Intorduction

Intorduction#

This section deals with ideas from mathematics that I found useful.

Sets of numbers#

Name

Symbol

Definition

Examples

Natural numbers

\(\mathbb{N}\)

Counting numbers (sometimes includes 0)

\(1, 2, 3, 4, 5, …\)

Whole numbers

—

Natural numbers plus zero

\(0, 1, 2, 3, 4, 5, …\)

Integers

\(\mathbb{Z}\)

Whole numbers plus negatives

\(…, −3, −2, −1, 0, 1, 2, 3, …\)

Rational numbers

\(\mathbb{Q}\)

Fractions of integers

\(1/2, −4/3, 7\)

Irrational numbers

—

Cannot be expressed as a fraction

\(\sqrt{2}, \pi\)

Real numbers

\(\mathbb{R}\)

All rational and irrational numbers

Any point on the number line

Imaginary numbers

—

Multiples of \(i\) (where \(i^2 = -1\))

\(i, 2i, −3i\)

Complex numbers

\(\mathbb{Q}\)

Numbers of form a + bi (a, b ∈ ℝ)

\(2 + 3i, −1 − i, 4\)

Square root#

The square root of a number \(x\) is the number \(y\) such that \(x=y^2\).

By definition of the square root, it always have to have two solutions. Since any pair of opposite numbers (i.e., \(a = -b\)) satisfy \(a^2 = b^2\), squaring each yields the same result.

Principal square root (\(\sqrt{}\))#

The principal square root (rus. арифметический корень) is the nonnegative square root of a number.

Principal suqre root usually denoted as:

\[y = \sqrt{x}.\]

Note: The symbol \(\sqrt{}\) is called a radical.

Note: It’s common to use the term “square root” when refferting to the principal square root. Remember that the notation \(\sqrt{x}\) refers to the principal square root.

It follows that when you are solving equation like:

\[y = x^2.\]

Remeber that the expression \(\sqrt{y}\) will correspond to \(x\) value in both signs:

\[x = \pm \sqrt{x}.\]

Quadratic function#

Consider a function of the form:

\[ f(x) = ax^2 + bx + c \quad \text{where } a, b, c \in \mathbb{Q}, a \neq 0 \]

This function is called a “quadratic function”.

A common practical task is to find the values of \(x\) for which \(f(x) = 0\). The corresponding equation is called quadratic and is displayed explicitly below:

\[ ax^2 + bx + c = 0 \quad \text{where } a, b, c \in \mathbb{Q}, a \neq 0 \]

Equation solution#

Multiply the equation to \(1/a\):

\[ x^2 + \frac{b}{a}x + \frac{c}{a} = 0 \]

Perform the following sequence of transformations to apply the sum-of-squares formula:

\[x^2 + 2\frac{b}{2a}x + \left( \frac{b}{2a} \right)^2 - \left( \frac{b}{2a}\right)^2 + \frac{c}{a} = 0\]
\[\left(x + \frac{b}{2a}\right)^2 = \frac{b^2}{4a^2} - \frac{c}{a}\]
\[x + \frac{b}{2a} = \pm\sqrt{\frac{b^2 - 4ac}{4a^2}}\]

Take a closer look at the expression on the right side of the equation:

\[\pm\frac{\sqrt{b^2 - 4ac}}{2 |a|}\]

Expression \(\pm|a| = \pm a\) is correct as a concise way to represent that \(|a|\) and \(a\) differ only in sign, and both \(\pm|a|\) and \(\pm a\) denote the set of two numbers: \(a\) and \(-a\). So:

\[x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 -4ac}}{2a}\]

Finally, both solutions to the equation can be expressed as follows:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

The expression \(b^2 - 4ac\) is called the descrimimnant of a quadratic equation and is denoted by \(D\). Obviously, \(D\geq0\) for there to be solutions in the rational numbers.

Logarithm#

Logarithm of a number \(x\) with base \(a\) is the number \(y\) such that \(a^y = x\). The logarithm of \(x\) with base \(a\) is denoted as \(\log_a(x)\).

Restrictions#

The logarithm is defined only in the following cases:

\[\begin{split} \begin{cases} x > 0; \\ a \neq 1; \\ a > 0. \end{cases} \end{split}\]

\(a \neq 1\), since \(1^y = 1\) for all real numbers \(y\). So for this case the function cannot be defined uniquely.

\(0^y = 0\) for all \(y \geq 0\) and not defined for \(y < 0\). So for this case the function cannot be defined uniquely.

If \(a < 0\), \(a^y\) changes the sign for even and odd values of \(y\). Therefore, for sertain values of \(x\), it is impossible to find such \(y\) that corresponds to the \(\log_a x\).

As \(a > 0\) it powerd to any number will return positive number - \(a^y > 0 \Rightarrow x>0\).

