Complex numbers - practice problems
Complex numbers extend the real number system by including the imaginary unit i, where i² = -1, allowing for solutions to equations like x² + 1 = 0. A complex number has the form a + bi, where a is the real part and b is the imaginary part. Complex numbers can be represented on the complex plane with the real axis horizontal and imaginary axis vertical. Operations include addition, subtraction, multiplication, and division following algebraic rules with the property i² = -1. They can be expressed in polar form r(cos θ + i sin θ) or exponential form re^(iθ), connecting to trigonometry through Euler's formula. Complex numbers are essential in electrical engineering, quantum mechanics, signal processing, and advanced mathematics.Instructions: Solve each problem carefully and provide a detailed solution for every item.
Number of problems found: 59
- Simplify complex expr
Perform the indicated operations and write the results in the form of a + bi: (2 + 3i)³ (ii) (1 + i)⁴ - Complex number z
Let z = -5 + i and w = 4 - 2i. Compute z - w and write your final answer in a + bi form. x = z - w - The expression 2 The expression (3+i)(1+2i) can be written in the form a+bi, where a and b are integers. What are the values of a and b?
- Evaluate complex eq If z=x+yi and 3x+(3x-y)i=4-6i then z=
- Complex equation If c - d = 2 and c = 12 + 7 i, find d. Write the result in the form of a + bi.
- Complex expr with fractions
Find 1½ of 16 ÷2⅓+(2¼ of ⅑). Use the correct order of operations. The order is PEMDAS: Parentheses, Exponents, Multiplication, Division (from left to right), addition, and Subtraction (from left to right). - Suppose 10
Suppose 4+7i is a solution of 5z²+Az+B=0, where A, B∈R. Find A and B. - Cplx sixth power
Let z = 2 - sqrt(3i). Find z6 and express your answer in rectangular form. if z = 2 - 2sqrt(3 i) then r = |z| = sqrt(2 ^ 2 + (- 2sqrt(3)) ^ 2) = sqrt(16) = 4 and theta = tan -2√3/2=-π/3 - Find all 2 Find all the complex solutions (write the answer in the form x+iy) of the system. {|z-12|/|z-8i|=5/3 ; |z-4|=|z-8|
- Midpoint between conjugate
Find the midpoint between two roots: 2+3.464i and 2 - 3.464i - Subtract polar forms
Solve the following 5.2∠58° - 1.6∠-40° and give answer in polar form - Conjugate coordinates If the rectangular conjugate of the polar vector 12 angle 35 degrees is equal to x+yi, find the sum of x and y.
- EE school boarding Three vectors, A, B, and C, are related as follows: A/C = 2 at 120 deg, A + B = -5 + j15, C = conjugate of B. Find C.
- Vertical components
Find the horizontal and vertical components of the vector which has magnitude 750 as shown in the following figure. - Turtles 2
A box turtle hibernates in the sand at 11 5/8 ft. A spotted turtle hibernates at 11 16/25 feet. Which turtle is deeper? Write the answer as number 1 or 2. - Laurent series
Expand the function f (z) into the Laurent series at point z0 for the given annulus. f (z) = 1 / ((z-2) * (z-3)) a) z0 = 2, 0 3 - Mistake in expression
A student made a mistake while attempting to multiply the expression (2 - 5i)(5 + 2i). (2 - 5i)(5 + 2i) = 10 + 4i - 25i - 10i2 = 10 + 4(-1) - 25(-1) - 10(1) = 10 - 4 + 25 - 10 = 21 Complete the explanation and correct the error. Hint: The student incorrec - Suppose 5 Suppose z5=2+3i and z6=6+9i are complex numbers and 3 z5 + 7 z6= m+in. What is the value of m and n?
- Complex plane mapping
Show that the mapping w = z +c/z, where z = x+iy, w = u+iv, and c is a real number, maps the circle |z| = 1 in the z-plane into an ellipse in the (u, v) plane. - Two grandmothers
Two grandmothers went to the market to sell eggs, and they had 100. When they sold all the eggs, they made the same money. The first grandmother said to the second, "If I sold my eggs for your price, I would earn 15 crowns. " The other grandmother replied
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