Give a real number which is not rational.
Answers
In mathematics, the irrational numbers are all the real numbers which are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. ... For example, the decimal representation of π starts with 3.14159, but no finite number of digits can represent π exactly, nor does it repeat.
Answer:
Others have provided the answer, but my take on the following is excerpted from my book on Group Theory, Glimpses of Symmetry, Chapter 4 – Rationality and Reality:
So far in this Chapter we have expanded our definition of number from the Natural Numbers and Integers (ℕ and ℤ) to the Rational Numbers (ℚ). We did this by asking a question about whether numbers and multiplication formed a Group, a question firmly within the scope of this book. We asked for an answer to the equation:
What is the inverse under multiplication of 4?
In this section we are going to expand our definition of number further. Rather than having a specific Group Theoretical justification, I want to introduce a new set of numbers which we will need to use in later Chapters. These are the Real Numbers, denoted by ℝ. We’ll approach these by considering the solution to another question:
x^2 = 2, what is x? [7]
The trite response is of course that the answer is √2 [8]. But this is just shorthand for “number which when squared equals 2” so we have perhaps not advanced our cause so very far.
With a calculator (or spreadsheet) to hand, we can get an answer like:
√2 = 1.4142135623
However that equals sign is not entirely correct, what we have is an estimate of the value of √2 to ten decimal places [9]. It can be rigorously demonstrated that the actual value of √2 extends to an infinite number of decimal places, or more prosaically the numbers to the right of the decimal point never stop [10].
Can we use fractions to represent √2? This is equivalent to saying that we can find two Natural Numbers a and b such that:
a / b = √2
or equivalently:
√2 ∈ ℚ .
There are various ways to show that there are no such numbers a and b. Each requires rather more Mathematics than I am looking to include in the main text. Interested readers are directed to the notes section [11]. For those not intrepid enough to explore the proof provided, please take it on trust that we have come across a number which is not part of ℚ. This means that we need to extend our definition of number to capture this new exhibit.
We defined elements of ℚ as fractions of whole numbers (excluding zero as a denominator). How can we define this new set? Well we have come across one way already; we can define the set of Real Numbers, ℝ, as the set of all decimals expansions. This includes numbers like √2, as well as even more complicated numbers such as π ( 3.14159265358979323846…) and e (2.71828182845904523536…).
We can see that any member of ℚ is a member of ℝ by noting how accustomed we are to forming equations such as:
½ = 0.5
¼ = 0.25
⅓ = 0.33333… where the 3s go on to infinity
The last equation points to a way to discriminate between Real Numbers that are members of ℚ and those that are not.
The former will have decimal expansions that repeat themselves at some point.
Either ½ = 0.5000… or ½ = 0.499999… depending how pedantic you want to be
1 / 7 = 0.142857142857… where 142857 repeats itself for ever
The latter, like √2, will have decimal expansions that never settle down in this manner.
[…]
As mentioned in Chapter 2, Mathematicians use a special symbol to show that something is a subset of another set (a set wholly contained in another set), this is ⊂. What we have shown so far with respect to numbers is:
ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ
It is of course tempting to consider whether this sequence can be extended further. One version of what might lie beyond ℝ is explored in Chapter 7.
Step-by-step explanation: