Mathematics

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“Most calculators will either return an error or state that 1.÷.0 is undefined. However, some graphing calculators will evaluate (1.÷.0)² to ∞.” [Wikipedia’s Division By Zero article, with no further explanation]

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“An off-by-one error is a logic error involving the discrete equivalent of a boundary condition. This problem could arise when a programmer makes mistakes such as using ‘is less than or equal to’ where ‘is less than’ should have been used, or fails to take into account that a sequence starts at 0 rather than 1.

A fencepost error is a specific type of off-by-one error. The following problem illustrates it: ‘If you build a straight fence 30 meters long with posts spaced 3 meters apart, how many posts do you need?’ The naïve answer of 10 is wrong. The fence has 10 sections, but 11 posts. The reverse error occurs when the number of posts is known and the number of sections is assumed to be the same. The actual number of sections is one less than the number of posts. Fencepost errors can also occur in units other than length. For example, the Time Pyramid, consisting of 120 blocks placed at 10 year intervals between blocks, is scheduled to take 1190 (not 1200) years to build, from the installation of the first block to the last.” ^[2]

[Pannenkoek] “An especially good example is how the 1800’s are the 19^th century, and the 1900’s are the 20^th century, etc.”

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“The Law of Large Numbers:  The average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.” ^[3]

“The Law of Truly Large Numbers:  With a sample size large enough, any outrageous thing is likely to happen.” ^[4]

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“In Euclidean geometry, a point is a primitive notion, meaning that it can’t be defined in terms of previously defined objects, but rather only defined in terms of some axioms it must satisfy ― in particular, that it’s zero-dimensional. A common interpretation is that a point is a unique location (as opposed to an object) in Euclidean space.” ^[5]

[Pannenkoek] “What about nothingness? Isn’t that also zero-dimensional? What makes a point more than nothing?”

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“By convention, 0 and 1 are considered neither prime nor composite.” ^[6]

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“The irrational numbers are those that cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being ‘incommensurable’, meaning that there’s no length, no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.

Among irrational numbers are π, e, the golden ratio, and √2. In fact, all square roots of natural numbers, other than of perfect squares, are irrational. The decimal expansion of irrational numbers neither terminates nor ends with a repeating sequence. Conversely, a decimal expansion that terminates or repeats must be a rational number.” ^[7]

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“The golden ratio (φ) is an irrational number equal to 1.618034….

Two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.

I remember, as a kid, noticing and thinking about how there’s some pair of numbers ―  around 0.6 and 1.6 ― that are reciprocals exactly 1 apart. That’s perhaps the simplest way to think about the golden ratio ― the number that’s one greater than its reciprocal.

Interestingly, the negative inverse of the golden ratio has this same property of being one greater than its reciprocal.

Due to its frequent appearance in geometry, the golden ratio has been studied by mathematicians since the time of ancient Greece. In the 16^th century, German mathematician Simon Jacob discovered that consecutive Fibonacci numbers converge to the golden ratio.” ^[8]

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“In 1545, the mathematician Gerolamo Cardano was working on one particular problem when something unusual happened. He reached a point where, in order to progress, he’d have to multiply 5.+.√-15 by 5.–.√-15.

So first he cross-multiplied, which left him with

(5 × 5) – (5 × √-15) + (5 × √-15) – (√-15)²

The first term obviously equals 25. But for the next three terms, it seems as though, in order to proceed, he’d have to take the square root of a negative number ― something which at the time was expressly forbidden. In other words, it appeared he’d reached a dead end.

However, Cardano had an idea. He reasoned that since the second term and the third term are the same, and since the second term is being subtracted while the third term is being added, then perhaps ― without having to figure out what the square root of negative fifteen is ― we can just assume that the two terms cancel each other out.

And similarly, for the final term, since we’re immediately just squaring the square root of -15, perhaps we can just assume that those cancel each other out, and it equals -15.

Then, overall, we’re left with (25).–.(-15), which equals 40.

