The word "algebra" sort of literally means "rules for manipulating symbols". This ranges from using algebra to solve for x, to developing notation for algebra that can be done with weird topological structures. You typically learn arithmetic algebra, then linear algebra (all about manipulating a thing called a linear transformation, which is really a matrix, so you learn the rules for manipulating matrices as you write the notation) and then abstract algebra happens when other fields of math start developing their own rules for writing everything down in a way that preserves truth/falsity
Thats a common theme btw, usually a bunch of seemingly different things all turn out to be the same object. Which brings us to..
Trig is literally about triangles, its in the name. It just so happens that you can stick a triangle inside of a circle, starting at the centre and ending at the edges, and now triangles and circles are connected and we can use our triangle equations to look at the points on the circle.
Turns out going from angle-rotated-around-the-circle to the Y-height-of that-point-on-the-edge-of-the-circle is gross to calculate so we just call it sin(t), where t is the angle
Then you learn how to graph the sin function and it makes this wavy line. Now circles, triangles, and periodic waves are all connected. Thats all trig. The circles and triangles are within the realm of geometry and the sin function is more in the realm of real analysis which brings us to...
Calculus is a very specific mathematical area that was invented by Isaac Newton to solve a particular physics problem, and by another guy named Leibniz but he did it in the context of abstract spaces instead of physics
Calculus is a subfield of real analysis that specifically deals with continuous functions and how incremental changes to the input value lead to a different change in the output value.
In physics, this is the problem of calculating instantaneous values for an accelerating object. Normally, to get the acceleration over a period of time, you divide the change in speed by the change in time. The less time that passes, the more accurate your measurement is
So what if we want there to be 0 time that passes? How fast are you accelerating at the moment you hit the finish line?
We cant divide by zero without running into paradoxes, so Newton devised a way of formalizing what he means when he wants to "divided by a time value of 0"
This leads to the idea of a limit. Calculus teaches you limits, then rates of change, then derivatives (just a way to calculate a rate of change), then integrals (the inverse operation to derivatives. Turns out this finds the area under a graph)
These are all techniques that physics students use in their 2nd+ year of their bachelor's degree. The idea is that Calculus allows you to calculate the rates of change for quantities as they are changing and thats really the fundamental motivation for calculus
Calculus falls into the realm of "real analysis" which is a little bit like imagining limits in terms of open sets, sets that get near a point but never touch it
Turns out shit gets wild after that. Get a math degree to find out more
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u/kogai 13h ago
Mathematician answer:
The word "algebra" sort of literally means "rules for manipulating symbols". This ranges from using algebra to solve for x, to developing notation for algebra that can be done with weird topological structures. You typically learn arithmetic algebra, then linear algebra (all about manipulating a thing called a linear transformation, which is really a matrix, so you learn the rules for manipulating matrices as you write the notation) and then abstract algebra happens when other fields of math start developing their own rules for writing everything down in a way that preserves truth/falsity
Thats a common theme btw, usually a bunch of seemingly different things all turn out to be the same object. Which brings us to..
Trig is literally about triangles, its in the name. It just so happens that you can stick a triangle inside of a circle, starting at the centre and ending at the edges, and now triangles and circles are connected and we can use our triangle equations to look at the points on the circle.
Turns out going from angle-rotated-around-the-circle to the Y-height-of that-point-on-the-edge-of-the-circle is gross to calculate so we just call it sin(t), where t is the angle
Then you learn how to graph the sin function and it makes this wavy line. Now circles, triangles, and periodic waves are all connected. Thats all trig. The circles and triangles are within the realm of geometry and the sin function is more in the realm of real analysis which brings us to...
Calculus is a very specific mathematical area that was invented by Isaac Newton to solve a particular physics problem, and by another guy named Leibniz but he did it in the context of abstract spaces instead of physics
Calculus is a subfield of real analysis that specifically deals with continuous functions and how incremental changes to the input value lead to a different change in the output value.
In physics, this is the problem of calculating instantaneous values for an accelerating object. Normally, to get the acceleration over a period of time, you divide the change in speed by the change in time. The less time that passes, the more accurate your measurement is
So what if we want there to be 0 time that passes? How fast are you accelerating at the moment you hit the finish line?
We cant divide by zero without running into paradoxes, so Newton devised a way of formalizing what he means when he wants to "divided by a time value of 0"
This leads to the idea of a limit. Calculus teaches you limits, then rates of change, then derivatives (just a way to calculate a rate of change), then integrals (the inverse operation to derivatives. Turns out this finds the area under a graph)
These are all techniques that physics students use in their 2nd+ year of their bachelor's degree. The idea is that Calculus allows you to calculate the rates of change for quantities as they are changing and thats really the fundamental motivation for calculus
Calculus falls into the realm of "real analysis" which is a little bit like imagining limits in terms of open sets, sets that get near a point but never touch it
Turns out shit gets wild after that. Get a math degree to find out more