Visualization of Irrational numbers

What is the proper definition of an irrational number? Will it be correct to define it always as "a number having a root as a factor"? Is it necessary for a number to have a root as a factor for being called an irrational number? How can I exactly visualize an irrational number?

• would you explain which "root" $\pi$ has as a factor? – dEmigOd Sep 7 '17 at 5:53
• Exactly. That's why I am asking for a definition. – user167573 Sep 7 '17 at 5:55
• The proper definition of an irrational number is that it is an element of $\Bbb R\setminus \Bbb Q$, that is to say in words a real number which is not a rational number. Remember that a rational number is a real number of the form $\frac{a}{b}$ where $a$ and $b$ are both integers and $b\neq 0$. The set of irrational numbers includes (among many other things) numbers like $\sqrt{2},\sqrt{3},\pi,e,0.1011011101111011111\dots$ and others. – JMoravitz Sep 7 '17 at 5:55

A number $r \in \mathbb{R}$ which could not be presented as a fraction $\frac{p}{q}$, with $p \in \mathbb{Z}$ and $q \in \mathbb{N}_+$.

• And how do I visualize it? – user167573 Sep 7 '17 at 5:58
• And suppose, given a number pi, how do I actually prove it cannot be expressed in p/q form? – user167573 Sep 7 '17 at 5:59
• @user167573 How do you visualize the rational numbers? – Charles Sep 7 '17 at 6:00
• – Charles Sep 7 '17 at 6:01
• @user167573 Proving that a number is irrational is rather hard, and must basically be done one number at a time. Some times you're lucky and can prove a bunch of them simultaneously (the proof that $\sqrt2$ is irrational works for any root of any prime), but I believe it's unknown whether, say, $\pi^e$ is irrational. – Arthur Sep 7 '17 at 7:12

An irrational number is defined as any real number which is not a rational number: that is, any real number which is not equal to any fraction $\frac{a}{b}$ where $a$ and $b$ are integers.

In particular, an irrational number doesn't need to have a formula involving a radical, or any of the other familiar "signs of irrationality" you may be used to seeing. In fact, most irrational numbers don't have any formula at all: there's no way to write down a finite expression for them.

So, how can you visualize an irrational number? Well, the real question is: how can you visualize a real number? There are many ways you might think about a real number. Geometrically, you could think of a real number as an arbitrary point on a number line. Or you could think of a real number as an infinite decimal expansion. An irrational number is just a real number which happens to not be rational. And actually, most real numbers aren't rational--it's a big coincidence if a random point you choice on the number line happens to be exactly equal to a fraction of two integers.

A number is irrational if it is not rational. That's the definition.

If you want the definition using symbols, then a number is irrational if it is an element of $\mathbb R$ and it is not an element of $\mathbb Q$.

Is it necessary for a number to have a root as a factor for being called an irrational number?

What do you even mean by "having a root as a factor"?

How can I exactly visualize an irrational number?

What do you mean by that? How do you visualize any number? How do you visualize "$1$"?

As you insist on visualization here is something close to that using the concept of length in plane geometry. Take two (finite length) line segments lying in a plane. I assume the concept of whole numbers. ($1,2,3,\ldots$ etc)

Suppose, there is a third line segment of smaller length than these two such that both the bigger segments are measurable as whole number times the third segment (for example, the first one could be 37 times the third segment and the second could be 14 times the third exactly). In this case the ratio of these two lengths is rational number.

On the contrary, if any choice of third line segment (however small) that makes the first segments length a whole number times the third fails to do the same for the second segment, now we have an irrational as the ratio of these two lengths.

Construction of examples: Take a rectangle whose breadth is 1 cm, and the length is a variable from among 2cm, 3cm, 4cm etc. Then the length of the diagonal of all these rectangles when expressed in centi metres will be an irrational number.