Numb and Number Trivia Quiz

NOTE - I strongly suggest you start reading the information from question 4 and then work both ways. The smallest three sets are more easily understood from the perspective of the larger, but more intuitively grasped, ones (and I have also written the info that way, so the first few paragraphs reference later ones).

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Mersenne primes are a special kind of prime number: They are just one less than a power of two. The smallest Mersenne prime is 3 (2^2-1), then the next are 7, 31, 127, 8191, etc. It can easily be proven that the exponent in the power of 2 has to be prime for the result of the calculation to be prime (if the exponent is composite, a very simple formula exists for finding a factor), but the opposite is not true: Not every prime exponent yields a Mersenne prime, however. The smallest example for this is 2^11-1 = 2047, which is 23 * 89.

Mersenne primes are used extensively in mathematical research, especially when it comes to finding new large primes as they are very easy to construct and represent in a computer (their binary representation is just a long string of 1s) and their special properties make some commonly used primality tests faster to execute. For this reason, the largest known prime at any one time has almost always been a Mersenne prime in the computer age. In 2018, a Mersenne prime with almost 25 million digits has been discovered, using the exponent 82,589,933. It was the 51st Mersenne prime to be found.

It is unknown whether there are infinitely many Mersenne primes, although research so far does suggest this is most likely true.

Oh, and just because Math is often beautiful: Remember perfect numbers from question 3? Each perfect number corresponds to a Mersenne prime - if 2^p-1 is a Mersenne prime, (2^p-1)*2^(p-1) is perfect - and this is also the only way to get a perfect number. So at any one time, there are exactly as many known Mersenne primes as there are perfect numbers.

You have almost certainly heard of prime numbers before - they are numbers that have exactly one proper divisor (the 1) or, to use the more common definition, are evenly divisible only by 1 and themselves. (The second definition is, however, slightly problematic, as it could suggest that 1 is prime, which it is not.)

It can easily be proven that there are infinitely many primes by assuming the opposite: Suppose there were only finitely many primes, there is a largest one among them. You could now multiply them all together and get a really large number. Now if you add 1 to that number (or subtract 1 from it, it doesn't matter), you have a number that is clearly not divisible by any of those smaller primes - which makes it a much larger prime than the largest prime. This way, you have proven that the assumption cannot be true - the existence of a largest prime gives you the means to construct a larger prime, which contradicts your initial assumption. Thus, the assumption must be false and there are infinitely many primes.

Now, if you read in the recommended order, it should be clear why I had to choose deficient in question 3 - all primes yield the sum of 1 when adding up their proper dividers, so they are all deficient. It was the only way I could keep the subset order intact!

Deficient numbers are rarely talked about as they are a relatively uninteresting subset of the natural numbers, but they relate to two interesting related concepts, namely perfect and abundant numbers.

If you write down all proper divisors of a number and then add them up, you can either wind up with less than the original number (deficient), exactly the original number (perfect) or more than the original (abundant). Note that proper divisors do include the 1, but never the number itself.

As an example for each type, 10 is deficient (1+2+5=8, which is less than 10), 6 is perfect (1+2+3=6) and 12 is abundant (1+2+3+4+6=16, which is more than 12).

It is well known and easy to prove just by providing examples that there are both odd and even deficient and abundant numbers and, in fact, infinitely many of each kind. However, so far, no odd perfect number has been found and neither can it be proven one cannot exist. It is also not known whether there are infinitely many perfect numbers or if there is a largest one. However, they are extremely rare.

Now you might ask, why I chose deficient numbers over their more interesting counterparts? Move on up to question 2 to find out!

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Natural numbers are the kind we most often encounter in daily life, representing things that can be counted. Each natural number identifies a positive quantity of whole things - one, two, three, et cetera. Even in their simplicity, natural numbers can get unbelievably large - a googol, for example, is 10 multiplied by itself 100 times (10 to the 100th power) and even that number is rather small. A famous *really* big number is Graham's number - a number for which the universe would be way too small to write it out.

If you add or multiply two natural numbers or raise one to the power of another (with the exception of the undefined 0 to the power of 0), you will always wind up with another natural number. This is not true for subtraction or division, however.

