A레벨_Further Mathematics(AS)_Further Pure 1 > A레벨
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Further Pure Mathematics 1
Further Pure Mathematics 1 extends horizontally from Pure Mathematics 1&2 to cover a wide range of topics suitable for those who are looking to progress to a highly numerate course in a STEM field. It serves as an introduction to more formal aspects of mathematics which are later built upon in Further Pure Mathematics 2&3
Chapter 1: Complex Numbers
Imaginary and complex numbers are introduced, along with their algebra and their applications to solving polynomials. Through Argand diagrams, the geometry of complex numbers is also introduced, and their relationship to vectors is made clear.
Chapter 2: Roots of Quadratic Equations
The concept of using the roots of quadratic equations – without explicitly calculating them – is introduced, and new equations are formed by manipulating expressions involving these roots.
Chapter 3: Numerical Solutions of Equations
The majority of functions cannot be solved analytically, and thus numerical methods must be used to find their roots. This chapter introduces three basic methods to estimate the roots of equations: Interval Bisection; Linear Interpolation; and the Newton-Raphson method.
Chapter 4: Coordinate Systems
Curves related to Conic Sections were studied as far back as the Ancient Greeks. Using the (relatively) modern Cartesian coordinate system, this chapter investigates two such curves: The Parabola and the Rectangular Hyperbola. Alongside their algebraic properties, their geometric characteristics are also developed, with the parabola being derived as a locus of points defined by a focus and a directrix.
Chapter 5: Matrices
At the most fundamental level a Matrix is just an array of numbers, but the fact they connect in powerful ways to almost all areas of mathematics makes their study essential. In this chapter, the algebra of Matrices is introduced, including finding inverses and determinants, and the non-commutative nature of their multiplication is appreciated.
Chapter 6: Transformations using Matrices
All linear transformations can be represented using matrices. Linear transformations are using throughout mathematics, and the sciences in general, making matrices indispensable tools. This chapter we looks at the geometric effects of matrices including reflections and rotations, inverse transformations, and how the determinant of a matrix represents its scaling factor.
Chapter 7: Series
Sigma notation for summations is introduced and standard results up to the sum of cubes are used to find both general formulae and evaluate explicit sums which are linear combinations of these standard results.
Chapter 8: Proof
Here we look at a powerful and formal method of proof known as Proof by Induction. Ironically, it is actually a form of deductive reasoning. Firstly the concept of logic of Proof by Induction is introduced, then it is applied to four different cases: Proving the validity of summation formulae; proving the divisibility of integer sequences; proving general term expressions derived from recurrence relations; and proving results involving the repeated multiplication of matrices.
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