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Complex Numbers Formulas

Definitions A complex number is written as a + bi where a and b are real numbers an i, called the imaginary unit, has the property that i2 = -1. The complex numbers a + bi and a – bi are called complex conjugate of each other. Formulas 1) Equality of complex numbers a + […]

Exponential Formulas

  \( a^p = \underbrace{a \cdot a \cdot \dots a }_p ~~~( \text{if} ~~ p \in \mathbb{N} ) \)   \( a^0 = 1 ~~~ (\text{if} ~~~ a \ne 0) \)   \( a^r \cdot a^s = a^{r+s} \)   \( \frac{a^r}{a^s} = a^{r-s} \)   \( \left( a^r\right)^s = a^{r \cdot s} \)   \( (a \cdot b)^r = […]

Solutions of Algebraic Equations

Quadric Equation: \( ax^2 + bx + c = 0 \) Solutions (roots): \( x_{1,2} = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \) If \( D = b^2 – 4ac \) is the discriminant , then the roots are 1. real and unique if D>0 2. real and equal if D=0 3. complex conjugate if D<0 Cubic Equation: \( x^3 + a_1x^2 + a_2x […]

Sets of Numbers Formulas

Definitions N: Natural numbers N0: Whole numbers Z: Integers Z+: Positive integers Z-: Negative integers Q: Rational numbers C: Complex numbers Formulas Natural numbers (counting numbers ) \( N = \{ 1, 2, 3, \dotsc \} \) Whole numbers ( counting numbers with zero ) \( N_0 = \{ 0, 1, 2, 3, \dotsc \} \) Integers […]

Set Identities Formulas

Definitions: Universal set: I Empty set: \(\emptyset\) Union of sets \(A \cup B = \{x|x \in A  or x \in B\} \) Intersection of sets \(A \cap B = \{x|x \in A  and  x \in B\} \) Complement \( A’ = \{ x  \in  l | x \in A \} \) Difference of sets \( B […]

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