The rectangular form of complex numbers is the first form we’ll encounter when learning about complex numbers. This form depends on its Cartesian coordinate, and you’ll actually learn why in the next section.
Rectangular forms of complex numbers represent these numbers highlighting the real and imaginary parts of the complex number.
Basic operations are much easier when complex numbers are in rectangular form. It’s more intuitive for us to graph complex numbers in rectangular form since we’re more familiar with the Cartesian coordinate system.
This article will refresh our knowledge on:
- The components that make up a complex number.
- Graphing complex numbers on a complex plane.
- Converting complex numbers in the rectangular form to polar form, and the other way around.
- Manipulating complex numbers in rectangular form.
Make sure to pull up your notes and review these concepts as we’ll be needing them as we learn more about complex numbers in rectangular form.
What is rectangular form?
The rectangular form is based on its name – a rectangular coordinate system. This means that these are complex numbers of the form $z = a + bi$, where $a$ is the real part, and $bi$ represents the imaginary part. Here are some examples of complex numbers in rectangular form.
- $-3 + 4i$ : $-3$ represents the real number part while $4i$ represents the imaginary part.
- $-6i$: This is an imaginary number that only contains an imaginary part, $-6i$.
- $5$: Since $5$ is a counting number and consequently, a real number, $5$ is still a complex number with its imaginary number part equal to $0$.
Complex numbers of the form $a + bi$ can be graphed on a complex plane simply by plotting $(a,b)$, where $a$ is the coordinate for the real axis and $b$ is the coordinate for the imaginary axis.
Here’s a graph of how $a + bi$ is graphed on a complex plane. As mentioned, $a$ represents the distance along the real axis, and $b$ represents the distance along the imaginary axis – a similar approach when we graph rectangular coordinates.
The distance formed by $a + bi$ from the origin is equal to $\sqrt{a^2 + b^2}$ or also known as the modulus or the absolute value of the complex number.
How to convert rectangular form?
As mentioned, rectangular form is the first form of complex numbers that we’ll be introduced to, but complex numbers can also be rewritten in their trigonometric or polar forms.
| Rectangular Form | Polar Form |
| $-3 + 3i$ | $3\sqrt{2}(\cos 135^{\circ} + i\sin135^{\circ})$ |
| $-2\sqrt{3} – 2i$ | $4(\cos 210^{\circ} + i\sin 210^{\circ})$ |
| $4 – 4i$ | $4\sqrt{2}(\cos 315^{\circ} + i\sin 315^{\circ})$ |
| $5 + 5\sqrt{3}i$ | $10(\cos 60^{\circ} + i\sin 60^{\circ})$ |
These are just some examples of pairs of complex numbers in their two forms: rectangular and polar forms. Let’s refresh what we’ve learned about writing complex numbers in these two forms.
How to convert rectangular form to polar form?
We’ve thoroughly discussed converting complex numbers in rectangular form, $a + bi$, to trigonometric form (also known as the polar form). Make sure to review your notes or check out the link we’ve attached in the first section.
This section will be a quick summary of what we’ve learned in the past:
- Find the modulus, $r = \sqrt{a^2 + b^2}$, of the complex number.
- Determine the argument, $\theta = \tan^{-1} \dfrac{b}{a}$, and make sure we choose the angle that lies on the right quadrant.
- Use these values and write the complex number of the form $r(\cos \theta + i\sin \theta)$.
How to convert polar form to rectangular form?
Changing complex numbers in polar form is much easier since it requires us to only evaluate cosine and sine at different values of $\theta$.
- When given complex number of the form $r(\cos \theta + i\sin \theta)$, evaluate the values of $\sin \theta$ and $\cos \theta$.
- Distribute $r$ to each of the evaluated values of $\cos \theta$ and $i\sin \theta$.
- Make sure to return the values of the form, $a + bi$.
Don’t worry. We’ve prepared some examples for you to work on to practice your knowledge of converting complex numbers in polar form.