So in order to arrive at the solution, Cardano had to pass through a region which was, so to speak, out of the familiar mental theatre of the mathematicians of the time. And this passage perturbed the people of the 16^th century.

Cardano described the arithmetic he’d performed as ‘so subtle that it’s useless’. I take this to be his way of saying that he hasn’t yet gotten this strange creature ― the square root of negative 15, or complex numbers in general ― into his mathematical repertoire. Yet he had an understanding that mathematicians had better begin to deal with it, despite the fact that it may be ‘subtle’ and ‘useless’. This is the first recorded instance in which someone complains bitterly and yet uses such a number.”^i[9]

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“In the real number system, y = √x is undefined for negative x.

Back in elementary school, kids would sometimes try taking the square root of a negative number on a calculator, only to receive an error message. The kids who’d done it unintentionally were often confused and surprised and excited, while the kids who did it intentionally usually treated it almost like a magic trick.

When I first found out about it, it blew my mind ― almost like I’d learned how to escape the number system through a secret back exit.

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The number i (the imaginary unit) is the solution to the equation x.=.√–1.

It extends the real number system ℝ to the complex number system ℂ.

The transition from the real number line to the complex plane strikes me as a quintessential example of the notion of jumping out of the system, or transcendence. I still more or less think of i as the secret back exit from the real number line.

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The Greek mathematician Hero of Alexandria, born in 10 AD, is noted as the first to have conceived of imaginary numbers. (Talk about ahead of your time!) Rafael Bombelli was the first to set down the rules for multiplication of complex numbers in 1572. The concept had appeared in print earlier, though, for instance in work by Gerolamo Cardano. At the time, imaginary numbers, as well as negative numbers, were poorly understood and regarded by some as fictitious or useless, much as 0 once was. The use of imaginary numbers wasn’t widely accepted until the work of Euler and Gauss in the 1700’s.

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There are two complex square roots of –1, i and –i.”^i[10]

Pannenkoek once gave me this riddle he thought of: “What number has an additive inverse equal to its multiplicative inverse?”

“Now i and –i aren’t the same number. They just have a property in common. The only trouble is that it’s the property which defines them! We have to choose one of them (it doesn’t matter which) and call it ‘i’. In fact, there’s no way of telling them apart! So for all we know, we could’ve been calling the wrong one ‘i’ for all these centuries, and it would’ve made no difference!” [Hofstadter]

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Have you ever wondered about iⁱ?

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“The number e (Euler’s number) is an irrational number equal to 2.71828….

It’s the base of the natural logarithm ― the unique number whose natural logarithm is equal to 1.

The discovery of the constant is credited to Jacob Bernoulli in 1683, who attempted to find the value of the following expression:

When I first saw this, it struck me as something that would equal either 1 or ∞, not something in between.

Bernoulli had been studying a question about compound interest: ‘An account starts with $1.00 and pays 100% interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be $2.00. What happens if the interest is computed and credited more frequently during the year?’

If the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial $1 is multiplied by 1.5 twice, yielding $1.00.×.1.5².=.$2.25 at the end of the year. Compounding quarterly yields $1.00.×.1.25⁴.=.$2.441…, and compounding monthly yields $1.00.×.(1.+.1/12)¹².=.$2.613…. If there are n compounding intervals, the value at the end of the year will be $1.00.×.(1.+.1/n)ⁿ. Bernoulli noticed that this sequence approaches a limit with larger n and, thus, smaller compounding intervals. Compounding weekly (n.=.52) yields $2.692…, while compounding daily (n.=.365) yields $2.715…, just two cents more. With continuous compounding, the account value will reach $2.7182818…, which equals e.

The number e can also be calculated as the sum of the following infinite series:

The function e^x is the unique nontrivial function which is its own derivative and therefore its own antiderivative as well.

This is also true of e^x multiplied by any constant.

