Complex numbers in MATLAB - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of complex number multiplication

What is the output of the following MATLAB code?

MATLAB

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

A 3 + 8i

B 11 + 2i

C 11 - 2i

D 7 + 10i

💡 Hint

Recall how to multiply complex numbers: (a+bi)(c+di) = (ac - bd) + (ad + bc)i.

Multiplying (3+4i) and (1-2i) gives (3*1 - 4* -2) + (3* -2 + 4*1)i = (3 + 8) + (-6 + 4)i = 11 - 2i.

🧠 Conceptual

intermediate

Understanding complex conjugate in MATLAB

Which MATLAB expression correctly computes the complex conjugate of z = 5 + 6i?

A abs(5 + 6i)

B conj(5 + 6i)

C real(5 + 6i)

D imag(5 + 6i)

💡 Hint

The complex conjugate flips the sign of the imaginary part.

The MATLAB function conj() returns the complex conjugate, which changes 5+6i to 5-6i.

🔧 Debug

advanced

Identify the error in complex number addition

What error does the following MATLAB code produce?

MATLAB

z1 = 2 + 3i;
z2 = 4 + 5j;
result = z1 + z2;
disp(result);

A No error, output is 6 + 8i

B No error, output is 6 + 8j

C Error: Undefined function or variable 'j'

D Error: Complex number notation mismatch

💡 Hint

In MATLAB, both i and j represent the imaginary unit by default.

MATLAB treats both i and j as the imaginary unit unless they are redefined. So adding 2+3i and 4+5j works fine, resulting in 6+8i.

📝 Syntax

advanced

Correct syntax for creating a complex number

Which option correctly creates a complex number with real part 7 and imaginary part -9 in MATLAB?

A z = complex(7, -9);

B z = 7 - 9i;

C z = 7 + (-9)i;

D z = 7 - 9j;

💡 Hint

The complex() function takes two arguments: real and imaginary parts.

Option A uses the complex() function correctly. Option A is valid MATLAB syntax; MATLAB accepts it but option A is the most explicit and error-free. Option A is invalid syntax. Option A uses j which is valid but option A is the safest and clearest.

🚀 Application

expert

Calculate magnitude and phase of a complex number

Given z = -3 + 4i, what are the magnitude and phase (in radians) computed by MATLAB's abs() and angle() functions?

MATLAB

z = -3 + 4i;
magnitude = abs(z);
phase = angle(z);
disp([magnitude, phase]);

A [1, 0.7854]

B [7, 0.9273]

C [5, -0.9273]

D [5, 2.2143]

💡 Hint

Magnitude is sqrt(real^2 + imag^2). Phase is the angle from the positive real axis.

Magnitude = sqrt((-3)^2 + 4^2) = 5. Phase = atan2(4, -3) ≈ 2.2143 radians (in second quadrant).

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