Podcast
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|.
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Write down the conjugate of (i) $-3 + \sqrt{-1}$ (ii) $i^3$
Write down the conjugate of (i) $-3 + \sqrt{-1}$ (ii) $i^3$
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Find the modulus of $|\frac{1 + i}{1 - i} - \frac{1 - i}{1 + i}|$
Find the modulus of $|\frac{1 + i}{1 - i} - \frac{1 - i}{1 + i}|$
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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$\left(\frac{z_1z_2}{z_1}\right)$.
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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.
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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) $x + 4iy = ix + y + 3$
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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.
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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.
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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 $(x^2 + y^2) = 1$.
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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.
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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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Description
This quiz focuses on the simplification and operations involving complex numbers. You'll learn to perform addition, subtraction, multiplication, and the use of conjugates, as well as representation in different forms. Prepare to calculate moduli and understand the significance of complex numbers in mathematics.
