01. Complex Numbers
Real and imaginary parts, conjugates, modulus, argument and complex arithmetic.
NUMBER SYSTEMS✦ ADVANCED MATHEMATICS SYSTEM
Further
Mathematics
Explore complex numbers, matrices, vectors, advanced calculus, differential equations and mathematical proof through clear lessons and interactive practice.
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02. Matrices
Matrix operations, determinants, inverses and solving simultaneous equations.
LINEAR ALGEBRA03. Vectors
Vector arithmetic, magnitude, direction, scalar products and geometric applications.
GEOMETRY04. Advanced Integration
Integration techniques, definite integrals and applications to area.
CALCULUS05. Advanced Differentiation
Rates of change, stationary points, product rules and chain rules.
CALCULUS06. Series
Arithmetic and geometric progressions, sums and convergence ideas.
SEQUENCES07. Binomial Expansion
Expand powers, calculate coefficients and work with general binomial terms.
ALGEBRA08. Differential Equations
Understand derivatives in equations and solve introductory differential equations.
MODELLING09. Mathematical Proof
Construct logical arguments, direct proofs and proof by contradiction.
REASONING10. Numerical Methods
Approximate roots, use iteration and understand numerical accuracy.
APPROXIMATION11. Further Trigonometry
Use identities, equations, radians and trigonometric relationships.
TRIGONOMETRY12. Mathematical Modelling
Translate real situations into equations, state assumptions and interpret results.
APPLICATIONSInteractive Lessons
Read the explanations and formulas before attempting the practice test.
01 — Complex Numbers
A complex number has a real part and an imaginary part. The imaginary unit \(i\) satisfies \(i^2=-1\).
z = a + bi
Re(z) = a
Im(z) = b
i² = −1
Re(z) = a
Im(z) = b
i² = −1
Example: if \(z=3+4i\), its real part is 3 and its imaginary part is 4.
|z| = √(a² + b²)
For \(z=3+4i\), the modulus is \(\sqrt{3^2+4^2}=5\).
02 — Complex Conjugates
The conjugate of a complex number is formed by changing the sign of its imaginary part.
z = a + bi
z̄ = a − bi
z z̄ = a² + b²
z̄ = a − bi
z z̄ = a² + b²
Example: the conjugate of \(2+3i\) is \(2-3i\).
Multiplying a complex number by its conjugate produces a real number.
03 — Matrices
A matrix is a rectangular arrangement of numbers. Addition requires matrices of the same dimensions.
A = [a b; c d]
det(A) = ad − bc
det(A) = ad − bc
For a 2 × 2 matrix, the determinant is the top-left entry times the bottom-right entry minus the other diagonal product.
A⁻¹ = 1/(ad − bc) [d −b; −c a]
The inverse exists only when the determinant is non-zero.
04 — Vectors
A vector has magnitude and direction. In two dimensions, its components can be written as \((a,b)\).
|(a,b)| = √(a² + b²)
Example: the magnitude of \((3,4)\) is 5.
(a,b) · (c,d) = ac + bd
If the scalar product is zero, two non-zero vectors are perpendicular.
05 — Differentiation
Differentiation measures the instantaneous rate of change of a function.
d/dx (xⁿ) = nxⁿ⁻¹
Example:
y = x³
dy/dx = 3x²
dy/dx = 3x²
At a stationary point, the first derivative is zero, though the point may be a maximum, minimum or stationary point of inflection.
06 — Integration
Integration reverses differentiation for many functions and can calculate accumulated quantities.
∫xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1
Example:
∫3x² dx = x³ + C
The constant \(C\) appears in indefinite integration because a constant disappears when differentiated.
07 — Arithmetic Progressions
An arithmetic progression has a constant difference between consecutive terms.
uₙ = a + (n − 1)d
Sₙ = n/2 [2a + (n − 1)d]
Here \(a\) is the first term and \(d\) is the common difference.
08 — Geometric Progressions
A geometric progression has a constant ratio between consecutive terms.
uₙ = arⁿ⁻¹
Sₙ = a(1 − rⁿ)/(1 − r), r ≠ 1
For an infinite geometric series, the sum exists when \(|r|<1\):
S∞ = a/(1 − r)
09 — Binomial Expansion
For a non-negative integer \(n\), the binomial theorem expands a power of a sum.
(a + b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳbʳ
Example:
(x + 2)² = x² + 4x + 4
The coefficients come from binomial coefficients, which can be calculated using factorials.
ⁿCᵣ = n! / [r!(n − r)!]
10 — Differential Equations
A differential equation relates a function to one or more of its derivatives.
dy/dx = ky
One general solution is:
y = Aeᵏˣ
The constant \(A\) is determined using an initial condition when one is given.
11 — Mathematical Proof
A proof is a logically complete argument showing that a statement follows from accepted facts or assumptions.
Direct proof: begin with known information and logically derive the result.
Contradiction: assume the statement is false and show that this leads to a contradiction.
A few examples supporting a statement do not, by themselves, prove a universal mathematical claim.
12 — Numerical Methods
Numerical methods can approximate solutions when exact algebraic solutions are difficult to obtain.
xₙ₊₁ = g(xₙ)
In fixed-point iteration, repeatedly apply the function \(g\) to the previous approximation.
Convergence depends on the function and starting value. Always consider the required accuracy and check whether the method converges.
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