slope

In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter m; there is no clear answer to the question why the letter m is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equation of a straight line as "y = mx + b" and it can also be found in Todhunter (1888) who wrote it as "y = mx + c".Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient ("rise over run"), giving the same number for every two distinct points on the same line. A line that is decreasing has a negative "rise". The line may be practical - as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan.
The steepness, incline, or grade of a line is measured by the absolute value of the slope. A slope with a greater absolute value indicates a steeper line. The direction of a line is either increasing, decreasing, horizontal or vertical.

A line is increasing if it goes up from left to right. The slope is positive, i.e.



m
>
0


{\displaystyle m>0}
.
A line is decreasing if it goes down from left to right. The slope is negative, i.e.



m
<
0


{\displaystyle m<0}
.
If a line is horizontal the slope is zero. This is a constant function.
If a line is vertical the slope is undefined (see below).The rise of a road between two points is the difference between the altitude of the road at those two points, say y1 and y2, or in other words, the rise is (y2 − y1) = Δy. For relatively short distances, where the earth's curvature may be neglected, the run is the difference in distance from a fixed point measured along a level, horizontal line, or in other words, the run is (x2 − x1) = Δx. Here the slope of the road between the two points is simply described as the ratio of the altitude change to the horizontal distance between any two points on the line.
In mathematical language, the slope m of the line is




m
=




y

2




y

1





x

2




x

1





.


{\displaystyle m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}.}
The concept of slope applies directly to grades or gradients in geography and civil engineering. Through trigonometry, the slope m of a line is related to its angle of incline θ by the tangent function




m
=
tan

(
θ
)


{\displaystyle m=\tan(\theta )}
Thus, a 45° rising line has a slope of +1 and a 45° falling line has a slope of −1.
As a generalization of this practical description, the mathematics of differential calculus defines the slope of a curve at a point as the slope of the tangent line at that point. When the curve is given by a series of points in a diagram or in a list of the coordinates of points, the slope may be calculated not at a point but between any two given points. When the curve is given as a continuous function, perhaps as an algebraic formula, then the differential calculus provides rules giving a formula for the slope of the curve at any point in the middle of the curve.
This generalization of the concept of slope allows very complex constructions to be planned and built that go well beyond static structures that are either horizontals or verticals, but can change in time, move in curves, and change depending on the rate of change of other factors. Thereby, the simple idea of slope becomes one of the main basis of the modern world in terms of both technology and the built environment.

View More On Wikipedia.org
  1. B

    Kitchens floor slope

    Hi all it’s My first post here newbie alert lol I’m fitting my kitchen and the floor is slanting towards the gas meter by roughy 10mm iv self levelled the area on the meter wall already but it’s not come up enough using 2 bags I am going to lay either tile or vinyl click and would like some help...
  2. MCTMike

    Slope problem in installed shower

    Looks like I didn't get the slope correct on this shower bed. Here's water not drained after 10 hours since turning off the shower. How atrocious is my fix going to have to be? Be honest. I can handle it.
  3. scottmcgaw5

    Creating a slope in a wetroom

    Hi all, have tiled small areas before but trying to create a wetroom.. I have read a lot of tips etc on here so far. I am creating a bench within an enclosure which I want to create a steam room in. Floor to ceiling glass, and I plan to put an long drain butted against the wall under the...
  4. S

    Front entrance tiling slope

    Can a front entrance tiling with no slope be corrected without removing everything and redoing the job? Basically, this is a new job( by a professional) but when it rains I have 2 puddles on my steps ... Thanks
  5. S

    Replacing cracked tile - slope on subfloor

    Hi All. I am replacing a cracked tile (porcelain 60x60). However I have noticed the subfloor is sloped slightly, the adhesive that was on before was a lot thicker at the bottom of the tile then the top so I don't think its recent movement. What is the best way of dealing with this? Should i...
  6. D

    New Home shower floor not sloped properly, tile company wants to tile over bad slope

    Hello, Bought a new construction home in January 2019 in Orlando. After living in the house for about 6 months, I realized that my master bathroom shower floor was not draining properly. Started seeing discoloration, and mildew around all the areas that were not draining properly. The shower is...
  7. A

    fixing slopes in a shower with decorated 20cm tiles

    Hello, We have chosen a painted decorated 20X20cm tiles for the bathroom with a walk-in shower and wanted to make the tiles pattern as seamless as possible. Are the tiles small enough to create fall to to drain without cutting them diagonally? I can settle for slopes that are 95% perfect and...
  8. B

    Slope for Wet Room Floor

    I posted on this in the main forum but responses although gratefully received are fairly slow coming and I need to make decisions quite soon. Below is a diagram for my bathroom renovation. New stud walls being erected and new concrete floor. Question: Will the slope shown for the tiled floor...
  9. Kyle Knowles

    Outdoor Tiling | Tiling outdoors

    I gave a step to tile it's 6ft x 6ft with to steps up to the main platform it's brick build then backfilled with concrete would it be acceptable to use something like kerdi membrane to cover the whole thing or even the old homelux wetroom fabric or sold I just go ditra drain or similar? ? The...
  10. M

    Do I need a tray former? how to create a slope for drainage?

    Hi, I have a screeded concrete floor in my bathroom and instead of a conventional shower tray I want to have a wet room style linear drain floor. The area I am using is a corner and I have found linear former trays but they all require an envelope style tiling system. Would be possible to just...
  11. A

    DIY Tiling Slope (metro Style)

    Hi all, I am currently tiling my bathroom in 200 x 100 white metro tiles. I have a sloped area to do approximately 45 degrees. The wall had tiles on previously that have been removed leaving the plaster in a reasonable condition. Can anyone advise if I am ok to reapply the new tiles without any...
  12. D

    adhesive for slope

    Hi I'm a real newbie to tiling and have a quick question for the experts... Is it possible to use a thicker layer of tile adhesive during tiling to make a slope to match variations in sub-floor levels (floorboards)? I'd rather not overboard with really thick ply in this situation as I'm...
  13. B

    Repairing/Improving Insufficient Mortar Slope

    Well I installed my first shower (in my own home) and carefully created a pre-slope, used a three-piece drain, weep hole guard and created a properly sloped mortar bed (almost). I have not yet laid the floor tiles. My question relates to the "almost" qualifier. Generally, the mortar bed is...
  14. A

    slope in floor

    i have a kitchen floor to do 4m long with a 9mm overall dip, am i right to go with the slope and start at the highest end. the floor is already plyed out and very sturdy. also customer has said whole house seems to fall slightly in the slope.
  15. S

    Tiling a slope

    Newbie tiler! We live in an old house and the area we want to tile has a slope approx 45 degrees. Any suggestions how to prevent the tiles slipping or falling off wihout having to stand there holding onto them for ages. Thanks
This website is hosted and managed by www.untoldmedia.co.uk. Creating content since 2001.

Replies you've not seen

UK Tiling Forum Stats

Threads
66,583
Messages
866,612
Members
9,500
Latest member
Dazzer
Top