Welcome

I know a lot of people who are intelligent. They really are -- any way you look at it, they're sharp. If you get in an argument with them, they're very quick at picking up discrepancies and finding ingenious counter-arguments. They can navigate the maze of ideas with ease.

But they all gave up on mathematics at one time or another, and even a mention of the word mathematics is a cue for them to start heading for the exit.

Here is an example of this mathematics-phobia. People rarely make a statement about coke (the drink), then the same one about diet coke, then repeat the same thing for pepsi and dr. pepper. Unless they're comparing tastes, they usually just call them all soft drinks and talk about soft drinks in general. The terms soft drinks and soda are abstractions that everyone seems comfortable with.

When taken literally, the word "soft drink" doesn't make any sense. A smoothie is perhaps, literally, a soft drink. But everyone understands that the word is just a label and that when the word is mentioned, we don't try to imagine a hard drink that's been softened, we just know it's a label and we go straight to the thing that it refers to.

But when a number is called irrational, which simply means "not a ratio of two whole numbers", somehow this term becomes mind-numbing and an obstacle to understanding. Perhaps the cause is the great authority mathematics has -- after all, you can't argue with something that's been proven, and in mathematics everything is either proven, or so complicated that no one has proven it yet. So this sense of ultimate authority carries over to the mathematical jargon, which is ironic, because it's actually anything but. It's very human -- in fact, littered with expressions of astonishment at the newly found concepts. Consider such words as irrational (beyond reason!), transcendental (beyond the realm of the senses!), and imaginary. All of them were labels for concepts that were new at the time. If we were to come up with new names for these concepts now, we'd probably choose much more mundane words.

So maybe it's the great mathematical authority and intolerance to error that creates a stumbling block for most of us. Perhaps, and here is another likely reason -- it is the really inflexible rate at which things get taught in school -- students who are a little too fast get bored, the ones who are a little too slow get clobbered, and all are forced to remember as many answers as they need to pass the exam. The way they get even with mathematics is lose all interest in it and forget all about it the moment their grade doesn't depend on it.

But sooner or later, they get curious: what are all these mysterious terms, like complex and transcendental numbers, what's so natural about the natural logarithm anyway? And what is a logarithm? (it's the number of digits in a number). Popular books on mathematics and the nature of the universe will all mention famous, intriguing equations like e^iπ+1 = 0. These people look at these equations like they were Greek ornaments, and don't appreciate the conceptual beauty, even though they want to. This purpose of this blog is to allow them to do it.

My mission is to explain all the steps leading up to the equation above to a normal, intelligent, non-mathematical person in a way that they will understand. It won't be a "dash for the finish" -- there is no time restriction on this, no exam at the end. The scenery is beautiful along the way, and can be appreciated for what it is.

This blog should be read top-to-bottom. Since each post depends a little bit on the previous post, I'm backdating them to make them appear in the right order. If you are a fellow traveler, what I would appreciate most of all is letting me know if you get lost by leaving a comment. Once this blog is finished, I'll just leave it sitting here.

I will revise, reorder, and modify these posts in response to the suggestions I receive.

Arrows

Have you ever wondered, why is it when you multiply any number by itself, the result is a positive number? We all know that "minus times minus is plus", but why?

To answer, let’s talk about arrows. For example, +1 looks like this.

We can reverse, or negate, the arrow and make it go in the opposite direction. Negative numbers are arrows pointing left.

To add two numbers, we place them one after another, like our feet.

+3 is the same as 1 * 3, because * means repeated addition.

Here is another example:

Here we see that +1 added to –1 leads back to where we came from – the sum is an arrow of zero length.

I think it’s very interesting that we had this notion of direction – right or left, and when we got zero we lost it, so that the direction is now unclear.

Subtracting an arrow is really adding a reversed arrow. The above example shows adding +1 to –1, but how is it different from 1-1, i.e. subtracting 1 from 1? It’s not – we just have this word called subtraction that refers to addition of a negated arrow.