The number e also has applications in probability theory. Suppose that a gambler plays a slot machine that pays out with a probability of 1 in n and plays it n times. Then, for large n (such as a million), the probability that the gambler will lose every bet is approximately 1/e.

The number e is of eminent importance, alongside 0, 1, π, and i. All five of these numbers play recurring roles across mathematics.” ^[12]

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“Euler’s identity is considered to be an exemplar of deep mathematical beauty, as it shows a profound connection between the most fundamental numbers in mathematics. Three of the basic arithmetic operations each occur exactly once: addition, multiplication, and exponentiation. It also links five fundamental mathematical constants: 0, 1, π, e, and i. Including equality, the equation contains nine unique basic mathematical concepts.

It’s considered by many to be the most beautiful equation in all of mathematics.”^.[13]

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“Just like there’s only one way to order a single object, there’s only one way to order nothing.”^i[14]

[Pannenkoek] “Even though 1! and 0! both equal 1, these two 1’s feel completely different to me: one feels like it has substance to it, since it comes from 1 itself; whereas the other feels empty, since it arises from nothingness.”

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“Here’s a problem: Find a smooth curve that connects the points (n,.f(n)) given by f(n).=.n! at the non-negative integer values for n.’

The (most popular and useful) solution to this problem is known as the gamma function. Or rather, the gamma function describes this curve but shifted one unit to the right. (I won’t try to explain or even include how the gamma function is actually defined. It’s too far beyond me.) So for any non-negative integer n,

The gamma function extends the factorial function so that it’s defined for all complex numbers except the non-positive integers.” ^[15]

I always love when the fundamental constants pop up serendipitously. But what might be even more interesting is that the curve connecting the ‘normal’ factorial values has its minimum at ½, exactly in the middle of 0 and 1.

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[Pannenkoek] “We all learned in school how to manipulate math equations, which is by performing the same operation on both sides. For example, we can add 5 to both sides of the equation. Doing this causes the value of both sides to change, but that’s okay because the equality itself still holds, and the equality itself is the fundamental attribute that we want to preserve. Often when we simplify an equation, our goal is to isolate one variable. Consequently, we perform operations on both sides of the equation that aim to cancel out any operations that apply to that variable. So while on one side of the equation, the newly added operations are canceling out with existing operations; on the other side of the equation, the newly added operations are building up. And so we end up simplifying one side of the equation at the expense of the other.

But now consider the case where you have an expression instead of an equation, but you still want to simplify it. This is a much less talked-about situation and is rarely tested in school. In this case, we can’t perform operations on both sides of the equation, because there is no equation to work with. If we try performing a non-trivial operation on just the expression, then it would change the value of the expression, which is bad because this time the fundamental attribute that we want to preserve is the value of the expression. So it might seem like there’s no way to do anything, but there is. We’re still allowed to perform trivial operations on the expression that have no net effect. For example, we could add and subtract 5 from the expression, which wouldn’t change the overall value.

Thus, manipulating equations and expressions looks like this:

However, our manipulation of the expression may seem pointless here, because on our very next step we could cancel out the +5 and -5, and we’d end up back where we started. In certain circumstances, though, we can use this technique to make progress.”

“Let’s say our goal is to simplify the expression i^i.

i^i
e^(ln(i^i))
e^(i*ln(i))
e^(i*π/2*i)
e^(-π/2)

And so even though our initial step of taking the natural logarithm and raising e to that power resulted in additional complexity, it ultimately gave us more tools to use to simplify the expression. More specifically, i^i was at a local minimum in terms of complexity, and we had to temporarily add complexity to reach a lower local minimum.”^i[16]

[Pannenkoek] “I’ve rarely heard anyone directly compare simplifying an equation to simplifying an expression. While simplifying an equation is the more common and thought about situation, simplifying an expression is rarer and more difficult. Simplifying an expression in a productive manner is almost like a secret technique that instructors will do only in special circumstances as a last resort.

And adding and subtracting the same number reminds me of virtual particle-antiparticle pairs appearing in the universe.”