Note: You can find such \(x < 0, a < 0\) that the logarithm is correct by definition, e.g. \(\log_{-2} -8 = 3\). However, it is difficult to use such logarithms in applications, se there is an argeement that \(x > 0\) and \(a > 0\).

Arithmetic progression#

An arithmetic progression is a sequence of numbers in which each element differs from the previous one by a constant amount.

Let’s denote:

  • \(a_1\): the fist element of an arithmetic progression.

  • \(d\): difference of an arithmetic progression.

By definition, each element of the progression can be counted as:

\[a_i = a_{i-1} + d, i=\overline{2,n}\]

The sequence also can be written using only \(a_1\) and \(d\).

\[a_1, a_1 + d, a_1 + 2d, ...\]

This leads to a more useful formula:

\[a_n = a_1 + (n-1)d\]

Sum#

Finding a sum of the first \(n\) members of an arithmetic progression is a very typical task. There is no issues to compute it by hands but there is a better approach - formula for \(S_n\):

\[S_n = \frac{n(a_1 + a_n)}{2}\]

To prove this formula, write the sum in the expanded form:

\[S_n = a_1 + (a_1 + d) + (a_1 + 2d) + \ldots + (a_1 + [n-1]d).\]

And the same sum in the reverse order:

\[S_n = (a_1 + [n-1]d) + (a_1 + [n-2]d) + \ldots + a_1.\]

Now, add those two sums together and group the :

\[2S_n = (2a_1 + [n-1]d) + (2a_1 + d + [n-2]d) + \ldots + (2a_1 + d + [n-2]d) + (2a_1 +[n-1]d).\]
\[S_n = \frac{n(a_1 + a_1 + [n-1]d)}{2}.\]

Finally as \(a_n = a_1 + [n-1]d\):

\[S_n = \frac{n(a_1 + a_n)}{2}.\]

Trigonometry#

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles, especially right-angled triangles.

There are three main functions in trigonometry:

  • sine: \(sin(\theta)\).

  • cosine: \(cos(\theta)\).

  • tangent: \(tan(\theta)\).

The following cell shows the most common values of the trigonometric functions.

Angle (°)

Angle (rad)

\(\sin(\theta)\)

\(\cos(\theta)\)

\(\tan(\theta)\)

0°

\(0\)

\(0\)

\(1\)

\(0\)

30°

\(\frac{\pi}{6}\)

\(\frac{1}{2}\)

\(\frac{\sqrt{3}}{2}\)

\(\frac{1}{\sqrt{3}}\)

45°

\(\frac{\pi}{4}\)

\(\frac{\sqrt{2}}{2}\)

\(\frac{\sqrt{2}}{2}\)

\(1\)

60°

\(\frac{\pi}{3}\)

\(\frac{\sqrt{3}}{2}\)

\(\frac{1}{2}\)

\(\sqrt{3}\)

90°

\(\frac{\pi}{2}\)

\(1\)

\(0\)

undefined

120°

\(\frac{2\pi}{3}\)

\(\frac{\sqrt{3}}{2}\)

\(-\frac{1}{2}\)

\(-\sqrt{3}\)

135°

\(\frac{3\pi}{4}\)

\(\frac{\sqrt{2}}{2}\)

\(-\frac{\sqrt{2}}{2}\)

\(-1\)

150°

\(\frac{5\pi}{6}\)

\(\frac{1}{2}\)

\(-\frac{\sqrt{3}}{2}\)

\(-\frac{1}{\sqrt{3}}\)

180°

\(\pi\)

\(0\)

\(-1\)

\(0\)

210°

\(\frac{7\pi}{6}\)

\(-\frac{1}{2}\)

\(-\frac{\sqrt{3}}{2}\)

\(\frac{1}{\sqrt{3}}\)

225°

\(\frac{5\pi}{4}\)

\(-\frac{\sqrt{2}}{2}\)

\(-\frac{\sqrt{2}}{2}\)

\(1\)

240°

\(\frac{4\pi}{3}\)

\(-\frac{\sqrt{3}}{2}\)

\(-\frac{1}{2}\)

\(\sqrt{3}\)

270°

\(\frac{3\pi}{2}\)

\(-1\)

\(0\)

undefined

300°

\(\frac{5\pi}{3}\)

\(-\frac{\sqrt{3}}{2}\)

\(\frac{1}{2}\)

\(-\sqrt{3}\)

315°

\(\frac{7\pi}{4}\)

\(-\frac{\sqrt{2}}{2}\)

\(\frac{\sqrt{2}}{2}\)

\(-1\)

330°

\(\frac{11\pi}{6}\)

\(-\frac{1}{2}\)

\(\frac{\sqrt{3}}{2}\)

\(-\frac{1}{\sqrt{3}}\)

360°

\(2\pi\)

\(0\)

\(1\)

\(0\)
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