Summary of rectangular form definition and properties
Why don’t we recap what we’ve learned so far about complex numbers in the rectangular form before diving into the different problems we’ve prepared?
- The general rectangular (or standard) form of the complex numbers is $a + bi$.
- We can convert complex numbers in rectangular form, by finding $r = \sqrt{a^2 + b^2}$ and $\theta = \tan^{-1} \dfrac{b}{a}$.
- Don’t forget, when working with equations involving complex numbers, the real number parts and imaginary number parts must be equal for the equation to be valid.
We can also do many things when given a complex number in rectangular form, and we’ve listed some that we’ve learned in the past. Don’t have your handy notes with you? No worries, we’ve added a few links for you to check out as well.
- It’s easier to add and subtract complex numbers in rectangular form since we combine the real and imaginary number parts.
- Yes, we can also multiply and divide complex numbers in rectangular form through algebraic manipulation.
- The product of a $a + bi$ and its conjugate, $a – bi$, is equal to $a^2 + b^2$, which can help simplify the quotient of two complex numbers.
Let’s apply everything that we’ve learned from this article and try out these sample problems.
Example 1
Graph the following complex numbers on the complex plane and include their corresponding absolute value number.
a. $6 – 6i$
b. $-4\sqrt{3} – 4i$
c. $-5i$
Solution
Since we also need the absolute value number of these three complex numbers, why don’t we start with that using the fact that $|a + bi| = \sqrt{a^2 + b^2}$?
| $\boldsymbol{a + bi}$ | $\boldsymbol{|a + bi| }$ |
| $6 -6i$ | $\sqrt{(6)^2 + (-6)^2} = 6\sqrt{2}$ |
| $-4\sqrt{3} -4i$ | $\sqrt{(-4\sqrt{3})^2 + (-4)^2} = 8$ |
| $-5i$ | $\sqrt{(0)^2 + (-5)^2} = 5$ |
Now that we have the absolute value of the three complex numbers let’s graph the three complex numbers on one complex plane.
- For $6 – 6i$, graph the coordinate $(6, -6)$ or $6$ units to the right and along the real axis and six units downward and along the imaginary axis.
- Similarly, we can graph $-4\sqrt{3} – 4i$ by plotting $(-4\sqrt{3}, -4)$ on the complex plane.
- Since $-5i$ only contains an imaginary number part, we graph $-5i$ on the imaginary axis and should be found $5$ units below the real axis.
Connect each complex number to the origin and label the segment with the corresponding absolute value number.
Example 2
Evaluate the following operations on the following complex numbers.
a. $(8 – 8i) + (-6 + 12i)$
b. $(-3\sqrt{3} + 5i) – (4\sqrt{3} – 6i)$
c. $(-4 + 2i)(-2 – i) + (2- 3i)$
Solution
Recall that adding and subtracting complex numbers is similar to adding and subtracting binomials. We combine the terms with the real numbers and imaginary numbers. It’s the same way that we combine “like terms.”
Let’s work on the first item first: $(8 – 8i) + (-6 + 12i)$.
$\begin{aligned} (8 – 8i) + (-6 + 12i) &= [8 + (-6)] + (-8 + 12)i\\&= 2 + 6i\end{aligned}$
Make sure to distribute the negative sign carefully when subtracting two complex numbers.
$\begin{aligned} (-3\sqrt{3} + 5i) – (4\sqrt{3} – 6i) &=-3\sqrt{3} + 5i – 4\sqrt{3} -(-6i)\\&= -3\sqrt{3} + 5i – 4\sqrt{3} + 6i\\&= (-3\sqrt{3} – 4\sqrt{3}) + (5 + 6)i\\&=-7\sqrt{3} + 11i \end{aligned}$
For the third item, multiply the two complex numbers first.
- Apply the FOIL method to distribute the terms.
- Replace $i^2$ with $-1$.
- Combine real and imaginary number parts.
