True or False? The modulus of a complex number is the distance from the origin to the complex number on an Argand diagram?

True. The modulus of a complex number is the distance from the origin to the complex number on an Argand diagram.

What is the argument of a complex number?

The argument of a complex number is the angle that it makes when measured counter-clockwise from the positive real axis when plotted on an Argand diagram.

What is an Argand diagram ?

An Argand diagram is a two-dimensional plane on which complex numbers can be represented.

Which type of numbers are plotted on the horizontal axis of an Argand diagram, real or imaginary ?

Real numbers are plotted on the horizontal axis of an Argand diagram.

True or False? The real part of the complex solutions to a quadratic equation will have the same value as the x -coordinate of the turning point on the graph of the quadratic.

True. The real part of the complex solutions to a quadratic equation will have the same value as the x -coordinate of the turning point on the graph of the quadratic.

IBMathsDPApplications & Interpretation (AI)HLFlashcards1. Number & AlgebraComplex Numbers

Complex Numbers (DP IB Applications & Interpretation (AI))

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FrontIntroduction to Complex Numbers
What is the imaginary number, $i$ ?

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What is the imaginary number, $i$ ?
The imaginary number $i$ is solution to $x^{2} = - 1$ . It is a square root of -1.
What is a complex number?
A complex number can have both a real part and an imaginary part.
For example, $2 + 5 i$ is a complex number.
What is denoted by $Re (z)$ ?
$Re (z)$ is used to denote the real part of the complex number $z$ .
For example, $Re (2 + 3 i) = 2$ .
What is denoted by $Im (z)$ ?
$Im (z)$ is used to denote the imaginary part of the complex number $z$ .
For example, $Im (2 + 3 i) = 3$ .
True or False?
$2 + 3 i$ and $4 + 6 i$ represent the same complex number.
False.
$2 + 3 i$ and $4 + 6 i$ are different complex numbers.
Complex numbers are only equal if their real parts are equal and their imaginary parts are equal.
What is the Cartesian form of a complex number?
The Cartesian form of a complex number is $a + b i$ where $a$ and $b$ are real numbers.
How do you add or subtract two complex numbers in Cartesian form?
To add or subtract two complex numbers in Cartesian form, you add or subtract the real parts and imaginary parts separately.
For example, $(2 + 3 i) + (5 - i) = (2 + 5) + (3 + (- 1)) i = 7 + 2 i$ .
How do you multiply two complex numbers in Cartesian form?
To multiply two complex numbers in Cartesian form, you expand brackets and use the fact that $i^{2} = - 1$ .
For example, $(2 + 3 i) (5 - i) = 10 + 15 i - 2 i - 3 i^{2} = 10 + 13 i - 3 (- 1) = 13 + 13 i$ .
If $z$ is a complex number, then what is denoted by $z *$ ?
If $z$ is a complex number, then $z *$ denotes the complex conjugate.
What is the complex conjugate of a complex number?
The complex conjugate of the complex number is where the sign of the imaginary part has been changed.
For example, the complex conjugate of $2 - 3 i$ is $2 + 3 i$ .
True or False?
For any complex number $z$ , $z + z *$ is a real number.
True.
For any complex number $z$ , $z + z *$ is a real number.
If $z$ is complex number, then is $z - z *$ a real number or an imaginary number?
If $z$ is complex number, then $z - z *$ is an imaginary number.
True or False?
For any complex number $z$ , $z \times z *$ is an imaginary number.
False.
For any complex number $z$ , $z \times z *$ is a real number.
How do you divide a complex number by another complex number in Cartesian form?
To divide a complex number by another complex number in Cartesian form:
- write as a fraction,
- multiply the numerator and denominator by the complex conjugate of the denominator,
- simplify both parts of the fractions.
For example, $(2 + 3 i) \div (5 + 4 i) = \frac{(2 + 3 i) (5 - 4 i)}{(5 + 4 i) (5 - 4 i)} = \frac{22 + 7 i}{41}$ .
What notation is used to denote the modulus of the complex number $z$ ?
$|z|$ is used to denote the modulus of the complex number $z$ .
True or False?
The modulus of a complex number is the distance from the origin to the complex number on an Argand diagram?
True.
