Complex numbers in MATLAB - Step-by-Step Execution

Concept Flow - Complex numbers

Define complex number z = a + bi

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Access real part: real(z)

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Access imaginary part: imag(z)

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Perform operations: +, -, *, /

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Use functions: abs(z), angle(z), conj(z)

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Display or use results

Start by defining a complex number, then access its parts, perform operations, and use built-in functions to analyze it.

Execution Sample

MATLAB

z = 3 + 4i;
realPart = real(z);
imagPart = imag(z);
magnitude = abs(z);
angleRad = angle(z);

This code creates a complex number and extracts its real part, imaginary part, magnitude, and angle.

Execution Table

Step	Variable	Action	Value
1	z	Assign complex number 3 + 4i	3 + 4i
2	realPart	Extract real part of z	3
3	imagPart	Extract imaginary part of z	4
4	magnitude	Calculate magnitude abs(z)	5
5	angleRad	Calculate angle angle(z) in radians	0.9273
6	-	End of execution	-

💡 All variables assigned and calculations done, program ends.

Variable Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	After Step 4	After Step 5	Final
z	undefined	3 + 4i	3 + 4i	3 + 4i	3 + 4i	3 + 4i	3 + 4i
realPart	undefined	undefined	3	3	3	3	3
imagPart	undefined	undefined	undefined	4	4	4	4
magnitude	undefined	undefined	undefined	undefined	5	5	5
angleRad	undefined	undefined	undefined	undefined	undefined	0.9273	0.9273

Key Moments - 3 Insights

Why does imag(z) return 4 and not 4i?

imag(z) returns the coefficient of the imaginary part as a real number, not including the 'i' symbol. See execution_table step 3 where imagPart is 4.

Why is the magnitude 5 for z = 3 + 4i?

Magnitude is calculated as sqrt(real^2 + imag^2) = sqrt(3^2 + 4^2) = 5, shown in execution_table step 4.

What unit is the angle returned by angle(z)?

angle(z) returns the angle in radians, not degrees. In execution_table step 5, angleRad is 0.9273 radians.

Visual Quiz - 3 Questions

Look at the execution table, what is the value of realPart at step 2?

A 3 + 4i

B 4

C 3

D 0.9273

💡 Hint

Check the 'Value' column for 'realPart' at step 2 in the execution_table.

At which step does the variable 'magnitude' get its value assigned?

A Step 4

B Step 3

C Step 5

D Step 2

💡 Hint

Look for the action 'Calculate magnitude abs(z)' in the execution_table.

If z was changed to 1 + 1i, what would be the magnitude value at step 4?

A 1

B 1.4142

C 2

D 0.7854

💡 Hint

Magnitude is sqrt(real^2 + imag^2). For 1 + 1i, sqrt(1 + 1) = 1.4142.

Concept Snapshot

Complex numbers in MATLAB:
- Define: z = a + bi (e.g., 3 + 4i)
- Access real: real(z)
- Access imag: imag(z)
- Magnitude: abs(z)
- Angle (radians): angle(z)
- Conjugate: conj(z)

Full Transcript

This visual execution trace shows how MATLAB handles complex numbers. We start by assigning z = 3 + 4i. Then we extract the real part using real(z), which gives 3, and the imaginary part using imag(z), which gives 4. Next, we calculate the magnitude with abs(z), resulting in 5, and the angle in radians with angle(z), which is approximately 0.9273. The variable tracker shows how each variable changes step by step. Key moments clarify common confusions like why imag(z) returns just the number 4, why magnitude is 5, and that angle is in radians. The quiz tests understanding of these values and calculations. This helps beginners see exactly how MATLAB processes complex numbers.

Practice

1. Which of the following correctly represents a complex number with real part 3 and imaginary part 4 in MATLAB?

easy

A. 3 + 4

B. 3i + 4

C. 3 + 4i

D. 3 + 4j + 1

Solution

1. Step 1: Understand complex number format in MATLAB

   Complex numbers are written as real part plus imaginary part times i or j, e.g., 3 + 4i.

2. Step 2: Check each option

   3 + 4i uses 3 + 4i which is correct. 3 + 4 misses imaginary part. 3i + 4 swaps real and imaginary incorrectly. 3 + 4j + 1 adds extra 1.