$\begin{aligned} (-4 +2i)(-2 – i) &= (-4)(-2)+ (-4)(-i) + (2i)(-2) + (2i)(-i)\\&=8 + 4i – 4i- 2i^2\\&=8 + 4i – 4i -2(-1)\\&=8 + 4i – 4i + 2\\&= (8 + 2) + (4 -4)i\\&=10 \end{aligned}$
Replace $(-4 + 2i)(-2 – i)$ with its product then simplify the expression further.
$\begin{aligned} (-4 +2i)(-2 – i) + (2 – 3i) &= 10 + (2 – 3i)\\&= (10 + 2) – 3i\\&= 12 – 3i\end{aligned}$
Listing down the results for the three operations, we have the following:
a. $(8 – 8i) + (-6 + 12i) = 2 + 6i $
b. $(-3\sqrt{3} + 5i) – (4\sqrt{3} – 6i) = -7\sqrt{3} + 11i $
c. $(-4 + 2i)(-2 – i) + (2- 3i) = 12 – 3i $
Example 3
Convert the following complex numbers in polar form to rectangular form.
a. $-4(\cos 90^{\circ} + i\sin 90^{\circ})$
b. $6\left(\cos \dfrac{\pi}{3} + i\sin \dfrac{\pi}{3}\right)$
c. $-\sqrt{3} \text{cis } \dfrac{3\pi}{4}$
Solution
Evaluate the cosine and sine values inside the parenthesis when converting complex numbers to rectangular form. Distribute the modulus into each of the values inside to simplify the expression to a form, $a +bi$.
Starting with $-4(\cos 90^{\circ} + i\sin 90^{\circ})$, $\cos 90^{\circ} = 0$ and $\sin 90^{\circ} = 1$. Replace the terms inside the parenthesis with these values then distribute $-4$.
$\begin{aligned} -4(\cos 90^{\circ} + i\sin 90^{\circ}) &= -4(0 + i)\\&=0 – 4i\\&= -4i\end{aligned}$
The second item will require us to perform a similar process, but this time, we’re working with angles in terms of $\pi$. Recall that $\cos \dfrac{\pi}{3} = \dfrac{1}{2}$ and $\sin \dfrac{\pi}{3}= \dfrac{\sqrt{3}}{2}$.
$\begin{aligned} 6\left(\cos \dfrac{\pi}{3} + i\sin \dfrac{\pi}{3}\right) &= 6\left( \dfrac{1}{2} + i\dfrac{\sqrt{3}}{2}\right)\\&=6 \cdot \dfrac{1}{2} – 6 \cdot i \dfrac{\sqrt{3}}{2}\\&= 3 – 3\sqrt{3}i\end{aligned}$
For the third item, make sure to rewrite $r \text{cis } \theta$ as $r(\cos \theta + i \sin \theta)$.
$\begin{aligned} -\sqrt{3} \text{cis } \dfrac{3\pi}{4} &= -\sqrt{3}\left(\cos \dfrac{3\pi}{4} + i\sin \dfrac{3\pi}{4}\right)\\ &= -\sqrt{3}\left( -\dfrac{\sqrt{2}}{2} + i\dfrac{\sqrt{2}}{2}\right)\\&=-\sqrt{3} \cdot -\dfrac{\sqrt{2}}{2} – \sqrt{3} \cdot i \dfrac{\sqrt{2}}{2}\\&= \dfrac{\sqrt{6}}{2}-i\dfrac{\sqrt{3}}{2}\end{aligned}$
Hence, we have the following complex numbers in their rectangular forms:
a. $-4(\cos 90^{\circ} + i\sin 90^{\circ}) = -4i$
b. $6\left(\cos \dfrac{\pi}{3} + i\sin \dfrac{\pi}{3}\right) = 3 – 3\sqrt{3}i$
c. $-\sqrt{3} \text{cis } \dfrac{3\pi}{4} = \dfrac{\sqrt{6}}{2}-i\dfrac{\sqrt{3}}{2}$
FAQs
What is rectangular form example? ›
In the rectangular form, a complex number can be represented as a point on a two dimensional plane called the complex or s-plane. So for example, Z = 6 + j4 represents a single point whose coordinates represent 6 on the horizontal real axis and 4 on the vertical imaginary axis as shown.
