# Why mathematicians defined slope?

• B
parshyaa
• Is it to show rate of change mathematically and geometically as the tan(theta) of a line, and to apply it to calculus , physics etc.

## Answers and Replies

Homework Helper
When the architect wants to tell the carpenter how steep to make the roof, it is useful to have a way to do so.

parshyaa
I knew this
When the architect wants to tell the carpenter how steep to make the roof, it is useful to have a way to do so.
that

parshyaa
Why architect didn't told them to incline the roof at 30° or 60° etc and told them the length of roof
When the architect wants to tell the carpenter how steep to make the roof, it is useful to have a way to do so.

Why architect didn't told them to incline the roof at 30° or 60° etc and told them the length of roof
He enjoys smooth structures.

Homework Helper
Why architect didn't told them to incline the roof at 30° or 60° etc and told them the length of roof
Because the carpenter has a framing square, not a protractor.

parshyaa
Okk then you mean that they tell them the ratio of raise to far like 3:1 or 2:1 and they use this ratio to make the roof, but then 800cm/400cm and 40cm/20cm will also be 2:1, then how they will decide the lengths
Because the carpenter has a framing square, not a protractor.
you

Homework Helper
Okk then you mean that they tell them the ratio of raise to far like 3:1 or 2:1 and they use this ratio to make the roof, but then 800cm/400cm and 40cm/20cm will also be 2:1, then how they will decide the lengths
What is it that you are really trying to ask?

Slope is what it is. Whether we choose to describe it as "rise over horizontal run", "rise over diagonal run", "angle from the horizontal", "angle from the vertical", "percent grade" does not change it. But at the end of the day we still want to come up with an agreement between the architect and the carpenter on how to describe it so that the carpenter builds what the architect intends.

None of which has much to do with the length of the rafters except that we may also need to come up with a way to communicate that. Cubits, rods, feet, inches, meters, centimers, ... And a way to estimate that (e.g. Pythagoras theorem).

Mentor
Okk then you mean that they tell them the ratio of raise to far like 3:1 or 2:1 and they use this ratio to make the roof, but then 800cm/400cm and 40cm/20cm will also be 2:1, then how they will decide the lengths
The Theorem of Pythagoras has been around for more than 2000 years. If the base and altitude of a right triangle are known, it's easy to compute the hypotenuse.

• parshyaa
parshyaa
"To make the roof" means how inclined it must be , if we knew the ratio we can tell hime and get the angle which architect wanted to make, as earlier he was not having protractor
The Theorem of Pythagoras has been around for more than 2000 years. If the base and altitude of a right triangle are known, it's easy to compute the hypotenuse.

Homework Helper
"To make the roof" means how inclined it must be
So tell us. How inclined must it be?

Well, if slope was not defined how it was, how would you describe f'(a) where f(x) is a differentiable function and a belongs to its domain?

parshyaa
So tell us. How inclined must it be?
You said that slope is used to tell how steeper is line or roof , you said that carpenter were not having protractor so that they can inclined them just by saying the angle , therefore architects defined slope to get the [approximate] idea of steepness. So I think that they tell them the rise and far ratio by telling them lengths of rise and far(as per the tan of the angle calculated by them) . With your opinion I think that architect may knew trigonometry at that time. so basically I think that they have introduced slope more importantly to display the rate of change graphically and mathematically. And steepness of roof or road may be its 2nd application, I dont have any evidance to prove it but I think introducing slope for rate of change is more appropriate, what's your opinion

parshyaa
I just
You said that slope is used to tell how steeper is line or roof , you said that carpenter were not having protractor so that they can inclined them just by saying the angle , therefore architects defined slope to get the [approximate] idea of steepness. So I think that they tell them the rise and far ratio by telling them lengths of rise and far(as per the tan of the angle calculated by them) . With your opinion I think that architect may knew trigonometry at that time. so basically I think that they have introduced slope more importantly to display the rate of change graphically and mathematically. And steepness of roof or road may be its 2nd application, I dont have any evidance to prove it but I think introducing slope for rate of change is more appropriate, what's your opinion
I just want to know that how founder of slope must have introduced this concept.

Homework Helper
Looking for a historical record of the first fellow who thought up the idea that a fixed steepness of a roof corresponds to a fixed ratio between the height of the roof ridge and the width between the eaves is an exercise in futility. It is a pretty obvious geometric fact that certainly predates Euclid. The folks who built the Pyramids had a fairly decent handle on such matters.

What is the motivation for the question? Why does it matter?

• symbolipoint
parshyaa
Looking for a historical record of the first fellow who thought up the idea that a fixed steepness of a roof corresponds to a fixed ratio between the height of the roof ridge and the width between the eaves is an exercise in futility. It is a pretty obvious geometric fact that certainly predates Euclid. The folks who built the Pyramids had a fairly decent handle on such matters.

What is the motivation for the question? Why does it matter?
Yes I got it but can you give me the example for the architect and the carpenter . How architect told carpenter to make the roof steeper by telling him the slope. I totally agree with your answer but an example can make it more clearer. I read on wikipedia that slope is applied to the road by telling the % , 100% means 45° Inclined. I think similarly architects may have used some way to define steepness to the carpenter

Gold Member
If a "mathematician" never existed, ordinary people would still be measuring slopes. Things change and it's common sense to measure the change. The changes happen at a certain rate and it's common sense to measure the rate of change. You have heard the statement "Mathematics is the language of science." In the case of slopes, "Mathematics is the language of common sense."

• symbolipoint and parshyaa
parshyaa
I
If a "mathematician" never existed, ordinary people would still be measuring slopes. Things change and it's common sense to measure the change. The changes happen at a certain rate and it's common sense to measure the rate of change. You have heard the statement "Mathematics is the language of science." In the case of slopes, "Mathematics is the language of common sense."
yes , therefore I think that slope may be introduced to show rate of change mathematically and geometrically.

houlahound
I believe slope was first formalized by Newton to solve mechanics problems.

• parshyaa
parshyaa
Mee too, rate of change

It's just a mathematical definition looking at a rate of change.

• parshyaa
LvW
I am not a mathematician but only an electronic engineer.
In electronics, very often we have to work with non-linear input-output relationships (diodes, transistors, thermistors,..).
In most applications, we select a certain "operational point" on this non-linear characteristic using DC quantities (voltages and/or currents).
Then, it is very important to know the SLOPE of the characteristic in this particular operational point (bias point).
Fore example, for bipolar transistor amplifiers the slope of the voltage-in and current-out characteristic is the so-called "transconductance" - a key parameter for gain calculations.

• parshyaa and FactChecker