Math Is Fun Forum

Hannibal lecter

Member

Registered: 2016-02-11

Posts: 392

Hi, is -∞ < 1 ?

Hello,
is -∞ < 1 ?

---

Wisdom is a tree which grows in the heart and fruits on the tongue

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,863

Re: Hi, is -∞ < 1 ?

Hi Hannibal lecter,

Yes. -∞ < 1

The list of integers is

-∞, -∞ + 1, -∞ + 2, -∞ + 3, .........., -3, -2, -1, 0, 1, 2, 3, ..........., ∞

---

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

Re: Hi, is -∞ < 1 ?

hi Hannibal lecter

The number line is usually shown running left to right with (say) 2 to the left of 3 etc.

The symbol < (which we read as "less than") actually means "is to the left of", so -4 < -3 etc.  As -∞ is the 'leftmost' place you could have on the line* -∞ < x for all x on the line.

* But as you cannot get to this place it is (strictly) not on the line.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

zetafunc

Moderator

Registered: 2014-05-21

Posts: 2,440

Website

Re: Hi, is -∞ < 1 ?

You have to be careful what you mean by the statement "

". What do the symbols

and

mean here?

is not a real number, but we can treat it like a number by adding it on to our usual collection of real numbers. That is, if

is the set of all real numbers, then you can 'add on' infinity and negative infinity by considering the extended real numbers,

. Then, not only can you define arithmetic with these infinities (and this has significant geometric implications -- see for instance stereographic projection, which you can read about), but you can even define an ordering relation on the extended real number line so that it makes sense to say something like

, for any

.

Hannibal lecter

Member

Registered: 2016-02-11

Posts: 392

Re: Hi, is -∞ < 1 ?

thanks you so much,

and is 1/-∞ < 1 too?

---

Wisdom is a tree which grows in the heart and fruits on the tongue

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

Re: Hi, is -∞ < 1 ?

The calculation 1/-∞ doesn't really exist as the rules of arithmetic fail for ∞ .

I suppose you could ask what happens to 1/x as x tends to -∞ .

eg 1/-1000 < 1

1/-1000000000 < 1

etc.

It looks like 1/x tends to zero from below (from the left) as x tends to -∞  and we know that 0 < 1.

So I suppose the answer to your question (ignoring the rules for a moment) is YES.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob