Consider the number \(n\) in the paragraph above. Can you find another irrational number \(m\) so that \(n+m\) is a rational number? What about an irrational number \(k\) such that \(n+k\) is irrational? Finally, is it possible to have a rational number \(\ell\) such that \(n+\ell\) is rational? Explain (remember that all rational numbers can be expressed as terminating or repeating decimals).
Section 8.1 Understanding Irrational Numbers
This section explores the concept of irrational numbers, which can be viewed in two ways: decimals that neither repeat nor terminate, or numbers that cannot be written as a fraction (with integer numerator and denominator).
Subsection 8.1.1 Examples of Irrational Numbers
Common examples include the number pi (\(\pi\)) and certain square roots like \(\sqrt{2}\text{.}\) While it is outside the scope of this course to show why \(pi\) is irrational (the easiest method requires a little bit of calculus, but it is definitely understandable to most 2nd year mathematics students) but we can indeed show that "roots" of integers, which we will define in more detail later, are irrational in almost all cases, and the cases where they are rational numbers only the cases where it is obvious.
Another example is the number \(n=12.0100100010001 \ldots \) where the number of zeros preceding the next 1 increases by one each time. Although there is a pattern here, there is no repeating series of decimal digits and we know by Section Section 6.3
Checkpoint 8.1.1.
The set of all rational and irrational numbers together form the set of real numbers, denoted by a blackboard bold R (\(\mathbb{R}\)). We can view the sets of numbers we have studied so far in the following Venn diagram. Note that irrational numbers are "negatively defined", as in they are numbers that do not have a certain property. So when we try and show a number is irrational, we usually use a proof by contradiction; that is, we will pretend that the number is rational and show that this leads to a logical fallacy.
INSERT PIC HERE
Subsection 8.1.2 TypesOfIrrationalNumbers
There are many ways to classify irrational numbers, but the most common is to group them according to whether they are solutions to a polynomial equation with integer coefficients, like \(2x^3-4x^2-5=0\text{.}\) We call numbers that are solutions to such equations algebraic numbers and real numbers that are not algebraic we call transcendental numbers. The set of algebraic numbers can be quite complex to understand, so we will focus our attention to a subset of those numbers.
Definition 8.1.2.
A real number \(r \in \mathbb{R}\) is algebraic if there is a polynomial with integer coefficients for which \(r\) is a solution to that polynomial. We call an algebraic number a simple root if it is a solution to a polynomial equation of the form \(x^n=a\) for some \(n \in \mathbb{n}\) and \(a \in \mathbb{Z}\text{.}\) We call these solutions \(n\)th roots.
Not every polynomial equation has solutions that are real numbers. For example, \(x^2+1=0\) has no real solutions, since in order for this equation to be true we need that \(x^2 < 0,\) and we know that squaring real numbers gives us a positive number. We could expand our number system to include solutions to equations such as these (called the complex numbers) but this is beyond the scope of the course.
Some famous transcendental numbers are \(\pi\text{,}\) which is the ratio of the circumference of any circle to its diameter, and \(e,\) which is a number that is very important in calculus; in short and imprecisely, it is a number related to exponential growth, and in fact the function \(e^x\) is the only non-zero type of function where the growth rate is always equal to the actual growth.
Subsection 8.1.3 Functions Briefly
In order to talk about simple roots, we need to understand a little bit about functions. A function is a rule that turns an input element into exactly one output element. For example, the rule "add one to a number" can be expressed in the following way:
\begin{equation*}
f(x)=x+1.
\end{equation*}
Here, we name the function "add 1 to a number" with the letter \(f\text{,}\) and \(x\) is the variable we’re using to describe the rule using algebra. So if we wanted to add 1 to, say, 5, then we would write \(f(5)=5+1=6\text{,}\) where we are substituting \(x=5\) in to our function \(f\text{.}\)
In the example above, we could input any real number we like into this function and we get exactly one answer, and this answer makes sense. We call all elements we are allowed to input into our function the domain of our function, and all possible outputs the \(range\) of the function. Note that sometimes a function has its domain specified when it is being defined.
Example 8.1.3.
Let’s say the rule we want as a function is "multiply a number by itself and add one". If we want to name this rule \(m\) we can write \(m(x)=x^2+1\text{.}\) Here, the function makes sense for any real number as an input, so its implied domain is all of \(\mathbb{R}\text{.}\) Noting that \(x^2\) is never negative, and only \(0\) when \(x=0,\) we can say that the smallest number our function can output is when \(x=0\) which makes \(m(0)=0^2+1=1\text{.}\) So our range is all real numbers greater than or equal to \(1\text{,}\) or in set notation \(\{x \in \mathbb{R} \ | \ x \geq 1 \}\text{.}\)
Checkpoint 8.1.4.
For the following rules, write an algebraic expression for a function. Are there any numbers that cannot be inputted into this function? If so, list them.
-
Multiply a number by 2
-
Square a number and then subtract 1
-
Add 3 to a number and then divide the result by 3
-
Divide 5 by the chosen number.
