Is Math Cumulative & is Memorization Important?

  • #26
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Hey - I appreciate your effort man! To be honest, this might currently be a bit over my head. I think I get the gist of what you might be saying, but of course it's hard to tell. You're saying that there are truths of math that other truths can be based on?

What I didn't get was what you meant by saying you can prove the quadratic formula with axioms. We did learn the quadratic formula in my classes thus far, but nothing about axioms.




OK, this is actually very interesting to me! I'm a HUGE LOTR fan!!!

Having said that, I honestly don't think I understand what you mean. :smile: You were saying that there are axioms that are true and that we base other math on earlier, but then you say later here that these don't have to be true? ...and that we can invent our own truths like 1 + 1 = 0 :bugeye: Did I hear that correctly??? You need to come talk to my math teacher!! hahaha

Thanks for the memorization info. though. Very intriguing!

Yes, you heard that correctly. Axioms are true because we take them to be true. So in the rest of what we do, the axioms are true. However, they might not be true in real life!

Since we both like Lord of the Rings, let me give another example. For example, we might ask ourselves if Tom Bombadil could keep the ring safe and away from Sauron. An answer like "none of this is real" clearly is lame and insufficient here. Indeed, for the sake of argument, we have assumed that everything in lord of the rings is real.

In the same way, when we propose ##6+6=0##, we assume for the sake of argument that this is true, and then we see what else we can deduce from this.

Of course, if the axioms aren't applicable to real life, then none of the deductions will be either. However, if the axioms happen to be applicable to real life, then so will the consequences. Clearly, something like the Pythagorean theorem of the quadratic theorem is something applicable to real life. This is because we use it in physics and engineering all the time and it yields good results.

Something like ##6+6=0## is not applicable to real life because it's clearly not true. Or is it? It is definitely not the usual arithmetic we work with in our daily life. But this is an arithmetic that we use in reading the clock. Indeed, if it is now 6 hours, then 6 hours later it will be 0 hours. And if it is now 7 hours, then 11 hours later it will be 6 hours. So 7+11=6. So something like 6+6=0 isn't nonsense at all, it is actually useful. It's just that its uses are clearly different from what we typically use arithmetic for (like counting money).

Then again, there are some weird versions of arithmetic which aren't useful in real life at all. Still, they make up satisfactory theories which are mathematically acceptable.

So math really gives a many different theories (all based on different axioms). But only one (or a few) will be applicable to real life. In the same sense that if I give you three books, namely "The Lord of the rings", "The Furies of Calderon" and "World History of ancient times". All three are fascinating to read, but only one will truly be about our world. The rest are about worlds which are internally consistent, but are not real.

Which math theory is real (and which axioms are real) is something we must find out through physical experimentation and common sense (both of which can be deceiving).
 
  • #27
jbunniii
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You were saying that there are axioms that are true and that we base other math on earlier, but then you say later here that these don't have to be true? ...and that we can invent our own truths like 1 + 1 = 0 :bugeye:
Indeed, this is how your computer adds binary digits!

0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0

This is also known as the "exclusive or" operation. Of course when multiple binary digits are used to represent a number, the computer will "carry the 1" if there is enough room to do so, resulting in 1 + 1 = 10 (the binary representation for 2), instead of 0.
 
  • #28
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We did learn the quadratic formula in my classes thus far, but nothing about axioms.

Probably there was SOMETHING about axioms, but you didn't know it. You may have heard of something like the distributive property, associativity, commutativity, additive inverses (-5 is the additive inverse of 5), and multiplicative inverses (1/4 is the multiplicative inverse of 4). Those are examples of axioms, although you can actually deduce them from more basic axioms and definitions. Axioms are basically just starting assumptions. So, you can start with assumptions like associativity and call those axioms, or maybe you think there should be even more basic principles that those are based on, so you start with simpler assumptions and see if you can prove associativity and all that from those simpler axioms. In an elementary algebra class, it's not really foundational math, so you you'd start with things like associativity and so on.

The proof of the quadratic formula involves a calculation something like this:

x^2+2bx
= x^2+2bx + b^2 - b^2
= (x + b)^2 -b^2

Which relies on that sort of stuff, plus some idea of the number 2 and how it behaves. You also end up taking a square root a couple steps later, the existence of which would be a bit annoying to prove (not to mention it would normally require another somewhat complicated axiom) and out of the scope of your algebra class, so you definitely wouldn't derive the whole thing carefully from scratch at that level. But that's what's involved in it.

There's a more sophisticated way of looking at it, due to Lagrange, which, in essence, says you can try to solve polynomial equations by seeking symmetry. I find that a little more poetic than the calculation above ("completing the square"), but it's also quite a bit deeper and harder to actually understand.
 
  • #29
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Read math book and become pro active having a paper/pen ready. It is rather sad when I see classmates with books they have used for a full 3 months with no writing.

Question the material being read. Think of mathematics as reading Sir Bacon, " 4 idols." You have to ask question and write in the margins to understand.
 
  • #30
QuantumCurt
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It really is quite cumulative. Generally speaking, you're never going to stop using algebra in a math class. People that struggle in calculus are often struggling with the relevant algebra more than they're struggling with the calculus itself.

Some people are of the opinion that math has little or nothing to do with memorization. I can appreciate this sentiment. I've always found that understanding the process of arriving at a conclusion is a much better method of doing math than simply jumping to the conclusion. Clearly though, memorization is a very large part of math. It would be silly for a math, physics, or other math-heavy major to not take the time to memorize things like the multiplication tables, addition/multiplication algorithms, basic factoring formulas, the quadratic formula, and tricks for finding integrals or derivatives more quickly. Simple things like the shortcuts for factoring expressions like ##x^2-a^2## can and should be memorized. It's silly to work through something like that formally when you KNOW that the factors will always be of the form ##(x-a)(x+a)##. The same can be said of differential equations. My physics professor last semester gave me some valuable advice - "The best way to solve a differential equation is to skip the work and just write down the solution." This of course is a reference to the many different common types of differential equations that have standard solutions into which one can simply plug their constant values or otherwise manipulate as needed. My differential equations professor last semester didn't seem to be of the same opinion. He liked to see the work. ;)

edit - My physics professor did stress (after that fact) that it's important to see where these solutions come from and not just take them for granted. One should work through the differential equations for circuits, oscillations, waves, etc. when one first encounters them. After that though? You already did your work the first time you solved it. Now those solutions are part of your toolkit.
 
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