The modulus of a complex number is the distance from the origin to the complex number on an Argand diagram.
If $z = a + b i$ , what is the formula for $|z|$ ?
If $z = a + b i$ , then $|z| = \sqrt{a^{2} + b^{2}}$ .
True or False?
The modulus of $2 - 3 i$ is $\sqrt{2^{2} - 3^{2}}$ .
False.
The modulus of $2 - 3 i$ is $\sqrt{2^{2} + {(- 3)}^{2}}$ , which is just $\sqrt{2^{2} + 3^{2}}$ .
What is the formula that connects a complex number and its conjugate with its modulus?
The formula that connects a complex number and its conjugate with its modulus is $z \times z * = {|z|}^{2}$ .
True or False?
If $z_{1}$ and $z_{2}$ are any two complex numbers then $|z_{1} + z_{2}| = |z_{1}| + |z_{2}|$ .
False.
If $z_{1}$ and $z_{2}$ are any two complex numbers then, in general, $|z_{1} + z_{2}| \neq |z_{1}| + |z_{2}|$ .
True or False?
If $z_{1}$ and $z_{2}$ are any two complex numbers then $|z_{1} \times z_{2}| = |z_{1}| \times |z_{2}|$ .
True.
If $z_{1}$ and $z_{2}$ are any two complex numbers then $|z_{1} \times z_{2}| = |z_{1}| \times |z_{2}|$ .
True or False?
If $z_{1}$ and $z_{2}$ are any two complex numbers then $|\frac{z_{1}}{z_{2}}| = \frac{|z_{1}|}{|z_{2}|}$ .
True.
If $z_{1}$ and $z_{2}$ are any two complex numbers then $|\frac{z_{1}}{z_{2}}| = \frac{|z_{1}|}{|z_{2}|}$ .
What is the argument of a complex number?
The argument of a complex number is the angle that it makes when measured counter-clockwise from the positive real axis when plotted on an Argand diagram.
What notation is used to denote the argument of a complex number $z$ ?
$\arg (z)$ is used to denote the argument of the complex number $z$ .
True or False?
$\arg (0) = 0$ .
False.
$\arg (0) \neq 0$ , $\arg (0)$ is undefined.
True or False?
If $z_{1}$ and $z_{2}$ are any two complex numbers then $\arg (z_{1} \times z_{2}) = \arg (z_{1}) \times \arg (z_{2})$ .
False.
If $z_{1}$ and $z_{2}$ are any two complex numbers then $\arg (z_{1} \times z_{2}) = \arg (z_{1}) + \arg (z_{2})$ . If you multiply two complex numbers you add their arguments.
Write a formula for $\arg (\frac{z_{1}}{z_{2}})$ in terms of $\arg (z_{1})$ and $\arg (z_{2})$ .
$\arg (\frac{z_{1}}{z_{2}}) = \arg (z_{1}) - \arg (z_{2})$ .
How do you find the argument of the complex number $z = a + b i$ ?
The argument of the complex number $z = a + b i$ is a solution to $\tan θ = \frac{y}{x}$ .
You might need to add or subtract $π$ from the principal angle, depending on where the complex number lies in the Argand diagram.
What is an Argand diagram?
An Argand diagram is a two-dimensional plane on which complex numbers can be represented.
Which type of numbers are plotted on the horizontal axis of an Argand diagram, real or imaginary?
Real numbers are plotted on the horizontal axis of an Argand diagram.
True or False?
The complex number $2 - 3 i$ can be thought of as the point with coordinates $(2, - 3)$ on an Argand diagram.
True.
The complex number $2 - 3 i$ can be thought of as the point with coordinates $(2, - 3)$ on an Argand diagram.
Just remember to interpret the coordinates $(x, y)$ a a complex number $x + y i$ .
What determines if a quadratic equation has complex roots?
A quadratic equation has complex roots if it has no real roots.
Therefore, it has complex roots if its discriminant is negative, $(b^{2} - 4 a c < 0)$ .
What is the relationship between complex roots of a quadratic with real coefficients?
The complex solutions for a quadratic with real coefficients will occur in complex conjugate pairs.
I.e. $z = p + q i$ and $z = p - q i$ , where $q \neq 0$ .
True or False?
The real part of the complex solutions to a quadratic equation will have the same value as the x-coordinate of the turning point on the graph of the quadratic.
True.
The real part of the complex solutions to a quadratic equation will have the same value as the x-coordinate of the turning point on the graph of the quadratic.
Given that $z = p + q i$ and $z = p - q i$ are two solutions to the quadratic $0 = a z^{2} + b z + c$ , how can the quadratic be written in its factorised form?
Given that $z = p + q i$ and $z = p - q i$ are two solutions to the quadratic $y = a z^{2} + b z + c$ , the quadratic can be rewritten as: $a z^{2} + b z + c = (z - (p + q i)) (z - (p - q i))$ .

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