For math geeks: Even though natural numbers are the simplest type of numbers, they are also the only set which has two different possible meanings in mathematics: Most mathematicians agree that 0 is a natural number (and this is also the meaning standardized by ISO), while for some purposes, it is actually better to exclude zero, so this set is also called "natural numbers". This does not really make a difference in practice, as the name is rarely written out, rather using the symbol: a capital N with the central backslash line doubled. If the set with a zero is intended, a subscript (or superscript - both are allowed) "0" is added to this letter.

If you want to see how to use natural numbers to make other, bigger, number sets, keep reading forward from here - if you would rather want to look at some interesting subsets of the natural numbers, go up towards question 1! (and regardless which way you chose, come back here and do the other one afterwards :) )

Natural numbers, while easily understood, have several deficiencies. The biggest one of these is that you cannot always subtract one number from another and get another natural number. The solution for this is negative numbers - effectively "debt". If I give you three dollars and then you give me five, I owe you two dollars for us to be even again, so one can say that, in this scenario, I own negative two dollars.

In fact, money lending was among the earliest application of negative numbers - lenders and merchants had to process payments to and from individual customers and represent the balance. This was made much easier and less error-prone by replacing "in my favor" and "in their favor" by always looking at it from one side (which is, almost invariably, the customer side) and using a negative number.

While whole numbers always yield a whole number on addition, subtraction and multiplication, they still do not do so on division, and unlike natural numbers, they also don't always support exponentiation: 2 to the power of -2 is not a whole number. Let's take another step forward.

(Nitpick alert #1: Most people would equate integers and whole numbers, but in maths, this is not always true. Some of those who prefer "natural numbers" as not including zero use "whole numbers" for the natural numbers including zero. So "integer" is strictly the correct term.)

Rational numbers are, simply speaking, fractions. This is once again a rather intuitive concept when you look at many real life objects: If you slice through the middle of an apple, you have two things that together make a whole apple and, obviously we use the term "half an apple" for that.

All rational numbers can be obtained by dividing one integer by another integer (except zero, which cannot be divided by, but including 1). Now you might say what happens if I divide a rational number by another rational number? The answer is nothing special: You just get another rational number.

Rational numbers can also be written as a decimal fraction such as 0.27 or 0.333333333333, although one needs to be aware that a decimal representation with a finite number of digits is not always the exact value as a fraction - 1/3 is 0.3333333333333333... with infinitely many threes.

All decimal representations of fractions are periodic, but the period can be far longer than one digit: 1/7, for example, is 0.14285714285714... with a repeating group of six digits. At most, this length can be one fewer than the denominator of the fraction, so for example, 1/8191 has a period length of at most 8190. (It could be shorter, however).

All natural world measurements are understood to be in rational numbers, because measurements are inherently imprecise - while they can be extremely accurate, they are never perfect.

Rational numbers thus always allow the four basic arithmetic operations, but, again, exponentiation will not always yield a rational number: 2 to the power of 0.5 is the square root of 2, which is not rational.

Algebraic real numbers (I'll tell you in question 9 why I wrote it this way) are the next step up from rational numbers - they include square roots, cubic roots, any higher roots you may be able to think of as well as combinations thereof. Any number that is the solution to a polynomial, regardless of how high the exponents or how many terms it has, is algebraic.

So, you found a number that satisfies x^196723-48 x^77135+19283746 x^13=88136294863986? Congratulations, it's algebraic. It's even algebraic if I replace those coefficients with other algebraic numbers and put in non-zero coefficients for the other 196720 terms that have an exponent of less than 196723.

Remember that real polynomials must have natural numbers as exponents (you can however relatively easily convert any rational exponents to meet that criterion - irrational ones, however, won't work).

However, while you may think of these roots of all kinds as the same as "irrational numbers", they're only a small part of those. We'll take a look.

Real numbers are probably as far as you've ever ventured into number space unless you've taken some university-level math courses. Pretty much everything that can be measured, calculated or expressed in the real world can be expressed with a real number - even that pesky ratio between the circumference and the diameter of a circle we call "pi" or the base of the natural logarithm we call "e".

Both of these latter numbers are what we call transcendental - they are not algebraic. Yet you can use all arithmetic operations on them as you can with any of the previous types of numbers. Also, real numbers are still sortable - you can always state which of two is greater than the other (unless, of course, they're equal).

