Topics
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
The imaginary number i is the square root of −1:
In general, imaginary numbers are of the form bi,b∈R∖{0}. Notice that
Conclusion: the square of any imaginary number is negative.
A complex number
is the sum of a real number a and an imaginary number bi. We call this the Cartesian form for a complex number.
When a quadratic
has
it has no real roots since the square root in
is not a real number. Instead, the square root will give an imaginary number, making the roots complex.
For a complex number z=a+bi, we call a the real part of z and b the imaginary part of z:
For example,
is a complex number with a real part Re(z)=2 and imaginary part Im(z)=−3.
Real numbers are a subset of complex numbers a+bi where b=0. Imaginary numbers are also a subset of complex numbers with a=0.
The product of two complex numbers in Cartesian form is
Complex numbers can be visualized in the complex plane, also known as the Argand Diagram.
To plot a complex number, the real part determines the x-coordinate and the imaginary part determines the y-coordinate. Therefore the complex number a+bi has coordinates (a,b) on the plane.
It is conventional to use arrows from the origin to the point (a,b) to represent complex numbers.
The conjugate of a complex number z is the complex number with the same real component and the opposite imaginary component:
Since the real components of z and z∗ are the same, and the imaginary components are opposite, on the complex plane z∗ is the reflection of z in the x-axis.
The following properties hold for complex conjugates:
Fractions with complex denominator can be made real using a process analogous to rationalizing the denominator. For a fraction with a complex denominator c+di, we multiply both the numerator and the denominator by the conjugate c−di to get the fraction in a more workable form:
This allows us to split z into its real and imaginary components.
Example
We can solve complex equations involving z and z∗ by using the fact that a complex number z takes the form a+bi. Recall that for complex numbers z1 and z2,
We use this fact to equate the real and imaginary parts of both sides, which creates a solvable a system of two equations in two unknowns from one given equation.
The complex modulus ∣z∣ is a measure of the size of a complex number:
On the complex plane, z=a+bi has coordinates (a,b). Therefore
represents the distance of z from the origin:
Notice that
The following properties apply for the complex modulus:
The argument of a complex number is the angle that it forms with the real (x) axis on the complex plane:
By noticing a right angled triangle, we can say that
When a>0:
Note on convention: By convention, the argument is usually given in radians in the range [−π,π]. It will be made clear by the IB which range is preferred in a given question, and often both will be accepted.
If a<0, then argz is in the second or third quadrant, which are not in the range of arctan. We therefore need to add or subtract π to get the correct argument:
When a<0:
When z is in the second quadrant, we add π; when z is in the third quadrant, we subtract π.
Note on convention: By convention, the argument is usually given in radians in the range [−π,π]. It will be made clear by the IB which range is preferred in a given question, and often both will be accepted.
If a=0, then tan(argz)=ab is undefined. tanθ is also undefined for θ=2π,23π… So when a=0 we have
This can be seen on the complex diagram by remembering that bi lies on the yi axis:
Note on convention: By convention, the argument is usually given in radians in the range [−π,π]. It will be made clear by the IB which range is preferred in a given question.
The modulus ∣z∣ and argument argz uniquely define the complex number z. That means we can represent any complex number using its modulus and argument instead of a+bi:
It is conventional to call r=∣z∣ and θ=argz. Using trigonometry, we deduce that
And we use the shorthand cisθ=cosθ+isinθ:
There is one more way to express complex numbers:
For example, we can write z=−1=1⋅cis(π)=eiπ. This leads to the classic result
We call this Euler's form (or sometimes exponential form) because of the presence of Euler's number, e.
The main advantage of Euler's form is that it makes multiplying complex numbers much easier:
In words, when we multiply two complex numbers the arguments add and the moduli multiply.
Similarly for division:
In words, when we divide one complex number from another, we subtract the arguments and divide the moduli.
In polar form this becomes:
If z=rcisθ=reiθ, then the conjugate z∗ is
The distance in the complex plane between two points representing z and w is given by ∣z−w∣, which is the same as ∣w−z∣.
This is a reflection of the familiar distance formula
except that x1 and x2 are the real components of z and w, and y1 and y2 the imaginary components of z and w.
The angle between to complex numbers on an Argand diagram is the absolute value of the difference of their arguments:
For any two complex waves f(x)=r1sin(ax+α1) and g(x)=r2sin(ax+α2) with the same frequency but different phase, we can use Euler's form of complex numbers to express h(x) as a single sine function:
So we can write
where
Note: In most questions about adding complex waves, you can use your graphing calculator to find the phase shift instead of deriving it by hand.