Back to the question: why is it when you multiply two negative numbers, you get a positive number? You know the rule, we all learned it in school, but could you show where that rule follows logically from?

When we negate an arrow, we turn it 180 degrees. You can confirm this by looking at the picture above. 1 and –1 are the same arrow, just pointing in the opposite direction. Negation is rotation.

Each of our arrows so far has had a direction and a length.

Above, we have an arrow of length 3 pointing left (180°), corresponding to the number -3, and an arrow of length 2 pointing in the default direction (0°), corresponding to number +2.

I assigned the angle of 0° to positive numbers, but his choice does not affect anything. I could have called positive numbers 90° and negative 270°. Alternatively, instead of assuming that the "full circle" is 360°, we could say that it's really 2, then positive numbers would correspond to 0°, and negative to 1°.

To multiply 3 (180°) by 2 (0°), we use repeated addition for the length (3 * 2), and add the angles.

3 (180°) * 2 (0°), which is

3*2(180°+0°), which is

6 (180°)

That’s an arrow of length 6 pointing in direction of 180°, which makes it a negative number. So multiplication is also rotation, plus repeated addition of lengths.

Similarly, if we multiply 2(180°) by 2(180°), we get 4(180° + 180°), which is the same as 4(360°), which is 4(0°) or +4. Why is 360° the same as 0°? Because you make a full circle and come back to where you were. 360° is the same as 720°, same as 1080°, same as 3600° – they all represent the same direction.

We got a glimpse of multiplication as rotation from arrow reversal – after all, it’s true that reversing something is rotating it by 180°.

Now we know why multiplying by –1 is the same as negating:

a * -1 = -a

That’s because we’ve just added 180° to the number’s angle, which makes it point in the opposite direction without any scaling.

The angles also explain the way the signs behave when we multiply numbers:

positive * positive = positive, because 0° + 0° = 0° = positive

negative * negative = positive, because 180° + 180° = 360° = 0° = positive

positive * negative = negative, because 0° + 180° = 180° = negative

negative * positive = negative, because 0° + 180° = 180° = negative

This covers all the combinations of all signs, and we see the addition of angles agrees with what we learned in school.

You may well wonder, 0° and 180° is good, but what about all the other angles, do they go unused? The answer is of course not, and that’s the topic for the next post.

Where arrows are rotated

Arrows correspond to numbers; positive numbers point right, negative point left, and they rotate each other when you multiply them. When we multiply two numbers, we add their angles and scale their magnitudes (arrow lengths).

How do we get other arrows, with angles other than 0° and 180°? First, nothing prevents us from constructing one directly; here's one: 4(45°). The picture of it is below:



This is an arrow of length 4. If we multiply this arrow by itself, we will get 16(90°). If we multiply that by itself again, we get 256(180°), which is -256.

So we found a number, an arrow, which, when multiplied by itself four times, gives -256. That's pretty amazing, because normally, when we multiply something by itself, we get a positive number, but here we multiplied something by itself twice, and we still got a negative number. If we multiply -256 by itself, we'll get 65536, and by then the interesting part is over -- all the numbers will be positive from then on.

A side on terminology: just like repeated addition has a word (multiplication), repeated multiplication is called exponentiation. Also, the way to say "X multiplied by itself four times" is "X to the power of 4".

What are all the arrows that, to the power of 4, give 256(180°)? One is 4(45°), that much we already know.

