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Graph and label the points (x1, y1) and (x2, y2) to all graphs where they are referenced.


Can the wording be adjusted for clarity? "It is also always the same thing as how many rises in one run." to something like: "The "Slope" is the number of rises in relation to each run." --anon

Slope-point formula-y=mx+b

I know the m stands for slope in the formula... But why did they use the letter m? This is an extra credit problem on my homework.. If anyone knows, please tell me :) --anon

It is the first letter in the word for slope in the mathematician's language. Or at least that's what I learned. Jamesr66a 19:53, 19 March 2007 (UTC)

This is because we get the concept of slope and the applied mathematics from France. In French, the word for "go up" is "Monter." —Preceding unsigned comment added by (talk) 02:58, 27 September 2008 (UTC) Cool Men Are Cool Frogs To EAt.

Slope and gradient -- request for comment

I was exclusively about a line y=ax+b in the plane, and nothing else. Then the recently inserted paragraph is probably not very accurate. Oleg Alexandrov 15:44, 11 Mar 2005 (UTC)

Nitpicky semantics

currently says: The slope of a vertical line is not defined (it does not make sense to define it as +∞, because it might just e im around themas well be defined as -∞).

This isn't really an explanation of why it isn't defined. Is the (ambiguous) square root of 4 not defined since it could be 2 or -2? Of course not, its just both. Similarly, is 90 degrees not defined because it could just as well be 360+90? Even more similarly, is saying that one line is 90 degrees from another 'senseless' because it could also be 270 degrees? I'm sure there is a good way of explaining why we don't define it, but just saying 'because it could also be this' isn't really valid. --Intangir 20:47, 2 January 2006 (UTC)

I guess the slope of a vertical line cannot be defined in any meaningful way.
The slope is +∞ or −∞ depending on whether you travel on that vertical line up or down. Tricky business. :) Oleg Alexandrov (talk) 00:50, 3 January 2006 (UTC)
The fact that both work doesn't make it unmeaningful. +∞ slope means vertical line as does −∞. The things which are unmeaningful are like 0/0 or ∞/∞. Those things are unmeaningful because they could mean any number. Thats why we don't define them. However, a positive infinite slope has one meaning. It just so happens that it has the same meaning as negative infinite slope. The more I think about it, the more it seems that there is no reason for leaving a vertical slope undefined. --Intangir 02:03, 3 January 2006 (UTC)
Would you please provide references for defining the vertical slope? I don't think the concept is standard (neither useful for that matter) and I would not agree with including such a thing without references. Thanks. Oleg Alexandrov (talk) 02:27, 3 January 2006 (UTC)
Sure, I found one from Math Forum and also this random quiz as well as this example of the concept applied usefully to programming. The math forum answer clearly points out that there is no real number slope for a vertical line. Hence if you require the slope to be a real number, one is out of luck since ∞ is not real, duh. But it would be a silly justification indeed to say that there is no such thing as an infinite slope because infinity is not defined for reals. Under a simple compactification of the reals like the extended real number line or the real projective line there is no trouble. Also, I don't see how saying that 'all lines except vertical ones have slope' is more useful than just allowing for infinite slope. There is no reason to be afraid of infinity here. An infinite slope behaves correctly. Consider the line y=∞x+b. Its reflection across the y axis is the same as for every other line y=-mx+b. In this case either itself, y=∞x+b or y=−∞x+b depending on which compactification is used. Its reflection about the line y=x is still y=x/m+b or y=x/∞+b ,ie y=0x+b. Certainly there is nothing unuseful about a vertical slope if it behaves exactly as it should. Furthermore, it is useful because it allows every line to be written in slope intercept form. --Intangir 04:20, 3 January 2006 (UTC)
Errata- vertical lines don't all have a y-intercept, so some of them can't be expressed in slope-intercept form no matter what. However, every line can be written in point-slope form iff unbounded slopes are allowed.

