01. Coordinates, Graphs and Lines

Dated: 30-10-2024

What is Calculus?

Study of rate of change of one quantity with respect to other quantity.

Pictorial Summary of Hierarchy of Real Numbers

Here, the following sets¹ are represented:

\(\mathbb{C}\) for complex numbers.
\(\mathbb{R}\) for real numbers.
\(\mathbb{Q}\) for rational numbers.
\(\mathbb{Q}^{\prime}\) for irrational numbers.
\(\mathbb{Z}\) for integers numbers.
\(\mathbb{N}\) for natural numbers.

Coordinate line

Order Properties

When we compare 2 distinct numbers, one of them is going to be either \(>\) or \(<\).

Inequalities

They can contain multiple possibilities. Here's an example:

\[2 \le 6\]

This inequality consists of 2 possibilities.

\[2 < 6\]

\[2 = 6\]

Since the first one is true so the inequality itself is true.

Theorems

\[a < b \land b < c \implies a < c\]

\[a < b \implies a + c < b + c \implies a - c < b - c\]

\[a < b \implies ac < bc \text{ where } c > 0\]

\[a < b \implies ac > bc \text{ where } c < 0\]

\[a < b \land c < d \implies a + c < b + d\]

\[a < b \implies \frac{1}{a} > \frac{1}{b}\]

Intervals

We can define them as sets¹ in the following way:

For closed intervals:

\[[a, b]=\{x:a \le x \le b\}\]

For open intervals:

\[(a, b)=\{x:a < x < b\}\]

Geometrically, we can show it as:

Here, the red line represents the whole set¹ of values which are elements of the interval we have defined.

Solving Inequalities

Take for example:

\[3 + 7x \le 2x - 9\]

Then we apply the theorems of inequalities on it.

\[3 - 3 + 7x \le 2x - 9 - 3\]

\[7x \le 2x - 12\]

\[7x - 2x \le 2x - 2x - 12\]

\[5x \le - 12\]

\[\frac{1}{5} \cdot 5x \le - 12 \cdot \frac{1}{5}\]

\[x \le \frac{-12}{5}\]