What is a rectangular form? ›Rectangular form, on the other hand, is where a complex number is denoted by its respective horizontal and vertical components. In essence, the angled vector is taken to be the hypotenuse of a right triangle, described by the lengths of the adjacent and opposite sides.
What is rectangular form equation? ›A rectangular equation, or an equation in rectangular form is an equation composed of variables like x and y which can be graphed on a regular Cartesian plane. For example y=4x+3 is a rectangular equation.
What is polar form and rectangular form? ›Both polar and rectangular forms of notation for a complex number can be related graphically in the form of a right triangle, with the hypotenuse representing the vector itself (polar form: hypotenuse length = magnitude; angle with respect to horizontal side = angle), the horizontal side representing the rectangular “ ...
How do you simplify a rectangular form? ›Complex Numbers: Rectangular Form - YouTube
How do you find the rectangular form of a vector? ›Vectors in rectangular form - YouTube
What is rectangular form in complex numbers? ›The rectangular form of a complex number is given by. z=a+bi . Substitute the values of a and b . z=a+bi =rcosθ+(rsinθ)i =r(cosθ+isinθ) In the case of a complex number, r represents the absolute value or modulus and the angle θ is called the argument of the complex number.
How do you divide a rectangular form? ›How to Divide Complex Numbers in Rectangular Form ? we have to multiply both numerator and denominator by the conjugate of the denominator.
How do you convert an equation to rectangular form? ›To convert from polar coordinates to rectangular coordinates, use the formulas x=rcosθ and y=rsinθ.
How do you add a rectangular form? ›To add complex numbers in rectangular form, add the real components and add the imaginary components. Subtraction is similar. To multiply complex numbers in polar form, multiply the magnitudes and add the angles. To divide, divide the magnitudes and subtract one angle from the other.
How do you find a rectangular point? ›
Step 1: Find the x -coordinate for the rectangular coordinate form of the point by using the equation x=rcos(θ) x = r cos . Step 2: Find the y -coordinate for the rectangular coordinate form of the point by using the equation y=rsin(θ) y = r sin .
How do you convert from rectangular form to phasor? ›Phasor form of vector a+jb is, v = V∠θ. To convert to rectangular form, calculate the horizontal and vertical axis values for the vector V. Rectangular form of vector V∠θ is, v = a+jb.
What is polar form? ›The polar form of a complex number is a different way to represent a complex number apart from rectangular form. Usually, we represent the complex numbers, in the form of z = x+iy where 'i' the imaginary number. But in polar form, the complex numbers are represented as the combination of modulus and argument.
Is Cartesian form rectangular form? ›Cartesian form and rectangular form are two different names for the same system. A complex number “z = a + bi” form is called cartesian form or rectangular form. Read more: What is Cartesian Coordinate System?
How do you divide complex numbers in rectangular form? ›How to Divide Complex Numbers in Rectangular Form? - YouTube
How do you multiply complex numbers in rectangular form? ›Multiplication of complex numbers in Cartesian form - YouTube
How do you convert from rectangular form to exponential form? ›Exponential forms of numbers can be converted into their rectangular equivalents by the following formulas. The real value (the x value) of the rectangular form can be obtained through the formula, x= r cos(θ). The imaginary value (the y value) of the rectangular form can be obtained through the formula, y= r sin(θ).
What is rectangular unit vector? ›Answer: Rectangular unit vectors are unit vectors of unit magnitude in the direction of their respective perpendicular axis in rectangular coodinate system. Explanation: A rectangular coordinate system is a system used to locate the position of a point in space using a set of 3 mutually perpendicular lines.