I am not really impressed by the references provided. :) I would like a calculus book, or mathworld, or something more serious :) . But OK, I see your point. I would agree with a remark, or a section, or subsection somewhere in the article discussiong the concept. But I would not agree in putting that front and center in the introduction or something. I've been teaching math at college level for the last 6 years and never had to deal with infinite slope. :) So I don't know how important the concept is. Cheers, Oleg Alexandrov (talk) 05:27, 3 January 2006 (UTC)

The issue is not whether a vertical line should be defined as having infinite slope, but whether it is defined that way, which would require an authoritative reference, none of the above suffice. Paul August 05:36, 3 January 2006 (UTC)

Actually, the original issue was whether or not the reason given for why vertical lines have no slope makes sense. It seems it doesn't, no one is suggesting that it does, so I'm going to reword it. This is a matter of incoherence, not a matter of authority or existance. Of course, I might also like to make some statement affirming the sanity and utility of 'infinite slope'. However, I won't unless I can find a reputable enough source which uses it. If there isn't such a source then I agree that the article doesn't need to waste space talking about an unused concept. --Intangir 14:30, 3 January 2006 (UTC)

All I've found is information on projective geometry and infinite slopes. However, I don't really care about the one point compactification very much at the moment, so I'm not going to bother writing a section on infinite slopes. As a final thought, the statement that y=∞x is the same as y=−∞x definitely does not imply that unbounded slopes are meaningless. The irrelevance of sign is just the insight that a vertical line is the one kind of line which can be thought of as both sloping down and sloping up. --Intangir 16:17, 3 January 2006 (UTC)


I have seen the term "angular coefficient" used for "slope" (e.g. in nuclear physics). Should that be mentioned in the definition?


My teacher asked me, and I) am wondering, why is M used to represent slope? Billvoltage 19:54, 29 September 2006 (UTC)

Why not? Seriusly, who knows, it's just a convention and as such has no real reason behind it. 14:47, 23 October 2007 (UTC)

Actually, there is a reason. These concepts come from France and in french the word for "go up" is "Monter." Like an ode to the history of slope, we use the letter "m." Funny fact though: if you go an look in a mathematics book from France, slope will be designated by a "s" because it has been Americanized.

The term "Americanized" means "made American", which refers to something from that continent. You probably meant the language spoken in the USA, and no, I have no idea of what term would apply for that =/ But since "slope" is from the English language (spoken in the American countries that use it, like USA or Canada, but also in the UK, Ireland, etc.) I'd also like to know how to express that. In Spanish at least it's called "anglificar" (anglify). (talk) 13:59, 26 October 2010 (UTC)

Why Calculus is Necessary

I'm confused about why the section is called that, or why there's a section separation at all —The preceding unsigned comment was added by (talkcontribs) 03:09, 9 October 2006 (UTC)

  • That makes sense to me - I went ahead and deleted the section header. Next time, you can feel free to be bold and do it yourself! (ESkog)(Talk) 03:20, 9 October 2006 (UTC)

0 or undefined

I cannot understan why it changes if the line's vertical or horizontal. The slope is either 0 or undefined. --Unsigned by User:AgentPeppermint at 11:02, 22 November 2006

Point-slope form

Shouldn't the section on point slope form in this page be moved to the point slope form page? Right now, this page contains (IMHO), off-topic information when a link could suffice, and the point slope form page is seriously lacking in necessary and relevant information --unsigned

You can copy some info from here to there. I don't think though that the point-slope formula is off-topic in the slope article. Oleg Alexandrov (talk) 02:16, 16 June 2007 (UTC)

who idcovered slope and wen very important plz help immediately —Preceding unsigned comment added by (talk) 01:25, 9 December 2008 (UTC)


Do you think y=kx should be added to represent the slope without the intercept? I mean, it's still used as an equation. Write back with what you think on this. :-)

Geometry Section

The following statement should be qualified. Two lines are parallel if and only if their slopes are equal and they are not coincident or if they both are vertical and therefore have undefined slopes.

It should read something like this: For a single plane, two lines are parallel if and only if their slopes are equal and they are not coincident or if they both are vertical and therefore have undefined slopes.

Otherwise, the reader must infer from x and y that we are restricted to a single plane. However, there are an infinite number of planes through the z-axis for which lines with the same slope would not be parallel.

