Practice With Proofs? (Algebra, Trig, and Calc)

[Ascendant0]

I'm trying to brush up on my algebra, trig, and calculus, and one thing I know I was always weak on before was proofs. I was never sure what equations would suffice as "proof," and which equations did not. Maybe this is an inane question, and maybe there is a really simple answer to this. I simply never asked before.

Can any math "law" or "rule" or "identity" or "formula" that is written in the textbooks be considered "proofs"? Are there any that are specifically excluded?

 

[PeroK]

I'm trying to brush up on my algebra, trig, and calculus, and one thing I know I was always weak on before was proofs. I was never sure what equations would suffice as "proof," and which equations did not. Maybe this is an inane question, and maybe there is a really simple answer to this. I simply never asked before.

Can any math "law" or "rule" or "identity" or "formula" that is written in the textbooks be considered "proofs"? Are there any that are specifically excluded?

I don't understand your question. Everything in maths can, in principle, be proved from the fundamental axioms of mathematics (set theory). That said, most mathematics assumes the basic properties of numbers and sets without delving into the foundations. For example, we assume that for any two numbers $a,b$:
$$a + b = b + a, \ \text{and} \ ab = ba \ \text{and} \ a(b + c) = ab + ac \dots$$I don't know if that is what you are asking?

 

[Ascendant0]

I don't understand your question. Everything in maths can, in principle, be proved from the fundamental axioms of mathematics (set theory). That said, most mathematics assumes the basic properties of numbers and sets without delving into the foundations. For example, we assume that for any two numbers $a,b$:
$$a + b = b + a, \ \text{and} \ ab = ba \ \text{and} \ a(b + c) = ab + ac \dots$$I don't know if that is what you are asking?

Ok, so for example:

For "b" in this problem for example, seemed easy to me, because it uses the definition of the tangent function and the double angle formulas in the chapter just prior to the questions. So, I could convert tangent into its respective sin and cos counterparts, use the double angle formulas, and make both sides look exactly the same.

However, this is what I'm asking... I can use the definition of tan (tan = sin/cos) because I was exposed to that definition already, and I know it would be an acceptable answer. However, if I'm trying to "prove" this to someone who only knows basic algebra, it's going to be completely lost on them. It won't "prove" anything to them other than there is some math they know absolutely nothing about. For them, they would need further "proof" - explaining exactly what sin means, what cos means, what tan means, and give them that basis of understanding first.

So, in this question, I know I'm allowed to use the tan definition and the half angle formulas to prove it. But, if I'm trying to prove something similar to that, can I utilize any "definition," any "formula" and such to prove it? At what point would you be required to prove what you used to prove it, if that makes sense?

Maybe another analogy if that is confusing... If someone who is from an alternate universe with completely different laws of physics asked me to prove that gas is flammable, *if* they already learned and understood the internal mechanics and combustion engine of a vehicle, I could place gas in a gas tank, turn the car on, and the engine is running, so for someone with that basis of knowledge (understanding how a combustion engine is designed and functions), it would most likely suffice as proof to them. But, if that being had no idea what the insides of a car is like, that wouldn't prove gas is flammable, because they don't understand the process that's happening in the vehicle. I would have to break it down to something simpler, like making a puddle of gas on the ground and igniting it with a match.

With that said, what I'm trying to get as is how far you need to break something down as proof. That if there is any math textbooks anywhere that state something is a "rule" or "formula" or "an identity," I can use that in any mathematical proof, or if a proof is supposed to only be based on what has been taught to you and established already by the teacher, textbook, or other resource that's asking the question. Does that make more sense now?

 

[Hill]

The point is not "to prove X to somebody", but "to prove that X is true." To do so, one uses A, B, C, ... which either assumed to be true or have been already proven to be true without assuming that X is true.

 

[PeroK]

With that said, what I'm trying to get as is how far you need to break something down as proof. That if there is any math textbooks anywhere that state something is a "rule" or "formula" or "an identity," I can use that in any mathematical proof, or if a proof is supposed to only be based on what has been taught to you and established already by the teacher, textbook, or other resource that's asking the question. Does that make more sense now?

This depends on the course objectives. In some formal pure maths courses, you must explicitly state the justification for every line in a proof. This might be one of the hypotheses, a definition, an axiom or a theorem which you have already proved. Most courses don't require that level of detail.

In general, you have to judge whether something is acceptable. For trig identities, there should be an agreed set of standard identities that you can use. And, definitely, something like $\tan x = \frac{\sin x}{\cos x}$ is always going to acceptable.

In my first year at university I tended to put too much detail and had to cut back on justifications and intermediate steps. I remember it was in Dr Wilkinson's complex numbers class. Learning how much detail you need is part of maturing as a maths student. This is where you need feedback from your professors.

 

[Ascendant0]

This depends on the course objectives. In some formal pure maths courses, you must explicitly state the justification for every line in a proof. This might be one of the hypotheses, a definition, an axiom or a theorem which you have already proved. Most courses don't require that level of detail.

In general, you have to judge whether something is acceptable. For trig identities, there should be an agreed set of standard identities that you can use. And, definitely, something like $\tan x = \frac{\sin x}{\cos x}$ is always going to acceptable.

In my first year at university I tended to put too much detail and had to cut back on justifications and intermediate steps. I remember it was in Dr Wilkinson's complex numbers class. Learning how much detail you need is part of maturing as a maths student. This is where you need feedback from your professors.

Thank you for that clarification. You covered exactly what I was thinking about. I really appreciate it!

 

[MidgetDwarf]

If you want to get good at proofs i recommend working through two books and possibly working through a third, to understand basic arguments of proofs.

Edwin E. Moise/Downs: Geometry

Pommersheim: A Lively Introduction To Theory Of Numbers.

HAMMOK:Book of Proofs.

Moise book is a rigorous intro to high geometry. Concepts are simple, and its easier to prove things you have familarity with. It is very readable, and no dreaded two column proofs.

Pommersheim is a gentle book on number. Goes very slow and everything is explained well. Exercises are not too difficult, with a about 2 to 3 tough ones every section.

Hammok is a typical intro books proof. Very readable and cheap.