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 ↓


Operations with Complex Numbers


Complex numbers may seem like a difficult concept at first glance, but they are actually quite simple to understand once you understand them. In this comprehensive guide, we will explore what complex numbers are, how to perform operations with them, and provide numerous examples as well.

Understanding complex numbers

At the root of complex numbers is the imaginary unit i, which is defined as the square root of -1. Therefore, i2 = -1. A complex number is a number that has two parts: a real part and an imaginary part. It can be represented as:

    A+ Bye

Here, a is the real part, and bi is the imaginary part. For example, 3 + 4i is a complex number where 3 is the real part, and 4i is the imaginary part.

Visual representation of complex numbers

Complex numbers can be easily visualized on the complex plane. The complex plane is similar to the Cartesian plane, where the horizontal axis (x-axis) represents the real component and the vertical axis (y-axis) represents the imaginary component.

3 + 4i Real axis Imaginary axis

In this diagram, the point (250, 150) represents the complex number 3 + 4i. The x-coordinate (250) is 3 units from the imaginary axis, and the y-coordinate (150) is 4 units from the real axis.

Operations on complex numbers

Complex numbers can be added, subtracted, multiplied, and divided just like regular numbers. Let's look at each operation in detail.

Sum of complex numbers

To add two complex numbers, you just need to add their corresponding real parts and their corresponding imaginary parts. Consider the complex numbers (3 + 4i) and (1 + 2i).

    (3 + 4i) + (1 + 2i) = (3 + 1) + (4i + 2i) = 4 + 6i

So the sum of 3 + 4i and 1 + 2i is 4 + 6i.

Subtraction of complex numbers

Subtracting complex numbers follows the same process as addition. You subtract the corresponding real parts and imaginary parts. Consider (5 + 6i) and (2 + 3i).

    (5 + 6i) – (2 + 3i) = (5 – 2) + (6i – 3i) = 3 + 3i

Therefore, the difference is 3 + 3i.

Multiplication of complex numbers

To multiply complex numbers we need to use the distributive property, treating i as a regular variable with the special rule that i2 = -1. Let's multiply (2 + 3i) and (4 + i).

    (2 + 3i) * (4 + i) = 2*4 + 2*i + 3i*4 + 3i*i
                      = 8 + 2i + 12i + 3(-1)
                      = 8 + 14i – 3
                      = 5 + 14i

Hence the product of 2 + 3i and 4 + i is 5 + 14i.

Division of complex numbers

Dividing complex numbers involves multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of the complex number a + bi is a - bi.

Consider dividing (1 + 2i) by (3 + 4i). First, find the conjugate of the denominator 3 + 4i, which is 3 - 4i, and multiply both the numerator and denominator by this conjugate.

    (1 + 2i) / (3 + 4i) * (3 - 4i) / (3 - 4i)
    = [(1 + 2i) * (3 - 4i)] / [(3 + 4i) * (3 - 4i)]

Now, let's multiply the fractions:

    (1 + 2i)(3 - 4i)
    = 1*3 + 1*(-4i) + 2i*3 + 2i*(-4i)
    = 3 - 4i + 6i - 8i2
    = 3 + 2i + 8 (since i2 = -1)
    = 11 + 2i

Next, we multiply the denominators:

    (3 + 4i)(3 - 4i)
    = 3*3 + 3*(-4i) + 4i*3 + 4i*(-4i)
    = 9 - 12i + 12i - 16i2
    = 9 + 16 (since i2 = -1)
    = 25

Ultimately the division becomes like this:

    (11 + 2i) / 25

Thus, the result of dividing 1 + 2i by 3 + 4i is 11/25 + (2/25)i.

Conjugate of a complex number

The conjugate of a complex number is obtained by changing the sign of its imaginary part. If the complex number is a + bi, then its conjugate is a - bi. As demonstrated earlier, the conjugate plays an important role in the division of complex numbers.

Polar form of complex numbers

In addition to the standard form a + bi, complex numbers can also be expressed in polar form, where they are represented as magnitudes and angles relative to the positive real axis. The polar form is written as:

    r(cos(θ) + i sin(θ))

Here, r is the magnitude or modulus, which is the distance from the origin to the point, and θ (theta) is the argument, or the angle made with the positive real axis.

Example of conversion to polar form

Consider the complex number 3 + 4i. To convert it to polar form, first find the magnitude r:

    r = √(a2 + b2) = √(32 + 42) = √(9 + 16) = √25 = 5

Next, find the angle θ, which can be found using the arctangent function:

    θ = tan-1(b/a) = tan-1(4/3)

Now, the polar form of 3 + 4i is:

    5(cos(tan-1(4/3)) + i sin(tan-1(4/3)))

Cube and higher order roots of complex numbers

Just like real numbers, complex numbers can have roots. The nth root of a complex number can be found using de Moivre's theorem, which states that for a complex number in polar form:

    zn = [r(cos(θ) + i sin(θ))]n = rn(cos(nθ) + i sin(nθ))

For example, finding the cube root of 8 involves finding the roots of 8 in polar form, which are as follows:

    2(cos(0 + 2πk/3) + i sin(0 + 2πk/3)) for k = 0, 1, 2

Calculate each value to get the cube root.

Conclusion

Operations with complex numbers may seem challenging at first, but they are very simple once you understand the basics. We have covered many basic operations and given many examples. Remember that practice makes perfect, and working on different problems will strengthen your understanding of complex numbers and their operations.


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Operations with Complex Numbers

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