Notes for Further Studies Mathematics (AP math)

FURTHER STUDIES MATH
ALGEBRA




INEQUALITIES:


Intervals:
1. Open interval (between a and b where a and b are not included)
2. Closed interval (between a and b where a and b are included)

Rules:
1. If a < b then a + c < b +c
2. If a < b and c < d then a + c < b + d




HOW TO SOLVE:



1. Factorise to find critical values
2. Plot critical values on number line
3. Sub numbers around critical values and determine which numbers fulfil
the given formula
4. Give answers in interval notation

Note:
Never divide by x
Remember restrictions for rational inequalities
Infinity is not a number it is a representation of a very large number

, FURTHER STUDIES MATH
ALGEBRA




COMPLEX NUMBERS:


Imaginary numbers:
i is defined as i = √−1 and i^2 = −1
A solution is also known as a root and also known as a zero of a function.
If a + bi is a solution to an equation, then the conjugate is also a solution:
a − bi.
If c + √d is an irrational solution then c − √d is another irrational solution
Find the conjugate; the roots; the sum of the roots (coeff of x); the
product of roots (constant)
Alternatively place roots as factors and solve
To factorise, first find the roots and then place into factors.




HOW TO SOLVE:



For addition, subtraction and multiplication:
Use normal algebraic laws

For Division:
Use conjugate rule
Always write your answers in the form a+bi

Solving for a and b:
Equate real and imaginary parts and and solve separately