Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Algebra → Complex Numbers ↓


Complex Conjugates


Understanding complex numbers

Complex numbers are numbers that have a real part and an imaginary part. They are written in this form:

a + bi

Here, a is the real part and b is the imaginary part. i is a symbol that represents the square root of -1. Complex numbers are an extension of the regular numbers or real numbers, which we use every day. They give us ways to solve equations that do not have solutions in real numbers.

Introduction to complex conjugates

A complex conjugate is the partner of a complex number. If you have a complex number a + bi, its complex conjugate is a - bi. The conjugate simply flips the sign of the imaginary part.

Example of complex conjugates

Let's look at some examples:

  • If the complex number is 3 + 4i, then its complex conjugate is 3 - 4i.
  • If the complex number is -1 + 7i, then its complex conjugate is -1 - 7i.
  • If the complex number is 5 - 9i, then its complex conjugate is 5 + 9i.

Geometrical interpretation

Complex numbers can be viewed on a plane called the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.

        
        
        
        
        
        
        A+ Bye
        A - Bye
        Real
        Imaginary

In the diagram above, the blue line represents the complex number a + bi, and the red line represents its complex conjugate a - bi. You can see that they are symmetric with respect to the real axis.

Properties of complex conjugates

Some interesting properties of complex conjugates are:

  1. Conjugate of a sum: The conjugate of the sum of two complex numbers is the sum of their conjugates. If z = a + bi and w = c + di, then the conjugate of z + w is: 
    (a + c) - (b + d)i = (a - bi) + (c - di)
  2. Conjugate of a product: The conjugate of the product of two complex numbers is the product of their conjugates. If z = a + bi and w = c + di, then the conjugate of z * w is: 
    ((a * c) - (b * d)) - ((a * d) + (b * c))i
  3. Magnitude: A complex number and its conjugate have the same magnitude (or modulus). If the complex number z = a + bi, then the magnitude is: 
    |z| = |a + bi| = √(a² + b²)
  4. Multiplying a complex number by its conjugate: When a complex number is multiplied by its conjugate, the result is a real number: 
    (a + bi)(a - bi) = a² + b²

Uses of complex compounds

Complex conjugates are very useful in dividing complex numbers, finding the modulus and argument of a complex number, and solving polynomial equations where real coefficients give complex solutions.

Division of complex numbers

To divide two complex numbers, you can multiply both the numerator and the denominator by the conjugate of the denominator. Here's an example:

Let's say you want to divide 1 + 2i by 3 + 4i. You would do the following:

        (1 + 2i) / (3 + 4i) Multiply by conjugate of denominator: (1 + 2i) * (3 - 4i) / ((3 + 4i) * (3 - 4i)) = (3 + 2i - 4i - 8) / (9 - 16i²) = (-5 - 2i) / (25) = -1/5 - 2/25 i
        (1 + 2i) / (3 + 4i) Multiply by conjugate of denominator: (1 + 2i) * (3 - 4i) / ((3 + 4i) * (3 - 4i)) = (3 + 2i - 4i - 8) / (9 - 16i²) = (-5 - 2i) / (25) = -1/5 - 2/25 i

Conjugate root theorem

The complex conjugate root theorem states that if the coefficients of a polynomial are real, then the non-real complex roots lie in conjugate pairs. This means that if a + bi is a root, then a - bi will also be a root.

Example of using the conjugate root theorem

Consider the polynomial equation:

x² + 2x + 5 = 0

The differentiator is:

Δ = b² - 4ac = 2² - 4*1*5 = 4 - 20 = -16

Since the discriminant is negative, its roots are complex. Using the quadratic formula, the roots are:

        x = [-b ± √(b² - 4ac)] / 2a = [-2 ± √(-16)] / 2 = [-2 ± 4i] / 2 = -1 ± 2i
        x = [-b ± √(b² - 4ac)] / 2a = [-2 ± √(-16)] / 2 = [-2 ± 4i] / 2 = -1 ± 2i

These can be written as -1 + 2i and -1 - 2i, showing that the roots are complex conjugates of each other.

Practice problems

  1. Find the complex conjugate of -5 + 12i.
  2. If z = 7 - 3i, calculate zz*, where z* is the complex conjugate of z.
  3. 2 + 5i Divide by 1 - 3i.
  4. Show that the conjugates of 2 + 3i and 2 - 3i are equal.
  5. Use complex conjugates to find a quadratic equation with real coefficients that has roots 1 + i and 1 - i.

Complex conjugates simplify many operations with complex numbers, making it easier to find solutions to more advanced mathematical problems. By understanding and mastering them, you can step into the wide world of complex number applications with confidence.


Grade 11 → 1.2.4


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Complex Conjugates

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