What is rectangular coordinate vector? ›A rectangular coordinate system is defined, originating at the center point between the plates with z in the direction of plate separation, x in the width direction, and y in the length direction.
What is the difference between polar and rectangular coordinates? ›One big difference between polar and rectangular coordinates is that polar coordinates can have multiple coordinates representing the same point by adjusting the angle θ or the sign of r and the angle θ. conditions.
What is the rectangular form of 3pi 4? ›
The rectangular representation of the polar point (3,π4) ( 3 , π 4 ) is (3√22,3√22) ( 3 2 2 , 3 2 2 ) .
How do you convert complex numbers to rectangular polar form? ›- Complex numbers in the form a+bi are plotted in the complex plane similar to the way rectangular coordinates are plotted in the rectangular plane. ...
- The absolute value of a complex number is the same as its magnitude. ...
- To write complex numbers in polar form, we use the formulas x=rcosθ, y=rsinθ, and r=√x2+y2.
In rectangular form, the impedance is made up of a real part, called the resistance R (the same resistance that we know and love from DC!), and an imaginary part called the reactance X: Z = R + j X (Ω) Note that the unit of impedance is ohms.
How do you divide numbers? ›When we divide any number, divisor must be smaller than the number of dividend. Subtract the product from the number of dividend every time and then take the next number until all number finished. Quotient is always written above the line and product below the number of dividend.
How do you solve complex equations with division? ›Dividing Complex Numbers - YouTube
How do you solve operations with complex numbers? ›Operations With Complex Numbers - YouTube
How do you write in polar form? ›To write complex numbers in polar form, we use the formulas x=rcosθ, y=rsinθ, and r=√x2+y2. Then, z=r(cosθ+isinθ).
Which set of rectangular coordinates describes the same location as the polar coordinates? ›The set of polar coordinates that describes the same location as the regular coordinates (-5, 0) is (5, 180°).
How do you find the distance between polar coordinates? ›The Distance Formula in Polar Coordinates - YouTube
How do you convert rectangular to polar using scientific calculator FX 991ms? ›Casio FX-991ms Converting polar and rectangular - YouTube
How do you convert from rectangular to parametric? ›
Converting Parametric to Rectangular Equations - YouTube
What is the rectangular form of r − 2 sin θ )? ›r=−2sinθ=−yr . and so, r2=x2+y2=−2y .
What is phasor form? ›A “phasor” is a complex number that represents the complex amplitude and phase angle of a sinusoidal waveform or time-varying quantity. It can be represented in the mathematical: Rectangular, Polar or Exponential form. For example, (a + jb).
What do you understand by phasor? ›Definition of phasor
: a vector (as one representing an alternating current or voltage) whose vectorial angle represents a phase or phase difference.
Review the different ways in which we can represent complex numbers: rectangular, polar, and exponential forms.
What is Cartesian form? ›The cartesian form of complex numbers is represented in a two-dimensional plane. If a+ib is a complex number, then the point on the complex plane will be (a,b). Usually, the real part of a complex number is represented along the x-axis and the imaginary part is expressed along the y-axis.
What is polar form vector? ›In physics, a polar vector is a vector such as the radius vector that reverses sign when the coordinate axes are reversed. Polar vectors are the type of vector usually simply known as "vectors." In contrast, pseudovectors (also called axial vectors) do not reverse sign when the coordinate axes are reversed.
What are polar equations used for? ›From a physicist's point of view, polar coordinates (randθ) are useful in calculating the equations of motion from a lot of mechanical systems. Quite often you have objects moving in circles and their dynamics can be determined using techniques called the Lagrangian and the Hamiltonian of a system.
What is the difference between polar form and Cartesian form? ›This leads to an important difference between Cartesian coordinates and polar coordinates. In Cartesian coordinates there is exactly one set of coordinates for any given point. With polar coordinates this isn't true. In polar coordinates there is literally an infinite number of coordinates for a given point.