But there are others, in fact a total of four. That's because angles wrap around at 360. If we start, say, with an arrow whose angle is 135 degrees, after the first multiplication the angle becomes 270 = 135 + 135, then 405 = 135 + 135 + 135, then 540 = 135 + 135 + 135 + 135. But 540 is the same as 180, so after four rotations we arrive at the same result: 256(180°), but from a different starting point. Here is how you can find all four starting angles:

180 / 4 = 45
(180 + 360) / 4 = 135
(180 + 360 + 360) / 4 = 225
(180 + 360 + 360 + 360) / 4 = 315

Visually, they look like this:



So we see that arrows are more than "numbers"; they are more interesting, in fact. When you compute with arrows, you'll end up with a regular-looking number every once in a while, but only if the angle is 0 or 180. Arrows are to regular numbers are what a color monitor is to a black-and-white one: numbers only have two directions (black and white, left and right, negative and positive), but arrows have all the shades of gray in between.

Where multiplication is taken apart

The fact that positive and negative numbers are really right- and left-facing arrows, and that there is a whole continuum of arrows in between is truly beautiful. It's beautiful because it's true. For example, it's not an invention -- inventions can be different depending on who's inventing -- this continuum is really a discovery. If two people discover the same thing, they end up in full agreement despite their best efforts at being original, because there is something out there in the intangible space of ideas that is built in a very particular way, and it can be discovered by thinking.

Let's get back to the continuum between positive and negative numbers. Is it really there? Can we make it visible?

When you add two numbers, like 2 + 2, you fully expect that one big addition can be split into two smaller ones, for example 2 + 1 + 1. And if you wanted to look at things even more finely, you could say 2 + 1/2 + 1/2 + 1/2 + 1/2. This ability to do addition in whatever size pieces you like gives it a physical feel: you know that if you keep adding little bits of one quantity to another, eventually you'll add the whole thing.

But multiplying, say, a by -1, involves a jump -- there are no intermediate steps to be seen, and all of a sudden you're at -a; a long way from where you started. What's the continuous process that multiplies a by -1 in little steps? Does it exist? If we started multiplying a by -1, then stopped the process halfway, and looked at the result, where would we be?

Yes, the process exists, and it's visualized quite easily. When you multiply a by -1, you are adding 180° to a's arrows' angle. The arrow peels off the positive axis, rotates all the way to the diametrically opposite point, and stops there. If you stop halfway between a(0°) and a(180°), you're not nowhere, you're at a(90°), or a*i, as the illustration below shows:


Another way of saying "half way in the process of multiplying by -1" is to write -1^1/2, which is read as "minus one to the power 1/2". Using power 1/2, we gave -1 a multiplying power half as strong as it originally had. -1 raised to the power 1/2 multiplies other numbers half as much as it used to, in terms of this continuous process. And if we now wanted to restart the process from the midpoint, we could multiply the number we had so far by another -1^1/2, and we'd end up at -a.

But what if we stopped the process even earlier? We could stop halfway between a and ai, that would be a * -1^1/4 or a * i ^1/2. $We\; could\; keep\; going\; with\; ever-smaller\; powers.\; The\; smaller\; the\; power,\; the\; less\; powerful\; the\; multiplication.\; In\; fact,\; if\; we\; stopped\; the\; multiplication\; right\; at\; the\; start,\; we\text{'}d\; be\; at\; a\; *\; -1$⁰$.\; So\; you\; see,$$-1$⁰$$ is a powerless multiplier. In fact, any number to the power zero is a powerless multiplier, and is equal to 1.

What can be subdivided once, can be subdivided many times. We can now do any amount of multiplication. For example, we can perform half a multiplication and save the rest for later:
a * b^1/2
We can multiply twice:
a * b².
We can multiply 0.35 times, and then another 0.65 times, and the the original:
a * b = a * b^0.35 * b^0.65.
We can multiply zero times, which does nothing:
a * b⁰ = a.
We can multiply three times, and then undo one multiplication:
a * b³ * b^-1 = a * b²

If you like taxonomy, you'll be happy. There is a whole zoo of labels for these operations.

• b squared refers to b²
  
• square root of b refers to b^1/2, because b^1/2 * b^1/2 = b.
• division refers to b^-1
• exponentiation refers to all of these -- any time you change the power of multiplication.
  
