# What does the term “undefined” actually mean?

I have read many articles on many sites and in many books to understand what undefined means? On some sites of Maths, I read that it could be any number. and on some sites, I read that it may be some undefined thing; and there are more definitions. But they all have clashes with each other that all are defining the term “undefined” in different ways. So which concept is right for the word “undefined” in criteria of Maths among the following five?

1-A number divided by zero may be any number (real or imaginary)

2-A number divided by zero may be an entity which is not defined yet.

3-Division of a number by zero does not make sense.

4-If $$x=a/0x=a/0$$, then no solution exists!

5-It may give a third type of number other than real or imaginary [I have not read this definition in any book or site but it’s my thought.]

This query popped up into my mind while my teacher was solving a question from my book, I am showing it to you along with my teacher’s work.

Q- Prove that the roots of the following equation are real.

$$x2−2x(m+1m)+3=0x^2-2x(m+\frac{1}{m})+3=0$$ where, $$mm$$ is any real number.

Teacher’s attempt:

For roots to be real,

$$b2−4ac>0b^2-4ac>0$$

$$⟹4(m2+1m2−1)>0\implies 4(m^2+\frac{1}{m^2}-1)>0$$

$$⟹4(m2+1m2−2+1)>0\implies 4(m^2+\frac{1}{m^2}-2+1)>0$$

$$⟹4[(m−1m)2+1]>0\implies 4[ (m-\frac{1}{m})^2+1 ]>0$$.

My teacher let us write that inequality is satisfied for all $$mm$$ belongs to real number however if $$m=0m=0$$, $$1m\frac{1}{m}$$ is undefined. So if “undefined” means that “a number divided by zero may be any real or imaginary number” so then I can confess only for real numbers that inequality is satisfied for all $$mm$$ belonging to real numbers and not for imaginary numbers since we can’t make sense of a statement like this $$i>0i>0$$ but if the term “undefined”, in Maths, is defined as “Senseless” or “something else not known” then I strongly apprehend that why my teacher let us write that Inequality is satisfied for all $$mm$$ belonging to real numbers?

Saying that 1 divided by 0 is undefined, does not mean that you can carry out the division and that the result is some strange entity with the property “undefined”, but simply that dividing 1 by 0 has no defined meaning. That is just like when you ask whether the number 1.9 is odd or even: That is not defined. Or when you ask what colour the number 7 has.