Now that we know how to represent complex numbers in the form reiθ, we can understand De Moivre's Theorem, which gives us a shortcut to finding powers of complex numbers:
And since reiθ=rcisθ:
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
The imaginary number i is the square root of −1:
In general, imaginary numbers are of the form bi,b∈R∖{0}. Notice that
Conclusion: the square of any imaginary number is negative.
A complex number
is the sum of a real number a and an imaginary number bi. We call this the Cartesian form for a complex number.
When a quadratic
has
it has no real roots since the square root in
is not a real number. Instead, the square root will give an imaginary number, making the roots complex.
For a complex number z=a+bi, we call a the real part of z and b the imaginary part of z:
For example,
is a complex number with a real part Re(z)=2 and imaginary part Im(z)=−3.
Real numbers are a subset of complex numbers a+bi where b=0. Imaginary numbers are also a subset of complex numbers with a=0.
The product of two complex numbers in Cartesian form is
Complex numbers can be visualized in the complex plane, also known as the Argand Diagram.
To plot a complex number, the real part determines the x-coordinate and the imaginary part determines the y-coordinate. Therefore the complex number a+bi has coordinates (a,b) on the plane.
It is conventional to use arrows from the origin to the point (a,b) to represent complex numbers.
The conjugate of a complex number z is the complex number with the same real component and the opposite imaginary component:
Since the real components of z and z∗ are the same, and the imaginary components are opposite, on the complex plane z∗ is the reflection of z in the x-axis.
The following properties hold for complex conjugates:
Fractions with complex denominator can be made real using a process analogous to rationalizing the denominator. For a fraction with a complex denominator c+di, we multiply both the numerator and the denominator by the conjugate c−di to get the fraction in a more workable form:
This allows us to split z into its real and imaginary components.
Example
We can solve complex equations involving z and z∗ by using the fact that a complex number z takes the form a+bi. Recall that for complex numbers z1 and z2,
We use this fact to equate the real and imaginary parts of both sides, which creates a solvable a system of two equations in two unknowns from one given equation.
The complex modulus ∣z∣ is a measure of the size of a complex number:
On the complex plane, z=a+bi has coordinates (a,b). Therefore
represents the distance of z from the origin:
Notice that
The following properties apply for the complex modulus:
The argument of a complex number is the angle that it forms with the real (x) axis on the complex plane:
By noticing a right angled triangle, we can say that
When a>0:
Note on convention: By convention, the argument is usually given in radians in the range [−π,π]. It will be made clear by the IB which range is preferred in a given question, and often both will be accepted.
If a<0, then argz is in the second or third quadrant, which are not in the range of arctan. We therefore need to add or subtract π to get the correct argument:
When a<0:
When z is in the second quadrant, we add π; when z is in the third quadrant, we subtract π.
Note on convention: By convention, the argument is usually given in radians in the range [−π,π]. It will be made clear by the IB which range is preferred in a given question, and often both will be accepted.
If a=0, then tan(argz)=ab is undefined. tanθ is also undefined for θ=2π,23π… So when a=0 we have
This can be seen on the complex diagram by remembering that bi lies on the yi axis:
Note on convention: By convention, the argument is usually given in radians in the range [−π,π]. It will be made clear by the IB which range is preferred in a given question.
The modulus ∣z∣ and argument argz uniquely define the complex number z. That means we can represent any complex number using its modulus and argument instead of a+bi:
It is conventional to call r=∣z∣ and θ=argz. Using trigonometry, we deduce that
And we use the shorthand cisθ=cosθ+isinθ:
There is one more way to express complex numbers:
For example, we can write z=−1=1⋅cis(π)=eiπ. This leads to the classic result
We call this Euler's form (or sometimes exponential form) because of the presence of Euler's number, e.
The main advantage of Euler's form is that it makes multiplying complex numbers much easier:
In words, when we multiply two complex numbers the arguments add and the moduli multiply.
Similarly for division:
In words, when we divide one complex number from another, we subtract the arguments and divide the moduli.
In polar form this becomes:
If z=rcisθ=reiθ, then the conjugate z∗ is
The distance in the complex plane between two points representing z and w is given by ∣z−w∣, which is the same as ∣w−z∣.
This is a reflection of the familiar distance formula
except that x1 and x2 are the real components of z and w, and y1 and y2 the imaginary components of z and w.
The angle between to complex numbers on an Argand diagram is the absolute value of the difference of their arguments:
For any two complex waves f(x)=r1sin(ax+α1) and g(x)=r2sin(ax+α2) with the same frequency but different phase, we can use Euler's form of complex numbers to express h(x) as a single sine function:
So we can write
where
Note: In most questions about adding complex waves, you can use your graphing calculator to find the phase shift instead of deriving it by hand.
Now that we know how to represent complex numbers in the form reiθ, we can understand De Moivre's Theorem, which gives us a shortcut to finding powers of complex numbers:
And since reiθ=rcisθ: