Minstrel’s Log

You’d think it would be simple. Of course, at first, it is.

a = b

The same thing as saying b = a, it means (in simplified terms) that a and b are interchangeable, any place you could use one, you can use the other and get exactly the same result.

Okay, fine. but what happens when you get:

a=(sqrt(17)-2)/3

see, that’s nice and accurate, but just glancing at it, you don’t know about where it would come on the number line. So, of course, you write another line saying that it’s approximately equal to 0.707. Great, but how do you write that?

a ∼ 0.707
a ≈ 0.707
a ≅ 0.707

Well, the first one, ∼, according to the HTML specification means that it varies with, or is similar to. The Unicode standard just calls it the “tilde” operator. According to the Wikipedia article on Asymptotic Analysis (which refers to a couple of text books) it means that the functions are asymptotically equivalent - essentially that as they approach infinity, they scale in relatively the same manner. Of course, this would make no sense when declaring the numerical value of a variable.

Okay, the second one, ≈, is referred to by the HTML spec as meaning almost equal to, or asymptotic to. Somewhere on wikipedia it’s said to mean approximately equal to, and the Unicode standard says “almost equal to”.

The third one, ≅, has the HTML spec AND the Unicode standard call it “approximately equal to”, but wikipedia swears it’s “congruent to”.

That’s not even counting a couple of other Unicode symbols - including one that Unicode says is “asymptotically equal to”.

So which one do I use? I don’t want to end up saying that a is “congruent” to 0.707, but I’m not able to find a clear, established usage guide for them.

Grr.

Yes, it’s amazing! I’ve figured out a formula for subtraction!

Okay, well, actually, there are two of them, and they’re more like procedures, but the great thing is that they work!

Before I get into them, the question must be asked, Why? Why would one need a formula, or even any sort of strange procedure, just to subtract numbers? We learn that in Grade 1.

Sure we do, but then, after that, we learn about negative numbers, and everything become nasty.

It all has to do with the interaction of signs.

Addition, the first thing we do, is fairly simple - it’s just two numbers that you stick together, if they have the same sign, they keep their sign, if they have different signs, the result has the sign of the larger.

Multiplication and Division have no connection at all between magnitudes and signs, the signs interact on their own - if the signs are the same, the result is positive, if the signs are different, the result is negative.

In all three of these cases, the only signs you need to know about are the signs of the numbers themselves - its nice and simple.

Subtraction, however, is another story.

In its simplest form, when both numbers are positive, then if the second number is larger than the first, the answer is negative, otherwise the answer is positive.

However, if the first number is negative, and the second is positive, then the result will always be negative.

On the other hand, if the first number is positive, and the second is negative, then the result will always be positive.

Yet finally, if both numbers are negative, then if the first number is larger than the second, the result will be negative, and otherwise the answer will be positive.

It’s a dyslexic’s nightmare. Even if you get the sign correct, you have to make sure you didn’t subtract that negative number instead of adding it.

Which brings us back to the point - how do you subtract numbers without getting into the morass of if-or-else?

Simple:

If the signs are different, add the magnitudes* together, and take the sign of the first number.

Ex:

3-(-5) (signs are different)

3+5=8 (magnitudes)

answer: 8 (the first number is positive, therefore, the result is positive.)

If the signs are the same, find the difference of the magnitudes, and if the second number is larger, the sign is different than the two, otherwise it is the same.

Ex:

(-3)-(-5) (signs are the same)

5-3=2 (note that 3-5=-2, you can do it either way, we don’t care what sign we get - we figure out which sign we need in the next step)

answer: 2 (5, the second number, is larger than 3, the first, so our answer has a different sign than the two inputs - they were negative, so our answer is positive.)

Yarr!

* The magnitude is simply how far away it is from 0 - in either direction. Both 5 and -5 have the same magnitude: 5