Kananaskis (talk) 05:02, 9 July 2009 (UTC)

Link to Grade (slope)

{{#invoke:Hatnote|hatnote}} —was added at top of article. Took some time to get around disambiguation et al to find the correct article for term as used in nomenclature of earth sciences, environmental design, and civil engineering. Not vandalism, Thank you----Look2See1 t a l k → 20:41, 25 May 2010 (UTC)

Slope of a horizontal line

The introduction says: "Slope, as a practical term, is not defined for theoretically perfectly horizontal or vertical lines." I have no argument that the slope of a vertical line is undefined, as has been discussed on this page, but a horizontal line? Isn't the slope zero, as I was taught in high school? It is perfectly well defined by the equation on the page: Δy = 0 for Δx ≠ 0, so slope = Δy/Δx = 0. --ChetvornoTALK 19:01, 16 May 2012 (UTC)

I guess this sentence is part of the practical introduction which appears below it, using the example of the slope of a road. I assume what is meant is that a horizontal road is commonly described as having "no slope". However, I think this sentence is very misleading in the introduction of an article on mathematical slope. --ChetvornoTALK 19:17, 16 May 2012 (UTC)


why was the sequence for finding the slope through the tangent value removed? — Preceding unsigned comment added by (talk) 13:52, 24 April 2013 (UTC)

Because it was already in the article. Why did you add a comment about vandalism? Dbfirs 20:52, 24 April 2013 (UTC)

Im sorry about the vandalism comment, i got impatient waiting to see if my edit for finding the slope through the tangent function would be accepted. but i don't the text of what i did where i've looked. can you point to where what i did is shown e explicitly on wikipedia? — Preceding unsigned comment added by (talk) 04:00, 25 April 2013 (UTC)

at no point in the article does it say why m=tan or how to derive the function, if wikipedia is about passing knowledge to others than knowing how and why something is will pass on much more knowledge and skill to somebody than just memorizing a bit of text. — Preceding unsigned comment added by (talk) 04:04, 25 April 2013 (UTC)

The article contains the brief explanation:
"The angle θ a line makes with the positive x axis is closely related to the slope m via the tangent function:
If you think that this should be expanded slightly, then please put forward your suggestion, but please don't ramble on at great length. A simple diagram might be the best way to explain. Dbfirs 11:23, 25 April 2013 (UTC)

I used the diagram and written proof because together they say much more than they would separately,saying m=tan and the angle is equal to the cotangent of m doesn't tell someone how to derive a trigonometric function to be used in a cartesian plane. the two statements are equivalent, they are not very explanatory. — Preceding unsigned comment added by (talk) 18:11, 25 April 2013 (UTC) i already gave a suggestion, you deleted it for some reason — Preceding unsigned comment added by (talk) 18:13, 25 April 2013 (UTC) is there any way we can find a compromise on this? i want to add it because i used to come on wikipedia alot to learn math and i never understood how somethings could be so but not others, i just want to make sure others can learn as much as is offered, which is alot hopefully, would it be okay if i set up a link to a proofwiki under the m=tan argument to show the argument is valid? — Preceding unsigned comment added by (talk) 18:20, 25 April 2013 (UTC)

Your silly additions about the slope of the universe, and vandalism, made me think that you were just a troll.
Here is the serious part of your contribution:


The slope defines the rate of change of y with respect to x, and effort will show that the slope m can be found through trigonometric values.

The tangent value can be equated to the slope of a line by knowing basic trigonometric identities and algebraic manipulation,

then by replacing the x and y value with their respective trigonometric values

Where the line AC is equal to y and CE is equal to x and r equal to AE

When these functions are inserted into the equation, their r values will cancel. This is because the multiplicative law of algebra says:

where . This in turn gives us



Which was the desired result.

