Math Is Fun Forum

mathland

Member

Registered: 2021-03-25

Posts: 444

Slope

Let m = slope

For a vertical line, m = undefined.

Why is this the case?

For a horizontal line, m = 0.

Why is this the case?

Back in my school days, we were told to simply
memorize this fact of m. Most textbooks do not give a clear reason
for an undefined and zero slope.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

Re: Slope

Let's look at the horizontal case first as it's easy to see.

On the line y = a, take any two points, (b,a) and (c,a)

The gradient is the difference in y over the difference in x so we have

m = (a-a)/c-b) = 0/(c-b) = 0

For a vertical line, say, x = d, choose two points, (d,e) and (d,f)

m = (f-e)/(d-d) = (f-e) / 0

What do you get when you divide a number by zero?

Another way of answering this is to ask how many zeros must we add together to reach a total of (f-e).

It's just not a question we can answer. 

You could ask what is the limit of (f-e)/Δx as Δx tends to zero?

It tends to ∞.

In studying ∞, mathematicians have realised it's best not to regard ∞ as a number.  It doesn't obey the usual rules of numbers:

such as ∞ + ∞ = ∞

That's why the word 'undefined' is used for this gradient.

For a moment let's suspend the 'not a number' idea and play with the equation of the line anyway.

Suppose I write m = ∞ and make the equation y = ∞x + c

We know that (d,3) lies on the line so y = ∞d + c = 3

So c = 3 - ∞d.  How are you going to work that out?

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

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,837

Re: Slope

See the link: Slope of a straight line.

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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.

mathland

Member

Registered: 2021-03-25

Posts: 444

Re: Slope

Let's look at the horizontal case first as it's easy to see.

On the line y = a, take any two points, (b,a) and (c,a)

The gradient is the difference in y over the difference in x so we have

m = (a-a)/c-b) = 0/(c-b) = 0

For a vertical line, say, x = d, choose two points, (d,e) and (d,f)

m = (f-e)/(d-d) = (f-e) / 0

What do you get when you divide a number by zero?

Another way of answering this is to ask how many zeros must we add together to reach a total of (f-e).

It's just not a question we can answer. 

You could ask what is the limit of (f-e)/Δx as Δx tends to zero?

It tends to ∞.

In studying ∞, mathematicians have realised it's best not to regard ∞ as a number.  It doesn't obey the usual rules of numbers:

such as ∞ + ∞ = ∞

That's why the word 'undefined' is used for this gradient.

For a moment let's suspend the 'not a number' idea and play with the equation of the line anyway.

Suppose I write m = ∞ and make the equation y = ∞x + c

We know that (d,3) lies on the line so y = ∞d + c = 3

So c = 3 - ∞d.  How are you going to work that out?

Bob

Wow! You took time out to type all of this for me. Thank you very much.

mathland

Member

Registered: 2021-03-25

Posts: 444

Re: Slope

See the link: Slope of a straight line.

Thank you for the link. This is material I learned back in the 80s and 90s. It's good to revisit this stuff from time to time.