Lecture 1: All Real Numbers & the Real Line

Real Number - Can be represented as a decimal

  • Integers - 3-3,2-2, 1-1, 00, 11...
  • Non Negative Numbers - 00, 11, 22, 33...
  • Natural Numbers - 11, 22, 33
  • Rational Numbers - pq\frac{p}{q} where pp & qq are integers and q0q \neq 0
  • Irrational Numbers - Cannot be represented as a rational (ex. 2\sqrt 2, π\pi, 5\sqrt 5)

Interval - a set of all real numbers can be represented as a line on the number line

  • [include]
  • \leqslant \geqslant 
  • (exclude)
  • << >>


Number Lines

Absolute Value - denoted by x|x|, defined by formula : x={x,x0x,x0\left | x \right |=\left\{\begin{matrix} x, &x\geq 0 \\ -x, & x\leq 0 \end{matrix}\right.

  • a=a|-a| = |a|
  • ab=ab|ab| = |a||b|
  • ab=ab|\frac{a}{b}| = \frac{|a|}{|b|}
  • a+ba+b|a + b| \leq |a| + |b| triangle equality


Solving Inequality

x22x15x^2 - 2x \geq 15

x22x150x^2 - 2x - 15 \geq 0 thus we want positive value of values of x [++=+,=+][+ \cdot + = + , - \cdot - = +]

(x+3)(x5)0(x + 3)(x - 5) \geq 0

Case 1 : Both positive

x+30x + 3 \geq 0

x3x \geq -3

and

x50x - 5 \geq 0

x5x \geq 5

Case 2 : Both negative -

x+30x + 3 \leq 0

x3x \leq -3

and

x50x - 5 \leq 0

x5x \leq 5


All together \rightarrow ((-\infty, 3]3] \cup [5[5, )\infty)





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