I'll leave it for others to comment on whether it should be added to the article or elsewhere. Dbfirs 07:34, 26 April 2013 (UTC)

Thank you — Preceding unsigned comment added by (talk) 11:08, 26 April 2013 (UTC)

I've added a link to the definition of tangent. I think this is sufficient to explain the connection with slope angle. Dbfirs 11:59, 26 April 2013 (UTC)

what i'm saying is it's not just the tangent, its every other type of triangle equation that can be used to find the slope. would a third opinion be best here? — Preceding unsigned comment added by (talk) 03:32, 27 April 2013 (UTC)

I don't see the benefit of defining slope in terms of sine or cosine, but, like you, I was hoping that someone else would contribute to this discussion. Dbfirs 07:55, 27 April 2013 (UTC)

there really wouldn't be one, its just encouraging curiosity, which is what learning i think i supposed to do — Preceding unsigned comment added by (talk) 19:12, 27 April 2013 (UTC)

Some comments

  1. I absolutely love the bit with the images from the roads and railways. Fantastic, fun (and I learned something which I always wanted to know)!
  2. Very accessible article - readable and understandable.
  3. I translated this into Macedonian [1]. I am not done working on it, but I wanted to consult here. I can certainly translate any bit anybody wants into English if someone wants to know what the words say. If you go to this page:
A. I redid the deltay, deltax image (adding the angle and the point coordinates, but now it is png) and I redid and cleaned up the the animated gif (but in trying to give credit to the original creator I managed to name it some cyrillic gibberish which won't load here). These images do not include the letter of the slope which can be adapted in the caption. Feel free to use them if you want.
B. I added a bit on the fact that the slope determines both the "direction" (for want of a better word - please assist here) and steepness, i.e.

Direction: Positive slope means (monotonely) increasing (put your money in this bank) and negative slope means (monotonely) decreasing; 0 slope is just monotone, i.e. constant. (I always add the part about the bank because I am always getting the response that a decreasing function is "increasing to the left" - so to put a stop to this, I simple ask "would you put your money in this bank?", but I am sure that is not very encyclopedic :).

Steepness: A higher absolute value means the slope is steeper. Notice the addition of the word absolute, which makes the sentence mathematically correct, without (I hope) complicating it.

To explain this graphically, I added the four images you see at the top of my page, but I am not entirely sure that these images really clarify. Any comments here would be greatly appreciated (and of course if we want to add something like that here, I can change the slope letter to m and make any changes).

  • Finally, I am considering adding a bit about the letter used to denote the slope (often m as it is here). I find that many people ask why the letter m is used for slope (in Macedonia, in mathematics, we use the letter a).

Here are some references about this...

Lfahlberg (talk) 13:03, 1 July 2013 (UTC)

I agree that it's confusing for beginners to meet different pairs of letters used in the equation of a straight line. When I taught the topic, I always emphasised that "m" was the multiple of x in the standard equation, though I doubt whether that is the true origin. Dbfirs 07:25, 2 July 2013 (UTC)
Ok, I made some changes. I tried to stay very true to the style of the page (which I very much liked), just adding the important reference to increasing, decreasing and a couple of examples. I changed the first picture (with svg help) to include all the elements mentioned in the text... and the last picture to be more correct and easier to read for the color-blind. I hope it is okay. Lfahlberg (talk) 12:26, 11 August 2013 (UTC)

I think I forgot % sign in formula by road signs.

Hello. I think I forgot the % sign on the formula for slope when I edited it about 2 months ago.

The formula says:

I think it should be:

I hesitate to edit without writing here first.

Lfahlberg (talk) 19:12, 30 October 2013 (UTC)

I think your original formula looks OK because it is restricted in application to the percentage number. If you change this, then you need to change the other formula also. I suppose if you want to make it even clearer, you could specify that it applies to a P% slope, then use P in the formula. Dbfirs 08:25, 31 October 2013 (UTC)
Yes, for sure the percentage sign is needed in both formulas. Adding the % sign makes the formulas in the article consistent with the formulas above (and universally valid), whereas the plain 100 is mathematically incorrect (although clarified in the text).(On checking my notes, I see that I only added the example below the formulas; I did not alter these formulas at all.) Now, I have made these changes and slightly rearranged the word order for clarity and made other minor corrections. I hope it is okay.Lfahlberg (talk) 10:59, 1 November 2013 (UTC)
P.S. I see that I have added a hyphen for e.g. y-coordinate, y-intercept, y-axis but in text I did not write there is no hyphen. I do not know the wikipedia math convention for this. Lfahlberg (talk) 11:04, 1 November 2013 (UTC)

Lfahlberg (talk) 10:59, 1 November 2013 (UTC)