What is the Meaning of Gradient in Linear Slope?

[mister_jang]

Gradient as many knows is rise/run or the ratio between the change of Y over change of X

However, what's the meaning then?

It has no significant true meaning to a linear slope, the only true meaning is

- when i inverse tangent O/A(the gradient) ( right hand side of the picture ) i get the hypotenuse

- when i inverse sine O/H which i do not know the meaning ( left hand side of the picture ) i get the hypotenuse

Thanks in advance, did appreciate the help

 

[HallsofIvy]

If you really mean "what is the gradient of a line" (in the United States we would call it the "slope") it is exactly as you have defined it: "the amount y changes, with a certain change in x, divided by that change in x". That is just saying that it is the rate of change of y with respect to x. That has a specific meaning only when you give meaning to x and y. If x is, for example, time, and y is distance so that y(x) tells you far you have walked or ridden by time x, the gradient of the line is the "rate of change of distance with respect to time" which is just your speed. If y is your speed while x is still time, then the gradient is the rate of change of speed or acceleration.

But I can make no sense out of your "when i inverse tangent O/A(the gradient) ( right hand side of the picture ) i get the hypotenuse". No, you don't. The inverse of a trig function gives an angle- the measure of the angle at the lower right vertex. In this example, "O", the length of the opposite side, is 3 and the adjacent side is 4 so "O/A" is 3/4 and the arctan of 3/4 is the "36 degrees, 24 minutes" you have in the picture. The length of the hypotenuse is, of course, 5.

 

[mister_jang]

Firstly, i would like to thank you for the reply =)

Secondly, thank you for rectifying my ignorant with regards to the inverse matter

Thirdly, thank you for helping me understand the gradient is seemingly the measurement of the speed

However, the 24 minutes represent the 36 degrees which isn't the 24 minutes a value of the X axis?

And what does the 24 minutes means then?

Sorry, just realized i have a very weak basic which i need to make clear

Regards

 

[HallsofIvy]

You seem to have some massive misunderstandings. "speed" is one application or interpretation of the gradient. It is NOT correct to say that the gradient itself is "speed". That was one reason I also gave "accleration" as an application of a gradient.

Further, I am puzzled by "the 24 minutes represent the 36 degrees which isn't the 24 minutes a value of the X axis?". Perhaps this is a language problem. No "24 minutes" is not value of the X axis. Nor does the "24 minutes" 'represent' the 36 degrees. In fact, doing the calculation I should have done before, the "24 minutes" that I copied from your picture is wrong! The arctangent of 3/4 is approximately 36.870- that is 36 degrees and .870 degree. There are, by definition, 60 minutes in a degree so .870 minutes is (.870)(60)= 52 minutes, not 24. The angle is 36 degrees 52 minutes, not "24". The "52 minutes" (not "24 minutes") is part of the angle just as if a length were given as "4 meters and 23 cm." the "23 cm" would be part of the length measurement.

 

[mister_jang]

Oh all right, so the minutes itself is a representation of the decimal

And that the gradient is simply just a measurement of the slope

And does the 36 degrees 52 minutes a representation of something else or its just an angle that's all?

Thanks again =)

 

[Mentallic]

As Hallsofivy said, 36 degrees 52 minutes is just another representation of 36.87 degrees, so yes, it's all just an angle.

 

[mister_jang]

All right thanks =)