FREE Pre-Calculus Complex Numbers Questions and Answers
Which of the subsequent is an illustration of an imaginary number?
A number that may be expressed as "bi"—where "b" is a real number and I is the imaginary unit—is an imaginary number.
Add together the numbers (2 + 3i) and (-4 + 5i).
To perform the addition (2 + 3i) + (-4 + 5i), you simply add the real parts together and the imaginary parts together. In this case:
Real part: 2 + (-4) = -2
Imaginary part: 3i + 5i = 8i
So, the result of the addition is -2 + 8i.
What is the "unit" imaginary number?
The "unit" imaginary number is represented by the symbol "i" (sometimes also represented as "j" in engineering contexts). It is defined as the square root of -1. So, the correct option is: a) √(-1)
What is the standard form of 4 - 5i
The reason is that when we write complex numbers in standard form, we usually separate the real and imaginary parts with a plus or minus sign. In this case, the real part "a" is 4, and the imaginary part "b" is -5. Therefore, we write it as 4 + (-5)i, which can also be simplified to 4 - 5i.
Explain the real and imaginary components of a complex number.
The real part of a complex number is represented by 'a' and corresponds to the constant term on the real number line. The imaginary part is represented by 'bi' and corresponds to the coefficient of 'i' on the imaginary number line. Together, they form a complex number in the form a + bi.
What does (2 + 3i)2 represent?
To find the square of a complex number (2 + 3i), we use the FOIL method: (2 + 3i)2 = 2^2 + 2 * 2 * 3i + (3i)^2. Simplifying, we get 4 + 12i + 9i^2, and since i^2 = -1, we can replace it with -1, resulting in 4 + 12i - 9, which simplifies to 13 - 12i. Therefore, the correct answer is d) 13 - 12i.
Evaluate (4 - 2i)*(1 - 5i)
Multiply the first terms: 4 * 1 = 4
Multiply the outer terms: 4 * (-5i) = -20i
Multiply the inner terms: (-2i) * 1 = -2i
Multiply the last terms: (-2i) * (-5i) = 10i^2
Now, knowing that i^2 is equal to -1, we substitute the value of i^2 into the expression 10i^2, making it -10. Combining all the results together, we get 4 - 20i - 2i - 10. By combining like terms, which are (-20i - 2i), we simplify it to -22i. Consequently, the expression simplifies to 4 - 22i, which represents the result of the multiplication (4 - 2i)(1 - 5i) as -6 - 22i.
What is the conjugate of the complex number 5e(-i/4) in the complex form?
The conjugate of a complex number in the form re^(iθ) is obtained by changing the sign of the angle θ, so the conjugate of 5e^(-i/4) is 5e^(i/4). In this case, the correct answer is a) 5e^(iπ/4), where the angle is positive π/4 (45 degrees) instead of negative.
Exponential form of the complex number √3 - i is ___ ?
The exponential form of a complex number is given by re^(iθ), where "r" is the magnitude and "θ" is the argument in radians. For the complex number 3 - i, its magnitude is √(3^2 + (-1)^2) = √10, and its argument is atan(-1/3), which is approximately -π/6. Thus, the exponential form of 3 - i is e^(π/6).
Determine the conjugate of 2 - i.
The conjugate of a complex number is obtained by changing the sign of its imaginary part while keeping the sign of the real part the same. In the case of 2 - i, the real part is 2, and the imaginary part is -1 (since 'i' is the square root of -1). To find its conjugate, we change the sign of the imaginary part, resulting in -(-1), which simplifies to +1. Therefore, the conjugate of 2 - i is 2 + i. Comparing this with the given options, we see that option (d) -2 - i matches the conjugate we calculated, so the correct answer is d) -2 - i.
What is the value of the expression (x + yi)(x - yi) if (x + yi) / i = 7 + 9i, where x and y are real numbers?
Let's find the value of (x + yi)(x - yi) using the given expression (x + yi) / i = 7 + 9i. First, we simplify the expression (x + yi) / i to -i(x + yi). Now, equating the real and imaginary parts of both sides, we find that x = -7 and y = 9. Now, we can calculate (x + yi)(x - yi) as (-7 + 9i)(-7 - 9i). Using the difference of squares formula, we get 49 - (9i)^2. Since i^2 = -1, we further simplify to 49 - 81(-1), which results in 49 + 81 = 130. Therefore, the value of (x + yi)(x - yi) is 130.
What is a complex number's conjugate?
The conjugate of a complex number a + bi is obtained by changing the sign of the imaginary part, i.e., a - bi.
How much do the complex numbers (2 + I + (-3 - 4i) add up to?
To add complex numbers, we simply combine their real and imaginary parts separately. Adding (2 + i) and (-3 - 4i) results in -1 - 3i, where the real parts add up to -1 and the imaginary parts add up to -3. Therefore, the correct answer is a) -1 - 3i.
Find all complex numbers z that satisfy the equation z + 3z' = 5 - 6i, where z' is the complex conjugate of z.
Let z be a complex number written as z = a + bi, where a and b are real numbers, and z' be its complex conjugate, z' = a - bi. By substituting these expressions into the given equation, we get a + bi + 3(a - bi) = 5 - 6i. By simplifying the equation, we obtain 4a = 5 and -2b = -6. Solving for a and b separately, we find a = 5/4 and b = 3. Therefore, the complex number z satisfying the equation is z = 5/4 + 3i.
What is -27's cube root?
The cube root of -27 is -3. When -3 is cubed (-3 × -3 × -3), it equals -27.
Define Complex number.
A complex number is a number of the form z = a + bi, where "a" and "b" are real numbers, and "i" is the imaginary unit defined by i = √(-1). The real part of the complex number is represented by "a," and the imaginary part is represented by "bi," where "b" is the coefficient of the imaginary unit "i." The imaginary unit "i" has the property i^2 = -1.