Why Is It so Difficult to Reason About the Prime Numbers?

In the history of mathematics, the prime numbers are ancient as the invention of the natural numbers. To review a little, the natural numbers are the simple positive integers 1, 2, 3, ... —a series without end.

In this series of the natural numbers, when we are given some number, it is easy to tell what the next number is; just add 1 to the given number! (This explains why it is funny to ask elementary school kids for the biggest number he/she can think about, and then confront his answer by telling him/her to add 1 to the number he mentioned.)

On the other hand, the prime numbers are not so easy to visualize; their distribution is as if they were randomly distributed.

Statue of Euclid.

Let's review for a moment the standard definition of a prime number: a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. For example, the primes less than 100 are:

    2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

The number 1 is by definition not a prime number. Note how unusual is the beginning of this series: the first prime is even --the only even number that is prime; all the other following primes are odd. But not every odd number is prime. To the number 3 follows 5; to 5 follows 7, but to 7 does not follow the number 9.

The simple task of finding the prime number that follows another prime impossible. With the prime numbers it happens that given any prime number, there is no way to compute the next prime that follows—or precede—the given one.

However, the primes are very special in one aspect: every natural number is either a prime or it is the product of some primes. This is a far-reaching assertion because it surreptitiously states that the natural numbers are in one way or another all made of prime numbers.

Euclid (ca. 300 BC), and Eratosthenes (ca. 200 BC) were the first two Greek mathematicians to work extensively with the properties of the prime numbers. Euclid proved that the primes were infinite (no way to find the last prime), and Eratosthenes devised a method to sieve out the primes from the series of the natural numbers.

Since then, the primes numbers are a constant headache for number theorists and mathematicians in general.

Historical Quotations About Prime Numbers

However, mathematics is not a private property of mathematicians and philosophers of science. What about the public? They also have something to say. What does the layman have to say about the prime numbers? Do politicians have something to add to this arduous field of mathematics (they always have something to say!)?

Let's see how some imaginary examples of how some famous people would have reasoned about if the number 9 is prime or not.

Christopher Columbus: “3 is prime, 5 is prime, 7 is prime. According to some ancient manuscripts, 9 is not a prime number, but beyond the distant horizon of the oceans, in the New World that I am going to discover, there are surely lots of them.”

Dimitri Mendeleev: “3 is a prime, 5 is a prime, and 7 is a prime, but 9 is a noble prime that deserves a separate row in the periodic table of the primes.”

Charlie Chaplin: “3 is a prime, 5 is a prime, 7 is a prime, 9 is the next prime after 8.”

John F. Kennedy: “1 is not a prime number and 9 is not a prime number? Then ask not what the primes can do for you, ask what you can do for the primes.”

Stephen Hawking: “2, 3, 5 and 7 are prime numbers: 9 is not prime, but in the black holes, past beyond the event horizon, anything can happen.”

George W. Bush: “3 is prime, 5 is prime, 7 is prime, and 9 … well, any odd number can be prime as long as it is not 9.”


For more examples about how easily is to be misled when reasoning about the series of the prime numbers, then see the article:

Historical Quotations About Prime Numbers

Complex Numbers: How Complex Are They?

The "history" of the integer numbers is a simple one. From the natural numbers 1, 2, 3, ... we move to the positive integers 0, 1, 2, 3, ... then we add the negative integers ... -3, -2, -1, 0, 1, 2, 3, ....

Then we escalate to fractions and decimals and non-terminating decimals (although historically was not in this order). The ladder continues to the irrational numbers and to the algebraic and transcendental numbers. This is the "world" or universe of the real numbers.

Every real number has a specific place on the Number Line.

But mathematics is a product of our minds so this "universe" or field can be further expanded to suit our needs.

The next heaven after the real numbers field is the imaginary numbers; numbers that in combination with the reals make the complex numbers field.

But how complex are the complex numbers? Curiously, they are as simple as the "preceding" ones.

The negative numbers haunted the mathematicians and philosophers for many centuries; no wonder the misnomer "negative". Even the number zero took a long time before it was accepted in the kingdom of the mathematics (in Europe, where it was later accepted.) It was unacceptable to count "backward".

The imaginary numbers suffered the same fate: no wonder the epithet of "imaginary". The square root of minus 1 was impossible to compute because no number times itself is equal to minus 1.