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“In this game, the goal is to reduce a given starting number down to 0. The rules are:

• you can only use natural numbers (not including 0) to do it
• you can only use +, –, ×, ÷, ^
• you must apply each operation to (what used to be) the starting number

Let’s start with the number 10. We subtract 10, and we get 0. Pretty simple. How about ¾? We multiply it by 4, then subtract 3, and we get 0. How about √2? We square it, then subtract 2, and we get 0. How about i? We square it, then add 1, and we get 0. What about (√2.+.√3)? This one’s a little more complicated. We square it and multiply it out, which gives us 2.+.2√2√3.+.3_=_5.+.2√2√3. From here, we subtract 5, then square it again, then subtract 24, and we get 0.

All the numbers that satisfy this game belong to the family known as algebraic numbers.

What about the number e? Can we reduce it to 0 using the rules of our game?

Well, e has been around for about 400 years, and for the longest time, no one knew.

It turns out that it can’t be done. This is because e is a transcendental number.” ^[17]

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COUNTABLE AND UNCOUNTABLE INFINITES
(click here for the subsection)

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“A transcendental number is a real number (or complex number) that isn’t an algebraic number. While algebraic numbers are a root (a solution) of a nonzero polynomial equation with integer coefficients, transcendental numbers are not.

Though only a few classes of transcendental numbers are known, in part because it can be extremely difficult to show that a given number is transcendental, transcendental numbers aren’t rare. In fact, almost all real and complex numbers are transcendental.

This is because the real numbers and the complex numbers are both uncountable sets, and so if we remove the countable set of algebraic numbers, then we’re left with an uncountable set that consists of all the transcendental numbers. And since uncountable sets are (much) larger than countable sets, the set of transcendental numbers is (much) larger than the set of algebraic numbers.

All real transcendental numbers are irrational, but not all irrational numbers are transcendental.

The best-known transcendental numbers are e and π.” ^[19]

With π, it feels as if God found Himself with no choice but to pick a small real number at random.

Inspired by an equation in an XKCD comic, I once wrote this little poem:
On a scale from one to ten
I love you this much: (½)(e^π – π)
My love for you is transcendental

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“0 is a perfectly good number, but it’s also a dangerous number, and lot of things can go horribly wrong. Because it’s a slightly more unusual, nuanced number, you have to be a little more careful with how you handle it. And so there are some things you can’t do with it ― you can’t divide something by 0, and you can’t have 0⁰.

First of all, many people ask, why can’t you just say something divided by 0 is ∞? It’s because ∞ isn’t a number, and it can’t be treated like one. But if I misbehaved and said 1.÷.0.=.∞, then I’d just as easily get 2.÷.0.=.∞. And from here, it’s easy to see how we run into the problem that 1 seems to be equal to 2.

But what if we take the limit as x gets really close to 0?

Doesn’t this equal ∞? And therefore can’t we conclude that 1.÷.0.=.∞? I’ll show you why we can’t do that.

As x gets closer and closer to 0, y does get bigger and bigger and tend to ∞. But this is only true if you’re approaching 0 from the right. If you come in from the left, it goes racing down to –∞. You can’t get much more different than that. Maybe y goes all the way up and wraps the around the entire universe and then comes back up at the bottom. As far as I’m concerned, though, if you come in from one direction, you get one answer, and if you come in from the other direction, you get a different answer. There’s no one limit as you get closer and closer to dividing by 0, and that’s why it’s said to be undefined.” ^[19]

There’s something really beautiful about the concept of division by 0. It’s probably my favorite example of the notion of transcendence, or jumping out of the system.

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“People often get very annoyed about 0⁰. This is because when you’ve got anything at all to the power of 0, you always say it equals 1. And when you’ve got 0 to the power of anything, you always say it equals 0. So what happens when these collide?

People argue quite different things depending on what they need. In my experience, more often than not, they argue that 0⁰ = 1. Still, I can think of others who’ve insisted that it equals 0, which is of course equally insane.