• the word power refers to x in b^x
• the word base refers to b in b^x
• the word logarithm refers to power when it's not known. for example when you try to figure out which x satisfies b^x = c, given b and c, x is the logarithm of c.
• i refers to -1^1/2

Multiplication and beyond

Let's add some numbers, like this:

a + b + c + ...

Now let's start chopping things off the end of this list:

a + b +

Then

a +

What remains if we chop off the a? The original expression was really

0 + a + b + c + ...

So if we remove all the pieces, we get 0. And it makes sense, if we never add anything, we're left with nothing.

OK, let's now consider something very similar:

a * b * c * ...

We know that if we start chopping things off, at some point there will be nothing left. Is that nothing zero? Surprisingly, no, because our expression was really this:

1 * a * b * c * ...

So at the end, what's left is 1. In fact, 1 is to multiplication what 0 is to addition. Adding zero does the same as multiplying by 1 -- nothing. If you want an equation, here is one: a*1 = a+0.

We can write 1 * a * a using the shortcut notation as 1 * a², which is read as "a to the power of two". And as we chop things off the end, we first have 1 * a¹, then 1 * a⁰ -- which is another way at arriving at the fact that a⁰ = 1.

Note how each next series of operations squeezes more work into the same amount of space.
First, we had a+a+a+..., and we replaced that with the shortcut a*3 in order to understand the process better, to be able to think about things that are common to all such summations.
Then, we started having having multiplications like a*a*a*..., where each of the * signs stands the summation (a+a+a+...).
We replaced that with the shortcut a³.
The next step would be a^aa..., where the placing of one number above another stands for several multiplications (a*a*a*... etc).

An aside about keyboards

Multiplication in equal steps can be found on the modern piano keyboard.

On the piano, the sound frequency, or pitch, of each key can be found by multiplying the frequency of the previous key by 2^1/12. After 12 notes, or 12 multiplication steps, we increase the sound frequency by a factor of 2, which is an octave.

If we hold down one key and then press keys to the right of it, we'll hear the following intervals (an interval is the ratio of two frequencies).

2^1/12= 1.05946309... (the first few digits given for this and other numbers)
2^2/12= 1.12246205...
2^3/12= 1.18920712...
2^4/12= 1.25992105...
2^5/12= 1.33483985...
2^6/12= 1.41421356...
2^7/12= 1.49830708...
2^8/12= 1.58740105...
2^9/12= 1.68179283...
2^10/12= 1.78179744...
2^11/12= 1.88774863...
2^12/12= 2

The fact that each two subsequent notes have the same ratio of frequencies is referred to as "equal temperament". It means that you can take any melody and start playing it some number of keys to the left or to the right, and it will sound the same, just transposed.

Note that all the ratios listed above are complicated-looking numbers. Before equal temperament, the ancients preferred simpler frequency ratios such as 3/2 or 4/3 -- ratios you can easily get on a violin by holding down the string in the right place. Problem is, you can only tune the keyboard once, then you have to play it. Suppose your octave consists of 4 notes:

1
4/3
3/2
2

Now, holding down the middle two notes produces the interval (3/2) / (4/3), or 9/8. So since you're getting this interval anyway, you should add a key to your octave tuned at 9/8 with respect to the first key:

1
9/8
4/3
3/2
2

But now, the keys 4/3 and 9/8 give you the interval 32/27 (4/3 divided by 9/8), so you better add that one as well. But if you do, a new interval 256/243 will pop up right away, and so on. With pure ratios, you can never add the right number of keys so that you don't get out-of-tune-sounding intervals somewhere in the middle. It's OK to have an out-of-tune-sounding interval in the middle, because you can compose your melody around it. But the moment you try to shift your melody right or left, you'll hit that interval and it will sound wrong.

Interestingly, when Bach wrote his famous Well-tempered clavier, the keyboard he used didn't sound like the one we have now, because the keys were tuned differently. The well-tempered tuning system was sort of a compromise between the ratio-based and the modern equal-tempered tuning.