Take a read at this article: "The imaginary numbers are not so imaginary and the complex Numbers are not so complex" and you will see how easily and beautifully the complex numbers emerge out of the real numbers.

The 10 Top Myths About the Infinite

Myth 1: An infinite split by one half is no longer infinite

Let's us take the set of all natural numbers, i.e., the numbers we use to count, like 1, 2, 3, ... We will represent this set by the symbol Z. Each one of the natural numbers is either odd or even; the odd numbers being 1, 3, 5, ... and the even numbers 2, 4, 6, ... Note that the numbers we call even are those divisible by 2. Hence every natural number is either divisible by two or not. Those that are not divisible by 2 are the odd numbers.

Natural numbers = odd numbers + even number

Z = {1, 2, 3, 4, 5, 6, 7, ...} = {1, 3, 5, ...} + {2, 4, 6, ...}

The even numbers are infinite because there is no end to this series. Same with the set of odd numbers: there is no way to find and end to this series. So the infinite set of all natural numbers is the sum of two infinite series; the series of the odd numbers plus the set of the even numbers.

If you take away the infinite set of the even numbers from the infinite set of the natural numbers you are left with an infinitude of odd numbers.

{1, 3, 5, ...} = {1, 2, 3, 4, 5, 6, 7, ...} - {2, 4, 6, ...}

To a similar behavior we are faced if we take away the set of the odd numbers from the set Z.

Therefore, it is not necessarily true that if we split an infinitude in a half, the two parts are no longer infinite.

Myth 2: One infinite added to another infinite is a greater infinite

This one is the opposite of the above myth.

Myth 3: If we increasingly take away infinite elements from an infinite set, eventually, the remaining set is no longer infinite

This is not the same as Myth 1: there we were linearly taking away one integer for each one left.

Suppose that to the set of all natural numbers Z we remove numbers from it using this pattern:

1. Leave the number 1, but take away the next 2. We are left with {1, 4, 5, 6, ...}
2. Leave the number 4, but take away the next 5. We are left with {1, 4, 10, 11, ...}
3. Leave the number 10, but take away the next 11. We are left with {1, 4, 10, 22, ...}
4. Repeat the pattern over and over again.

Note that with each step we are taking more an more elements away from the original set of the natural numbers. The separation between the remaining integers is wider and wider. If we repeat this process indefinitely, we'll be progressively removing more an more elements. This is far from the first example above where we were removing even or odd numbers only, because in this schema we are removing from both types of numbers.

However, no matter how far we go or how many integers we remove, the remaining set will be always infinite because although the steps are infinite, the elements to be removed are always finite.

Myth 4: There are more fractions than natural numbers

This assertion might appear to be against our intuition because we assume that since every natural number can be expressed as a fraction, like

1 = 1/1,
2= 2/1 = 2/2,
3 = 3/1 = 6/2 = 9/3 ...

we can conclude that there are more ways of expressing fractions than the numbers themselves. However, note that in the pyramidal scheme above, we can count the fractions as follows:

1/1 = is the first

2/1 = is the second, 2/2 is the third

3/1 = is the fourth, 6/2 is the fifth, 9/3 is the sixth,

Hence, no integral fraction can escape our counting scheme. Therefore, the integral fractions are countable which means that there are not more integral fractions than natural numbers.

Myth 5: An infinitude of elements multiplied by another infinitude is always a grater infinitude

Myth 6: Since every fraction can be converted to a decimal then there are as my decimals as fractions

Myth 7: The segment of the line from 0 to 1 contains double the points as the segment from 0 to 1/2

Myth 8: The number of grains of sand is infinite.

This is a classic myth. Probably all of us, at some stage of our live, had think that the grains of sands are infinite.

Archimedes is the first documented one to tackle down the needed mathematics to show that it is impossible the for the sand to be infinite. Strictly speaking, what he showed was that we can count how many grains can a universe hold, no matter how big it is.

At his time the observable universe was up to Saturn, so what he did was to compute how many grains can fill a sphere the size of the orbit of Saturn. The mathematics needed to arrive at his conclusion were simple, but ingenuous extensions he devised for the arithmetic of his time was an enormous contribution.

You can download his all-time famous book The Sand Reckoner here.

Myth 9: If a vase is infinitely long, then it must have an infinite capacity

This is a beautiful one ...

Gabriel's Horn


Myth 10: If there were infinite universes out there, in some of them, or at least in one, should be an exact copy of our planet Earth