Let’s consider the function y = x⁰. What happens is, as you approach 0 from the right side, you hit 1. And as you approach 0 from the left side, you also hit 1.

Since in both cases, the function has the same limit, we should be able to call it 1, right? It’s slightly more complicated than that, though. This is only the real number line, where you can only approach something from two directions. If you consider the complex plane, though, there become many ways to approach 0. And these approaches give different limits. So it starts to fall apart when you go to the complex plane.” ^[19]

Unfortunately, I couldn’t find a visual showing how it falls apart on the complex plane.

Also, I wonder why he didn’t discuss the function y.=.0^x instead or in addition. Is it related to how this function is undefined when x is negative?

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“Russell’s Paradox: _Does the set of all sets that don’t contain themselves contain itself?

According to naïve set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself. And if R is a member of itself, then its definition dictates that it must not contain itself. This contradiction is Russell’s paradox.

Symbolically:_.Let.R.=.{x.|.x.∉.x}. Then.R.∈.R.⇔.R.∉.R.” ^[20]

What about the set of all sets that are members of themselves? Does that set contain itself?

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“Many-valued logic is a system of logic in which there are more than two truth values. Classical two-valued logic may be extended to n-valued logic for n greater than 2. Those popular include three-valued logic (which may accept the values ‘true’, ‘false’, and ‘unknown’) and infinite-valued logic.” ^[21]

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“The following equation of propositional logic may be pointed out: F.∧.X.=.T, where F is False, T is True, and X is a logical value which accepts one of the two states: True (T) or False (F). Within the frame of classical propositional logic, there’s no such value of X for which the requirements of this equation are satisfied.

An analogy to this may be sought in number theory, in which equations of the following type exist: x².=.–1; x.=.√–1. This equation has no solution within the frame of the theory of real numbers, since no real number exists that when squared gives a result of –1. This leads to extending the range of the real numbers and transitioning to the complex numbers through the introduction of the imaginary unit i.

The same approach may be used and a solution found by introducing an imaginary logical variable in the following way: F.∧.i.=.T. (Similarly: T ∨ ¬i = F.) State i and its negation ¬i are part of the set of possible imaginary states. This system is known as complex-valued logic.” ^[22]

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_{one imagining of}
The Mathematical Landscape

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It’s surprisingly difficult to come up with a definition that captures the essence of mathematics as a whole. These two are my favorites:

“mathematics: _the study of quantity, structure, space, and change” ^[24]

“mathematics: _the study of all possible patterns” ^[25]

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“In his novel, Contact, Carl Sagan suggested that the universe could have been designed, and that its designer could have encoded a message in a transcendental number such as π or e.” ^[26]

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“The rational numbers are like the stars in the night sky, and the irrational numbers are like the blackness.” ^[27]

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Citations

1. posted in the Overheard at CMU Facebook group
2. Wikipedia’s Off-By-One Error article
3. Wikipedia’s Law of Large Numbers article
4. Wikipedia’s Law of Truly Large Numbers article
5. Wikipedia’s Point (Geometry) article
6. leanprover.github.io/logic_and_proof/introduction
7. Wikipedia’s Irrational Number article
8. Wikipedia’s Golden Ratio article
9. Barry Mazur & Numberphile
10. (various sources)
11. Wikipedia’s Imaginary Unit article
12. Wikipedia’s e (Mathematical Constant) article
13. Wikipedia’s Euler’s Identity article
14. (various sources)
15. Wikipedia’s Gamma Function article
16. Simon Pampena & Numberphile
17. Matt Parker
18. The Divine Serpent from Yu-Gi-Oh
19. Wikipedia’s Transcendental Number article
20. Wikipedia’s Russell’s Paradox article
21. Wikipedia’s Many-Valued Logic article
22. V. Sgurev
23. (original source unknown)
24. Wikipedia’s Mathematics article
25. W. W. Sawyer
26. Brian Holtz
27. Cantor

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