The actual digits in the numbers above are not interesting. Given the first few digits of a number such as 1.41421356..., we actually know less than if given the real thing: 2^1/2. But somehow we feel better when given the actual digits of the number. It's true, we do know the answer is somewhere around 1.4, but the other digits are just not worth remembering. If we had eight fingers on each hand, the digits would have been different anyway.

Subdividing

Let's think about a line with notches. What I have in mind is something like this:



I have placed the middle notch somewhat randomly. If we label the left side as 0, and the right as 1000, the notch would correspond to the number 263:



Now, the labels 0 and 1000 are arbitrary. If I had labeled the left side 0, and the right 1, the middle notch would be 0.263 -- about 25% of the way from left to right.

That notch and its location aren't changing. What's changing is the way we describe it -- the labels are. And in fact, these labels only make sense relative to one another. There is no more information in the numbers 0, 263, 1000 than in the numbers 0, 0.263, 1. They only make sense as a way of describing the relationship of that middle point with respect to the whole.

Imagine that we have a template with 10 slots that we can place over the interval. The slots are labeled 0 to 9. When we place it over our line, we find that our notch is located in slot labeled 2.



OK good, now we zoom into that box, and where is our notch? It's somewhere inside. We apply the template again, and the notch is now in the slot labeled 6:



If we repeat this again, we'll see that the next number is 3, at which point we'll get the digits 2,6,3 as the address of our notch. Here, we stopped at 3 digits, but if this was some kind of number from nature, we could keep applying smaller and smaller templates to get a more and more accurate value for the notch. Perhaps at some point we'd realize that we can't make a template fine enough to improve the accuracy of our estimate. Then we'd be forced to stop.

We could also use a template with just 2 slots. It would look like this:



If we apply the same subdivision technique using this template, we'll find the address of our notch to be 0, 1, 0, 0, 0, 0, 0, 1, 1, 1. The first few steps of this process are shown below:



Just like 263 is the address of the notch when using the template with 10 slots, 0100000111 is the address of the notch when using the template with two slots. If we had labeled the two slots A,B instead of 0,1, we'd get ABAAAAABBB. It's the same notch, except its address is specified using a different rule.

Templates of size 2 are used in computers, because they are simpler: at each step, all you have to do is see whether the number is to the left or right of the middle, which is one decision.

With a template of size 10, we go left-to-right, looking to see if the number is in the slot we're in, and in the process above it took us 3 steps to find the first digit, followed by 7 steps for the second, and 4 steps for the final digit -- a total of 14 steps. With a template of size 2, it took us only 10 steps -- an improvement. Given two ways of doing things, one of which is simpler and faster, we go for that one.

In the post above, I used magic numbers, like 1.41421356... which people call "square root of two", and write as 2^1/2. The reason for the "..." is that no matter how long you apply the template, you keep getting digits out of it, so we are forced to stop at some point.

How was this number, 2^1/2, computed? I'd like to demonstrate. We can use the subdivision method to do it. Only of course we'll use subdivision by 2, because it's both simpler and faster. We'll call the number X. What do we know about X? It's definitely greater than 1, because 1*1 = 1. It's definitely less than 2, because 2*2=4, and we want X such that X*X=2. So our interval is between 1 and 2. We pick the middle, 3/2.

3/2 * 3/2 = 9/4, which is more than 2. So X is smaller than 3/2, and the first digit is 0.

Now the interval is narrowed down to 1 to 3/2. The middle is 5/4.
5/4 * 5/4 = 25/16, which turns out to be less than 2. So X is greater than 5/4, and the second digit is 1.

I'm obviously not going to continue by hand -- we have computers for that.But you can probably see how you can get any number of digits out of this process. The first ten bits of information are 0, 1, 1, 0, 1, 0, 1, 0, 0, 0. What do these mean? The same thing as 1.41... -- just an approximate address for the notch, nothing more.