entry that month. It was a little deflating, but the positive feedback helped. One story, "Cyber Where?", Danielle called the best story I'd written. I was elated to hear that. It meant I was improving, growing. Going somewhere.

This brings me to the past year or so. I tried writing flash fiction for a handful of markets out there. Not of them would pay more that twenty dollars, if that much, for 1,000 words or less. So far, I haven't been able to crack any of them.

I kept at them because I couldn't focus enough to write a short story, putting 3,000 to 5,000 words together. I was thinking in terms of scenes, and not acts.

By the time December rolled around, I had a plan of attack. A few months ago, there was a Kickstarter by Zombies Need Brains for three new anthologies. I back it for all three ebooks. Part of the plan was to have slots available for submissions. The odds aren't great, given the size of the slush pile, but a good, well-written story makes a difference.

I had ideas for two of the three anthologies. It came down to the final days, but I finished the two short stories I'd planned on, and both came in over 4,000 words. I wasn't sure that the second one would be written, but both had bounced around in my head for over a month, so I knew the acts as well as the scenes, and it wasn't just a bunch of dialogue.

I won't know until next month if either is accepted or both are rejected, but I'm happy that I got through it.

And if that wasn't enough of a high note to end the year on, there is one final note, but I can't mention it. Not that it's hush-hush or anything (then again, maybe it is), but until I see something announced, I don't know for sure that it's definite. At least not right now. But good things are coming, if I can keep at them.

Wednesday, January 01, 2020

Happy New Year 2020

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

Happy New Year! May all your characters stay viable!

The funny thing about today's New Years Day comic is that it isn't the one I might have done a week ago. What would I have done? I have no idea. I forgot -- didn't write it down. But something came to me that I liked.

I've done binary jokes, along with other bases, and even a modulo (remainder) function. And factoring when it was interesting. (In this case, 101 wasn't so interesting, at least not as a new character.)

Every year, in the last week of December, I see a post or tweet from some math person I follow online with a list of "fun facts" about the number of the New Year. The "fact" is that there are so many of them, you can always find something.

For example: Let's say you wanted to find a bunch of consecutive numbers that add up to 2020. Then see if the answer to any of the following are whole numbers:

a + a + 1 = 2a + 1 = 2020
b + b + 1 + b + 2 = 3b + 3 = 2020
c + c + 1 + c + 2 + c + 3 = 4c + 6 = 2020
d + d + 1 + d + 2 + d + 3 + d + 4 = 5d + 10 = 2020

I had unnecessary notation when words are fine. You can see the progression. The coefficient increases by 1 and the constant is the next triangle number.

A quick check online yields the following:

402 + 403 + 404 + 405 + 406 = 2020
249 + 250 + 251 + 252 + 253 + 254 + 255 + 256 = 2020

Additionally, many numbers can be written as the the sum of two squares. Most can be written as the sum of three squares, and all can be written as a sum or difference of three squares. Moreover, every square is the sum of two triangular numbers, so that just expands the possibilities.

It looks great, but makes for stale comics. The formula I used today is incredibly arbitrary and created backward from the solution. It doesn't have any particular meaning.

As for the 20/20 vision jokes, along with the Barbara Walters gag, have been old for months now. Which is why I did one on Monday, and not today. Not going to toss it out just because it's old if I can find a way to use it.

In any case, thank you for being one of the blog readers. I appreciate the ones who take the time to come here and read the posts, instead of just looking at the comic on line. Even moreso the people who comment here or on social media. (Note: the social media comments may drive some traffic here but those comments will be lost in the bit-storm like tears in rain.)

Have a Happy New Year. Here's to hoping that there is at least 100 new comics before it's over.

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