Further Maths ExamSolutions Maths Edexcel

This subject is broken down into 358 topics in 7 modules:

  1. Core Pure 149 topics
  2. Further Pure 1 53 topics
  3. Further Pure 2 40 topics
  4. Further Stats 1 30 topics
  5. Further Stats 2 24 topics
  6. Further Mechanics 1 33 topics
  7. Further Mechanics 2 29 topics
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This page was last modified on 28 September 2024.

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ExamSolutions Maths

Core Pure

Real and imaginary numbers

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Real and imaginary numbers

Real and Imaginary Numbers

Basics of Real and Imaginary Numbers

  • Real numbers include all the numbers that can be found on the number line. This includes both rational and irrational numbers.
  • Imaginary numbers are numbers that cannot be found on the number line. They're based on the imaginary unit, defined as i, where i² = -1.
  • The set of real numbers is denoted by R and the set of imaginary numbers by I.

Complex Numbers

  • Complex numbers are numbers that combine real and imaginary parts. They have the form a + bi, where a is the real part and bi is the imaginary part.
  • A complex number without an imaginary part is a real number. A complex number without a real part is an imaginary number.
  • The set of complex numbers is denoted by C. It brings together all real and imaginary numbers.

Modulus and Argument of Complex Numbers

  • The modulus of a complex number is the distance from the origin to the point represented by the complex number in the complex plane. It's calculated using Pythagoras’ theorem in the form |a + bi| = √(a² + b²).
  • The argument of a complex number is the angle the line drawn from the origin to the point makes with the positive real axis. It can be calculated using trigonometric equations.

Complex Conjugate

  • The complex conjugate of a complex number is created by changing the sign between the real and imaginary parts of the original number.
  • For any complex number a + bi, its complex conjugate is a - bi.
  • Multiplication of a complex number by its conjugate results in a real number.

Polar Form of Complex Numbers

  • Complex numbers can also be represented in Polar form, which is z = r(cos θ + i sin θ), where r is the modulus and θ is the argument.
  • This representation is particularly useful when multiplying or dividing complex numbers.

The Fundamental Theorem of Algebra

  • The Fundamental Theorem of Algebra states that every non-constant polynomial equation has at least one complex root. This includes both real and imaginary roots.

De Moivre’s Theorem

  • De Moivre’s theorem is a formula that relates the powers of complex numbers to the product of their modulus and the sum of their argument and is particularly useful when dealing with high powers or roots of complex numbers.
  • The theorem states that ((r(cos θ + i sin θ))^n = r^n (cos nθ + i sin nθ)) for any real number n.

Course material for ExamSolutions Maths, module Core Pure, topic Real and imaginary numbers

ExamSolutions Maths

Further Pure 1

Triple scalar product

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Triple scalar product

Understanding Triple Scalar Product

  • The triple scalar product is an operation involving three vectors. It could also be referred to as the volume product, as it is related to the volume of a parallelepiped defined by those vectors.
  • It combines the operations of both dot and cross products.
  • The mathematical representation of a triple scalar product is a.(bxc), where a, b, and c are vectors. Also, "." represents the dot product operation and "x" denotes the cross product operation.
  • The result of a triple scalar product is a scalar.

Calculating Triple Scalar Product

  • The triple scalar product a.(bxc) can actually be calculated as the determinant of a 3x3 matrix composed by the coefficients of the vectors a, b, and c.

  • The matrix set up in the calculation of the triple scalar product takes this form:

    |a1 a2 a3| |b1 b2 b3| |c1 c2 c3|

  • The determinant of such 3x3 matrix gives the result of the triple scalar product.

Properties of the Triple Scalar Product

  • The triple scalar product obeys some useful properties:
    • Antisymmetry: If any two vectors are interchanged, the sign of the scalar product is switched.
    • Linearity in any vector: If one of the vectors is scaled or added to another vector, the triple scalar product changes linearly.
    • Zero on parallel vectors: If any two of the vectors are parallel (or co-linear), the triple scalar product is zero. This corresponds to the geometric idea of a zero volume.

Applications of the Triple Scalar Product

  • Since the absolute value of the triple scalar product gives the volume of a parallelepiped defined by the three vectors, its application is significant in the field of geometry.
  • This operation is also used in physics for calculating the work done or potential energy when forces are acting in three dimensions.

Key Ideas

  • Understanding and calculating the triple scalar product is an essential skill in working with vectors in higher-level mathematics.
  • It is not only a mathematical operation, but also carries important geometric and physical significance. This operation is a tool to bridge algebraic computation and geometric understanding.
  • Familiarity with the properties of the triple scalar product is useful in mathematical reasoning and problem-solving in vector calculus.

Course material for ExamSolutions Maths, module Further Pure 1, topic Triple scalar product

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