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May, someone tell me, does one need to know the basics of algebra to calculus to understand the higher levels of mathematics, or do those rules don't apply?

I'm almost done with the basics, afterwards I plan on getting into the higher levels of mathematics and, then going to a college to test my way in.

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closed as too broad by Austin Mohr, астон вілла олоф мэллбэрг, iadvd, Leucippus, Daniel W. Farlow Feb 7 '17 at 3:05

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    $\begingroup$ yes indeed :D you need to understand the grounds and motivations of calculus and algebra if you want to understand higuer algebra and highr calculus which is based on easy mathematics $\endgroup$ – Jose Garcia Dec 18 '16 at 22:45
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    $\begingroup$ Many are not, strictly speaking, a prerequisite, but why would you want to learn "higher level math" before getting done with the "lower level math"? Whatever incentive you might have, you won't be able to get anywhere without having some genuine interest in understanding why certain things are the way they are; this, I believe, can only be obtained through a lot of exposure to elementary mathematics. $\endgroup$ – user384138 Dec 18 '16 at 22:50
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    $\begingroup$ Math has a neat way of building up on itself. When solving a calculus problem, you're probably going to be doing more algebra than actual calculus. You're going to want to be good at earlier problems if you're going to tackle calc. $\endgroup$ – Kaynex Dec 18 '16 at 22:58
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These basic knowledge are absolutely essential to understand higher levels, just as a writer, say Philip Roth, needs the alphabet for his novels. It would be impossible to understand, for example, the (not really of "high" level) law of quadratic reciprocity $$\left(\frac pq\right)\left(\frac qp \right)=\left(-1\right)^{\dfrac{p-1}{2}\cdot\dfrac{q-1}{2}}$$

if one could not handle the powers of $-1$ or fractional exponents.

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  • $\begingroup$ I don't mean to detract from your point, but those exponents are not really fractions: $p$ and $q$ are assumed to be odd integers. Of course, one has to be comfortable enough with such concepts to understand why such an expression is therefore an integer, so one needs a decent grounding anyway. And without a strong background in basic algebra, the proof would be reduced from a magic trick to a pile of symbols on a page. $\endgroup$ – Will R Feb 6 '17 at 19:14
  • $\begingroup$ You are absolutely right however that these exponents are not really fractions is clear and so is for the primes $p$ and $q$ being both odd. Anyway, that my English is weak is true. Thank you very much for your comment. $\endgroup$ – Piquito Feb 7 '17 at 20:13
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My guess (it's a guess, because it's not something you can prove rigorously, and honestly, I don't see how one would find a volunteer to test this empirically) is that although you could technically understand some part of higher mathematics without a thorough understanding of basic algebra, it would be quite a contortionist's act. It would be a little like walking without ever putting your left foot in front of the right. Sure, you could do it, but why?

And I suspect in many ways, it's worse than that, depending on what we mean by "a thorough understanding of basic algebra." If you don't never figure out how to complete the square, then OK, maybe you just understand some problems and not others. But basic algebra also comprises a fundamental understanding of things like "multiplication commutes." Not understanding that, and not being able to comprehend why it might not always be true, and when, are going to make making any real headway into advanced mathematics pretty difficult, I would say.

This isn't like many other fields, where we often employ a lie-to-children to make things behave temporarily, to be later replaced by a better understanding that will often produce the same behavior but which is often fundamentally incompatible with the earlier lie-to-children. (Note that the lie-to-children is often used with adults; the phrase is a term of art and is not to be taken literally.) In such cases, one may well be able to ignore the earlier material, because it's mostly invalid. That isn't the case in mathematics, though, for the most part.

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No, you don't. In principle you could learn (up to research level) formal logic/set theory/model theory, perhaps even set-theoretic and algebraic topology, without ever knowing how to solve $x^2 - 2x + 1 = 0$. Other fields, like analysis and number theory, basic algebra is necessary.

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    $\begingroup$ Without calculus, maybe. But learning algebraic topology without "basic algebra" (whatever it means)? $\endgroup$ – Peter Franek Dec 18 '16 at 23:09

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