Rational, Irrational, and Real Numbers
Episode #2 of the course Foundations of mathematics by John Robin
Welcome back to the foundations of mathematics. Now that we’ve been introduced to the natural numbers and integers, it’s time to learn about the further complexities of numbers.
You’re probably familiar with fractions:
3/4, 5/7, 23/17
Fractions are often a source of stress in math because of how difficult the rules can be for adding and multiplying them.
Let’s take a fresh look at fractions, though, in light of what we learned yesterday.
Recall that integers, Z, are all the negative numbers that go all the way to the left (to “negative infinity”) joined to all the natural numbers, N, that extend forever to the right (positive infinity):
…, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …
The next level of number is built out of integers. These are called rational numbers. We form them by taking integers and making every possible fraction out of them—i.e., a/b, where a is an integer and b is an integer (but b is not equal to zero). This is the familiar fraction we know from school. The number a is called the numerator and b the denominator.
The possibilities are infinite. You can pick any two numbers from …, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, … and put them together to make a rational number. For example:
-7/8, 1/5, -4/3, 3, -1, 0
You might have wondered how -1 and 3 and 0 are rational numbers. Aren’t they integers? This is just an area where the integers overlap with the rational numbers (think of this as analogous to how the natural numbers 0, 1, 2, 3, 4, 5, … overlap with the integers …, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …).
In the case of our example, we get -1, 3, and 0 whenever we have -1/1, 3/1, and 0/1. The denominator 1 has no effect on the fraction, so we omit it, leaving us with the integer.
Because of this property, you will find all the integers in the rational numbers. These are simply all numbers we form from …, -5/1, -4/1, -3/1, -2/1, -1/1, 0/1, 1/1, 2/1, 3/1, 4/1, 5/1, ….
The rational numbers have the symbol Q. Like with Z for integers, Q entered usage because an Italian mathematician, Giuseppe Peano, first coined this symbol in the year 1895 from the word “quoziente,” which means “quotient.”
There are many numbers we can make with rational numbers. We can make any fraction. But are these all the possible numbers? The answer is no, but let me show you why, by way of an example.
You might have seen this notation: √2.
This is the square root of 2. Finding the square root of a number means finding two numbers that are equal and, when you multiply them together, create the original number. For example, √4 is 2 because 2×2 = 4, i.e., two equal numbers that multiply together to make 4 are 2.
But √2 has no fraction answer. The proof for this requires some algebra. In fact, if you recall our prime numbers from yesterday, then it can also be proven that there are no rational square roots for any prime. It also has been proven that there are infinitely many primes. This means that there are infinitely many numbers that can’t be represented by fractions! We call these numbers irrational numbers. This name comes from the Greeks because they believed every number should be a fraction, so it was simply irrational to them to talk about numbers as though they weren’t.
We call the complete collection of numbers (i.e., every rational, as well as irrational, number) real numbers. They have the symbol R.
You can think of the real numbers as every possible decimal number. This includes all the rational numbers—i.e., 4, 3/5, 0.6783, and -86 are all decimal numbers. If we include all the irrational numbers, we can represent them with decimals that never terminate. For example 0.5784151727272… is a real number. (Note that when we use “…” after a decimal number, it means there are more digits on the right beyond where we truncated.)
With this foundation, we can now turn from numbers to some of the important things we can do with numbers. Tomorrow, we will start with the first of these: factoring.
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Proof That There Are Infinitely Many Primes
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