DrDelMath

College Algebra at Meramec
Complex Numbers
Exercises/Examples/Ilustrations
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As you study these exercises, move your cursor over arrows, equal symbols, light bulbs, and highlighted words. Always check in with the wise old owl and his little apprentice by moving your cursor over them. Study with an active cursor.

 
The definition that pops up when you move the cursor over a highlighted word is what should pop into your mind when you read, hear, or speak that word. The material that pops up when you move the cursor over a light bulb is a suggested strategy for solving the problem. You should always formulate a similar strategy when you attempt to solve a problem. When you move the cursor over an explanation of a step in a solution, the property that pops up is the mathematical justification for taking that action. You must always be able to provide such a justification for every step you take in mathematics.

Question: What is the real component and what is the complex component of the complex number 5 - 9i ?

Response: Refer to the definitions. To determine the real or complex components of a complex number, the number must be written in standard form. In this example 5 - 9i in standard form is 5 + (-9)i from which it is clear that the real component is 5 and the complex component is -9.

Question: What is the real component and what is the complex component of the complex number - 4 + 3i

Response: Refer to the definitions. To determine the real or complex components of a complex number, the number must be written in standard form. In this example the complex number is in standard form so comparison with the definitions indicates the real component is -4 and the complex component is 3.

Question: What is the conjugate of 3 - 7i ?

Response: The conjugate of a complex number is formed by replacing the complex component with its opposite. In this example, -7 must be replaced by 7. Therefore the conjugate of 3 - 7i is 3 + 7i.

Question: What is the conjugate of -4 + 23i ?

Response: The conjugate of a complex number is formed by replacing the complex component with its opposite. In this example, 23 must be replaced by -23. Therefore the conjugate of -4 + 23i is -4 - 23i.

Question: What is the norm of 4 + 5i ?

Response:The norm of a complex number is formed by adding the squares of the real and complex components. In this example we must add the square of 4 and the square of 5. Therefore the norm of 4 + 5i is 44 + 52 = 16 + 25 = 41.

Question: What is the norm of -3 - 2i?

Response:The norm of a complex number is formed by adding the squares of the real and complex components. In this example we must add the square of -3 and the square of -2. Therefore the norm of -3 - 2i is (-3)2 + (-2)2 = 9 + 4 = 13.

Question: What is the multiplicative inverse of 3 + 2i ?

Response: The multiplicative inverse of a complex number is it conjugate divided by its norm. In this case the multiplicative inverse of 3 + 2i is multiplicative inverser of a complex number .

Question: What is the multiplicative inverse of -4 + 5i ?

Response: The multiplicative inverse of a complex number is it conjugate divided by its norm. In this case the multiplicative inverse of -4 + 5i is multiplicative invers of -4 + 5i.

Question: What is the product of 3 + 2i and 5 + 4i ?

Response: Complex numbers are multplied as if they were binomials with the understanding that i2 = -1.
(3 +2i)(5 + 4i) = 15 +12i +10i +8i2 =15 + 22i - 8 =7 + 22i.

Question: What is the product of 8 - 2 i and -4 - i ?

Response: Complex numbers are multplied as if they were binomials with the understanding that i2 = -1.
(8 - 2i)(-4 - i) = -32 - 8i + 8i +2i2 = - 32 + 0i + 2(-1) = - 34

Question: What is the quotient (4 + 5i) ÷ (2 + 3i)

Response: In mathematics, division is always accomplished by changing the problem to multiplication of the dividend by the multiplicative inverse of the divisor ( see ). In this problem we have:
Steps for 4 + 5i divided by 2 + 3i

Question: What is the quotient (3 - 4i) ÷(-5 + 6i) ?

Response: In mathematics, division is always accomplished by changing the problem to multiplication of the dividend by the multiplicative inverse of the divisor. In this problem we have:
3 - 4i divided by -5 + 6i