Real numbers make up all possible and imaginable one-dimensional numbers, and, in fact, they are vastly more numerous than any of the previous sets.

One of the really weird parts of math is the concept of infinity. So far, we have seen only countable infinities - and, while one can be a strict subset of another, they are still, from an infinite view, the same size.

Let's take the primes and the natural numbers. 2 is the first prime, 3 is the second, 5 is the third and so on. Due to the fact that there are infinitely many primes, each natural number will get its own prime this way. You can also map the integers to natural numbers this way (0, 1, -1, 2, -2, 3, -3...) and do the same for positive fractions (1/1, 1/2, 2/2, 2/1, 1/3, 2/3, 3/3, 3/2... it doesn't matter that 1/1 and 2/2 yield the same result here). For algebraic numbers, it's less intuitive, but it still works.

Not so for real numbers including the transcendentals. Regardless of what you try, you can't find such a mapping. In fact, it can be proven there can't be one. They are infinitely more infinite than all the other numbers before them.

And yet, in spite of their completeness, real numbers still can't solve sqrt(-1).

We'll need something better for that. Next question!

If we want to solve the square root of negative 1, we have to look beyond real numbers. In reality, this just makes no sense because the square root essentially means "find the side of a square that has area x" and there is no such thing as negative area. But math can imagine even impossible things such as a negative area, and thus, the square root of -1 is called the imaginary unit, written as i.

When you use i in arithmetics, it behaves just like the x in a conventional equation:

(4 + 3i) - (7 - 2i) = -3 + i.
(4 + 3i) * (7 - 2i) = 28 + 21i - 8i - 6iČ = 28 + 13i - 6iČ. But iČ is -1, so we get 34 + 13i. Brilliant!

In fact, we can now almost do everything we want. We can add, subtract, multiply, divide and even raise a complex number to the power of another complex number and we'll get another complex number. We can even do all the fancy infinitesimal stuff like differentiation and integration.

Except we can't now say whether a number is bigger or smaller than another. 4i is neither bigger nor smaller than 3 and we can't even say 4 > 3 any more. Oh well.

And, on top of this theoretical limit, it's getting counterintuitive. You can represent these numbers as vectors (arrows) on a two-dimensional drawing, but how do you multiply arrows or raise one to the power of another? Beats me.

(Nitpick alert #2: Remember I said I had to write "algebraic real number" in question 7? That's because the qualifier "algebraic" does not prevent a number from being complex. 1+i is a perfectly viable algebraic number, but it is not real. In 99% of cases, when your math teacher talks of algebraic numbers and they've not explicitly mentioned complex before, however, they mean the real kind...)

Complex numbers sound about as... complex... as they can get, right? Well, for practical purposes, even theoretical mathematicians rarely need more. (That's a sentence!) However, it does get better: hypercomplex.

Hypercomplex numbers are not a single system, but various expansions of the complex numbers into dimensions higher than two (with appropriately even less intuitive representation).

It's possible to set up a number system that has not one "i" but rather three of them, called i, j and k. Each of them squared is -1, but they are not the same and they work in the way that i multiplied by j gives k, j by k gives i and k by i gives j (and ijk is also -1). However, in this system (called quaternions - four-dimensional vectors), multiplication is no longer commutative: ij = k, but ji = -k.

Still with me? (I won't be mad if you leave here... it does get WEIRD now and I'll be throwing around some technical terms without explaining them). If we take the next from quaternions to octonions (eight dimensions) via Cayley-Dickson construction (just one of MANY ways to make hypercomplex numbers), we can't even rely on associativity: (ab)c is not the same as a(bc). If we do it yet once more, for the 16-dimensional sedenions, we can even find two nonzero numbers that, if multiplied together, give zero.

Does this all sound too theoretical to be relevant? Well, yes, for almost all purposes in life you won't need Mersennes, deficient numbers or complex ones and in daily life, integers and the occasional rational will suffice. Even the vast majority of scientists won't ever touch hypercomplex ones. However, the theory behind some advanced particle physics makes use of hypercomplex number spaces, so maybe one of those highly theoretical constructs is hiding one of the secrets of the universe. You never know!