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Questions and Answers
Simplify $3(1 - 2i)$.
Simplify $3(1 - 2i)$.
-3 + 6i
Simplify $(2 + 3i)(4 - 5i)$.
Simplify $(2 + 3i)(4 - 5i)$.
23 + 2i
Express $(4 - 3i)^3$ in the form (a + ib).
Express $(4 - 3i)^3$ in the form (a + ib).
-44 - 117i
If $z = (\sqrt{2} - \sqrt{3}i)$, find Re(z), Im(z), \bar{z} and |z|.
If $z = (\sqrt{2} - \sqrt{3}i)$, find Re(z), Im(z), \bar{z} and |z|.
Write down the conjugate of (i) $-3 + \sqrt{-1}$ (ii) $i^3$
Write down the conjugate of (i) $-3 + \sqrt{-1}$ (ii) $i^3$
Find the modulus of $|\frac{1 + i}{1 - i} - \frac{1 - i}{1 + i}|$
Find the modulus of $|\frac{1 + i}{1 - i} - \frac{1 - i}{1 + i}|$
If $z_1 = (1 - i) $ and $z_2 = (-2 + 4i)$, find Im$\left(\frac{z_1z_2}{z_1}\right)$.
If $z_1 = (1 - i) $ and $z_2 = (-2 + 4i)$, find Im$\left(\frac{z_1z_2}{z_1}\right)$.
Find the least positive integral value of $m$ for which $\left( \frac{1 + i}{1 - i} \right)^m$ = 1.
Find the least positive integral value of $m$ for which $\left( \frac{1 + i}{1 - i} \right)^m$ = 1.
Find the value of $x$ and $y$, if (i) $2 + (x + iy) = (3 - i)$ (ii) $x + 4iy = ix + y + 3$
Find the value of $x$ and $y$, if (i) $2 + (x + iy) = (3 - i)$ (ii) $x + 4iy = ix + y + 3$
Show that $\left(\frac{\sqrt{7} + i\sqrt{3}}{\sqrt{7} - i\sqrt{3}} + \frac{\sqrt{7} - i\sqrt{3}}{\sqrt{7} + i\sqrt{3}}\right)$ is real.
Show that $\left(\frac{\sqrt{7} + i\sqrt{3}}{\sqrt{7} - i\sqrt{3}} + \frac{\sqrt{7} - i\sqrt{3}}{\sqrt{7} + i\sqrt{3}}\right)$ is real.
Find real values of $\theta$ for which $\left(\frac{3 + 2isin\theta}{1 - 2isin \theta}\right)$ is purely real.
Find real values of $\theta$ for which $\left(\frac{3 + 2isin\theta}{1 - 2isin \theta}\right)$ is purely real.
If (x + iy) = $\frac{(a + ib)}{(a - ib)}$, prove that $(x^2 + y^2) = 1$.
If (x + iy) = $\frac{(a + ib)}{(a - ib)}$, prove that $(x^2 + y^2) = 1$.
If z = (x + iy) and w = $\frac{1 - iz}{z - i}$ such that |w| = 1, then show that z is purely real.
If z = (x + iy) and w = $\frac{1 - iz}{z - i}$ such that |w| = 1, then show that z is purely real.
Flashcards
Complex Number Form
Complex Number Form
A complex number expressed as a + bi, where a, b are real numbers and i is the imaginary unit.
Conjugate of a Complex Number
Conjugate of a Complex Number
For a complex number a + bi, its conjugate is a - bi.
Modulus of a Complex Number
Modulus of a Complex Number
The modulus |z| of a complex number z = a + bi is calculated as √(a² + b²).
Real Part of a Complex Number
Real Part of a Complex Number
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Imaginary Part of a Complex Number
Imaginary Part of a Complex Number
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Multiplying Complex Numbers
Multiplying Complex Numbers
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Dividing Complex Numbers
Dividing Complex Numbers
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Purely Real Complex Number
Purely Real Complex Number
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Purely Imaginary Complex Number
Purely Imaginary Complex Number
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Complex Numbers Addition
Complex Numbers Addition
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Exponentiation of Complex Numbers
Exponentiation of Complex Numbers
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Real Values from Complex Expressions
Real Values from Complex Expressions
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Least Positive Integral Value
Least Positive Integral Value
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Complex Number Equalities
Complex Number Equalities
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Complex Number Inequalities
Complex Number Inequalities
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Expression Simplification
Expression Simplification
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Finding Imaginary Component
Finding Imaginary Component
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Real Values from Sinusoidal Functions
Real Values from Sinusoidal Functions
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Complex Modulus Simplification
Complex Modulus Simplification
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Study Notes
Complex Numbers Simplification and Operations
- Simplify complex numbers of the form (a + bi) involving multiplication, addition, subtraction, and powers.
- Example: 3(1 - 2i) - (-4 - 5i) + (-8 + 3i) = -9 + 4i
- Example: (2+3i)(4-5i) = 23 - 7i
Complex Number Representation
- Express complex numbers in different forms (rectangular, polar, exponential).
- Express complex numbers in the form a + bi where a and b are real numbers.
- Example (4 - 3i)³ can be expressed in a + bi form.
Conjugates
- Find the complex conjugate of a complex number.
- The conjugate of a + bi is a - bi.
- Example: Conjugate of (-3 + √−1) = (-3 - i)
- Example: Conjugate of i³ is -i
Modulus of a Complex Number
- Calculate the modulus (absolute value) of a complex number.
- |z| = √(x² + y²) where z = x + yi.
- Example: | √2 + i√3 | = √5
- Calculate the modulus of (1 + i) / (1 - i) = √2, which is the magnitude of the complex number.
Operations with Complex Numbers
- Calculate the real and imaginary parts of a complex number.
- Re(√2 - √-3) = √2, Im(√2 - √-3) = - √3
- Find the sum and difference of complex numbers.
- Find the product and quotient of complex numbers.
- Example: If z₁ = 1 - i and z₂ = -2 + 4i, find Im(z₁z₂/z₁) = 1
- Calculate (z₁z₂)/z₁
Solving for Integer Values
- Determine the least positive integer value of m that satisfies equation for a complex number.
- Example: Find m if (1 + i)^(m) / (1 − i)^(m) = 1
- Answer for this example is m = 4
Real Parts and Imaginary Parts
- Determine values of x and y using equations involving complex numbers.
- Example: if 2 + (x + yi) = 3 - i, then x = 1 and y = -1
- Example equations: x + 4iy = ix + y + 3
Real Numbers from Complex Number Operations
- Determine complex number expressions which simplify to real numbers.
- Show expressions such as (√7 + i√3) / (√7 − i√3) + (√7 − i√3) / (√7 + i√3) that are real.
Prove Equality of Expressions
- Prove an algebraic equality involving complex values such as (x² + y²) = 1
- if (x + iy) = (a + i b)/(a - i b).
Complex Numbers and Modulus
- If | w | = 1 and w = (1 - iz) / (z - i), show that z is purely real.
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