Related: In the equation y=ax+b, the values of all four variables are unknown, but x and y seem to be more unknown (more variable?) than a and b. It’s not clear what the difference is exactly.
Traditionally, the y variable is another way of defining the output of a function. y=f(x)=ax+b, where a and b are numerical constants. You may not know the constants, but they are specific numbers. The x is a variable you can change to get different values of y=f(x). y=ax+b has two unknown constants, a parameter, and an output.
Concrete example: Say I am looking at a hill with a straight line sloped side. For every x feet I walk forward I move ax feet up. The a has a single numerical value defined by topography. The b has a single numerical value defined by the altitude at the base of the hill where x=0. The x can be anywhere from zero (a point on the perimeter of the base of the hill) to however far horizontally it is from there to the top of the hill. As x varies, y goes from b to b + the height/prominence of the hill.
Note: y isn’t always intended to mean a function output. That depends on context. For example, say instead you have z=ax+by+c. This defines a plane that crosses the z axis at c, has slope a in the x direction, and has slope b in the y direction. It has two parameters (x and y) and three constants that generate the output, z=f(x,y)=ax+by+c
Parameters are also called the arguments of a function
Different concrete example from computer science: Say I want to write a program that multiplies a number by 4. I write code to take input from the user. I call the input x. I return x*4. y=4x. Or, instead, I could add a line in the code that defines a constant, a=4, and then I return ax. Why? Maybe I want to make it easy to see where to change a if I update the code in the future (I put a=4 at the top so I don’t have to read to hunt for the place where I use the number 4). Maybe I’m writing code for a more complicated function that uses a multiple times and I want to be able to make updates while only changing it manually in one place, for readability and lower risk of making mistakes. Basically the constant a is defined in the code, while the argument x is defined by the user.
All variables are equal, but some are more equal than others.
This is a quote from George Orwell’s unpublished manuscript The Theory and Practice of Algebraic Collections. He eventually split it into two separate novels which did see print. The 2+2=5 stuff went into 1984 and the “some are more equal than others” went into Animal Farm.
If you can let letters mean whatever you want then there’s nothing to stop you from doing the same with numerals. Let 5≡S(S(S(S(0)))).
Related: In the equation y=ax+b, the values of all four variables are unknown, but x and y seem to be more unknown (more variable?) than a and b. It’s not clear what the difference is exactly.
Traditionally, the y variable is another way of defining the output of a function. y=f(x)=ax+b, where a and b are numerical constants. You may not know the constants, but they are specific numbers. The x is a variable you can change to get different values of y=f(x). y=ax+b has two unknown constants, a parameter, and an output.
Concrete example: Say I am looking at a hill with a straight line sloped side. For every x feet I walk forward I move ax feet up. The a has a single numerical value defined by topography. The b has a single numerical value defined by the altitude at the base of the hill where x=0. The x can be anywhere from zero (a point on the perimeter of the base of the hill) to however far horizontally it is from there to the top of the hill. As x varies, y goes from b to b + the height/prominence of the hill.
Note: y isn’t always intended to mean a function output. That depends on context. For example, say instead you have z=ax+by+c. This defines a plane that crosses the z axis at c, has slope a in the x direction, and has slope b in the y direction. It has two parameters (x and y) and three constants that generate the output, z=f(x,y)=ax+by+c
Parameters are also called the arguments of a function
Different concrete example from computer science: Say I want to write a program that multiplies a number by 4. I write code to take input from the user. I call the input x. I return x*4. y=4x. Or, instead, I could add a line in the code that defines a constant, a=4, and then I return ax. Why? Maybe I want to make it easy to see where to change a if I update the code in the future (I put a=4 at the top so I don’t have to read to hunt for the place where I use the number 4). Maybe I’m writing code for a more complicated function that uses a multiple times and I want to be able to make updates while only changing it manually in one place, for readability and lower risk of making mistakes. Basically the constant a is defined in the code, while the argument x is defined by the user.
This is a quote from George Orwell’s unpublished manuscript The Theory and Practice of Algebraic Collections. He eventually split it into two separate novels which did see print. The 2+2=5 stuff went into 1984 and the “some are more equal than others” went into Animal Farm.
If you can let letters mean whatever you want then there’s nothing to stop you from doing the same with numerals. Let 5≡S(S(S(S(0)))).
Well that’s because it’s meant to be quantifying over linear equations.x and y are not meant to be replaced but a and b are.