Learning basic to advanced math

We offer practical tips for learning basic to advanced math. Above all, mathematics is a difficult and abstract subject, while a part of some students feels very good in this area.

Some tips for learning basic to advanced math

Mathematics is not created for itself, but to help other sciences such as physics, biology, chemistry, engineering, as well as social and economic sciences. Mathematics is a powerful tool and it is the language of science. In the following, we give some tips for loving math and changing your weird view of math.

We know very well that mathematics is like a monster for many students and even they doubt its usefulness in life. This is not the case, because mathematics is at the heart of all science. The question that arises is whether it is an effective method for understanding mathematics. To understand math, you need to have a clear mind and ideas to organize well.

Two classes of study must be distinguished here. The first is the one who chooses mathematics as the main course, and the second chooses mathematics as a complementary subject. This sharing is a necessity in the university. Perhaps the first class will suffer from some difficulties, especially with the licensed math.

Anyway, if you master math, you can easily master the other disciplines. So here are practical tips for learning math.

Think of math as a chain

Mathematics is a logical subject and we have to learn it step by step. But before going to the next step, we are forced to dominate the previous one. How to climb a staircase that misses at least one step? It’s difficult, right? For example, to solve second-order algebraic equations, I need to know the real numbers, rational and irrational.

On the other hand, you have to know the structure of math lessons. The latter always begins by defining a concept, then we give simple properties to check the conditions of the definitions, these are theorems, and propositions.

For example, let us discuss the concept of the limit of a sequence. Definition: we say that a sequence (x_n) converges to a real $\ell$ if we can find a natural number $N$ such that the distance between $x_n$ and $\ell$ is very small whenever $n\ge N$. The difficult part of this definition is how to determine the number $N$. In the exercises, it is difficult to use this definition to determine the limit of sequences. What we, in general, is to know some classical limit of particular sequences such as $x_n=\frac{1}{n}$ which has $0$ as the limit, and then use prove that $|x_n-\ell|\le \frac{1}{n}$. For example
\begin{align*}x_n=\frac{n+1}{n}=1+\frac{1}{n}.\end{align*} This shows that the limit is $1$. So to solve exercises do not use definition, but use related theorems and properties.

Practice math as much as you can

In class, the teacher simply gives you the definitions, certain properties, and some examples of applications. It’s up to you to do the rest at home to better understand the chapter. Always start with the easiest exercises, then try to solve a problem that requires a little intuition. Sometimes you get stuck in front of these difficult exercises. Do not worry, be patient, you will accumulate techniques to deal with this kind of exercise.

learning basic math requires clear ideas

Mathematics is based on logic. It’s like musical notes, you have to compose them well to produce good music. So if you are in front of a math exercise, you must first read it well to know what concept it is, after reviewing the course definitions. From there one can conceive the properties or the theorem which must be used to solve the problem. If fact, if we want to compute an integral, then we should think of using the integration by part and change of variables technique.

Use reasoning by induction

Reasoning by induction is a very simple and powerful technique to prove iterative properties. This means that recurrence is useful if we have to prove that the property $P(n)$ is valid for any natural n. What you need to do, is just check that this property is valid for the first term $n=0,$ sometimes $ n = 1$ or $2$, then assume that it is valid for the term $ n $, then try to prove that the order $ n + 1 $ is also true.

The absurd in proofs is the key to learning basic to advanced math

Sometimes, to prove certain mathematical problems, we often use an absurd argument. For example, if you want to prove that $\sqrt{2}\notin\mathbb{Q} $. We do not have enough information to directly address this issue. The only way we have is to use the absurd. We then assume that $ \sqrt {2}\in\mathbb{Q}, $ therefore by the definition of the set of rational numbers $\mathbb{Q}$, there exist integers $p$ and $q$ with the same sign such that $ q $ is different from zero and $\sqrt{2}=\frac{p}{q}$. Then we will use certain arithmetic properties to find a contraction.

Absurd is the most important practical tip for learning math and is an efficient technique to solve very had problems.

Remember classic formulas and inequalities

In many cases, you must rely on a classical formula or a known inequality to demonstrate a mathematical result. It is therefore advisable to learn these formulas by heart. Here we give some:

  • Binomial formula: For any $a,b\in\mathbb{R}$ and $n\in \mathbb{N},$ we have \begin{align*}(a+b)^n=\sum_{k=1}^n \frac{n!}{k!(n-k)!} a^k b^{n_k}.\end{align*}
  • Gauss Formula: for any natural number $n$, \begin{align*} 1+2+3+\cdots n=\frac{n(n+1)}{2}.\end{align*}
  • Geometric sum: For real $a$ with $a\neq 1$ and any natural number $n,$ \begin{align*}1+a+a^2+\cdots+a^n=\frac{1-a^{n+1}}{1-a}.\end{align*}
  • Trigonometric Identities: For any real number $\theta,$ \begin{align*}\sin^2(\theta)+\cos^2(\theta)=1.\end{align*}

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