3. Final Answer:

   3 + 4i -> Option C

4. Quick Check:

   Complex number = 3 + 4i [OK]

Hint: Use 'i' or 'j' after the imaginary part number [OK]

Common Mistakes:

• Forgetting to include 'i' or 'j' for imaginary part
• Swapping real and imaginary parts
• Adding extra numbers outside complex format

2. Which of the following MATLAB commands correctly creates a complex number variable named z with real part 5 and imaginary part -2?

easy

A. z = complex(5, -2);

B. z = 5 + 2i;

C. z = 5 + -2;

D. z = 5 - 2;

Solution

1. Step 1: Recall MATLAB syntax for complex numbers

   Using complex(real, imag) function is a clear way to create complex numbers.

2. Step 2: Analyze options

   z = 5 - 2; is just a real number. z = 5 + 2i; creates 5 + 2i (wrong imaginary part). z = 5 + -2; is a real number. z = complex(5, -2); uses complex(5, -2) correctly.

3. Final Answer:

   z = complex(5, -2); -> Option A

4. Quick Check:

   complex(5, -2) creates correct complex number [OK]

Hint: Use complex(real, imag) for clear complex number creation [OK]

Common Mistakes:

• Writing imaginary part without 'i' or 'j'
• Using wrong sign for imaginary part
• Confusing syntax of complex() function

3. What is the output of the following MATLAB code?

z1 = 2 + 3i;
z2 = 1 - 4i;
result = z1 * z2;

medium

A. 10 - 5i

B. 14 - 5i

C. 14 + 5i

D. 10 + 5i

Solution

1. Step 1: Multiply complex numbers (a+bi)(c+di)

   Use formula: (ac - bd) + (ad + bc)i. Here a=2, b=3, c=1, d=-4.

2. Step 2: Calculate real and imaginary parts

   Real: 2*1 - 3*(-4) = 2 + 12 = 14. Imaginary: 2*(-4) + 3*1 = -8 + 3 = -5. So result = 14 - 5i.

3. Final Answer:

   14 - 5i -> Option B

4. Quick Check:

   (2+3i)*(1-4i) = 14 - 5i [OK]

Hint: Multiply using (ac - bd) + (ad + bc)i formula [OK]

Common Mistakes:

• Mixing signs in multiplication
• Forgetting to multiply imaginary parts by i^2 = -1
• Adding instead of subtracting real parts

4. Identify the error in the following MATLAB code snippet:

z = 4 + 5k;
disp(z);

medium

A. Variable 'k' is undefined, should use 'i' or 'j' for imaginary unit

B. Missing semicolon after assignment

C. disp() cannot display complex numbers

D. Real part must be negative

Solution

1. Step 1: Check imaginary unit usage in MATLAB

   MATLAB uses 'i' or 'j' for imaginary unit, 'k' is not recognized.

2. Step 2: Analyze code error

   Using '5k' causes an undefined variable error. Semicolon is optional. disp() works with complex numbers. Real part can be any number.

3. Final Answer:

   Variable 'k' is undefined, should use 'i' or 'j' for imaginary unit -> Option A

4. Quick Check:

   Imaginary unit must be 'i' or 'j' [OK]

Hint: Use only 'i' or 'j' for imaginary unit in MATLAB [OK]

Common Mistakes:

• Using undefined letters for imaginary unit
• Assuming semicolon is mandatory
• Thinking disp() can't show complex numbers

5. You have two complex numbers a = 3 + 4i and b = 2 - 4i. Write MATLAB code to compute the magnitude of their sum and select the correct output.

sum_ab = a + b;
mag = abs(sum_ab);
disp(mag);

hard

A. 7

B. 4

C. 3

D. 5

Solution

1. Step 1: Calculate sum of complex numbers

   a + b = (3 + 2) + (4 - 4)i = 5 + 0i.

2. Step 2: Calculate magnitude using abs()

   Magnitude = sqrt(5^2 + 0^2) = sqrt(25) = 5.

3. Step 3: Check output of disp(mag)

   MATLAB displays 5.

4. Final Answer:

   5 -> Option D

5. Quick Check:

   abs(5 + 0i) = 5 [OK]

Hint: Use abs() to get magnitude of complex number [OK]

Common Mistakes:

• Adding magnitudes instead of complex numbers
• Forgetting to use abs() for magnitude
• Calculating magnitude incorrectly