How do you convert rectangular to polar in Matlab? ›[ x , y ] = pol2cart( theta , rho ) transforms corresponding elements of the polar coordinate arrays theta and rho to two-dimensional Cartesian, or xy, coordinates.
How do you divide a complex number by another complex number? ›
- Multiply the given complex number by the conjugate of the denominator on both the numerator and the denominator.
- Distribute the number in both the numerator and denominator in order to eliminate the parentheses.
- Simplify the powers of i.
Simplifying complex numbers rational expression (2-4i) / (1+3i) - YouTube
What is the first step in dividing complex numbers? ›The first step when dividing complex numbers is to rationalize the denominator of the division problem by multiplying both the numerator and denominator by the complex conjugate of the denominator.
What is the rectangular form of 3pi 4? ›The rectangular representation of the polar point (3,π4) ( 3 , π 4 ) is (3√22,3√22) ( 3 2 2 , 3 2 2 ) .
How do you add a rectangular form? ›To add complex numbers in rectangular form, add the real components and add the imaginary components. Subtraction is similar. To multiply complex numbers in polar form, multiply the magnitudes and add the angles. To divide, divide the magnitudes and subtract one angle from the other.
What is rectangular form in complex numbers? ›The rectangular form of a complex number is given by. z=a+bi . Substitute the values of a and b . z=a+bi =rcosθ+(rsinθ)i =r(cosθ+isinθ) In the case of a complex number, r represents the absolute value or modulus and the angle θ is called the argument of the complex number.
Is rectangular form the same as Cartesian form? ›Cartesian form and rectangular form are two different names for the same system. A complex number “z = a + bi” form is called cartesian form or rectangular form.
How do you convert to polar form? ›To convert from rectangular coordinates to polar coordinates, use one or more of the formulas: cosθ=xr, sinθ=yr, tanθ=yx, and r=√x2+y2.
How do you divide a rectangular form? ›How to Divide Complex Numbers in Rectangular Form ? we have to multiply both numerator and denominator by the conjugate of the denominator.
How do you divide rectangular form from rectangular form? ›1.4 Division: Rectangular Form
We can use the concept of complex conjugate to give a strategy for dividing two complex numbers, z1=x1+iy1 z 1 = x 1 + i y 1 and z2=x2+iy2.
How do you convert complex form to rectangular form? ›
It should be relatively easy to see that, if a complex number z has magnitude r and argument θ, then: z=r(cosθ+isinθ) This is called the polar form of a complex number. Thus, if you want to convert from polar form to rectangular form, remember that Re(z)=rcosθ and Im(z)=rsinθ.
How do you multiply complex numbers in rectangular form? ›Multiplication of complex numbers in Cartesian form - YouTube
How do you find rectangular coordinates? ›Step 1: Find the x -coordinate for the rectangular coordinate form of the point by using the equation x=rcos(θ) x = r cos . Step 2: Find the y -coordinate for the rectangular coordinate form of the point by using the equation y=rsin(θ) y = r sin .
How do you convert trigonometric form to rectangular form? ›Converting Complex Numbers From Trigonometric Form to Rectangular
What is the difference between rectangular and polar coordinate system? ›In a rectangular coordinate system, we were plotting points based on an ordered pair of (x, y). In the polar coordinate system, the ordered pair will now be (r, θ). The ordered pair specifies a point's location based on the value of r and the angle, θ, from the polar axis.
What is rectangular coordinate system in physics? ›A Cartesian coordinate system in two dimensions (also called a rectangular coordinate system or an orthogonal coordinate system) is defined by an ordered pair of perpendicular lines (axes), a single unit of length for both axes, and an orientation for each axis.
What is the Cartesian form? ›What Is Cartesian Form? The cartesian form helps in representing a point, a line, or a plane in a two-dimensional or a three-dimensional plane. The cartesian form is represented with respect to the three-dimensional cartesian system and is with reference to the x-axis, y-axis, and z-axis respectively.