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Problems for Practice - Complex Numbers | Algebra - Mathematics PDF Download

Complex Numbers - Practice Problems
Perform the indicated operation and write your answer in standard form.
Q.1. (4−5i)(12+11i)(4−5i)(12+11i)
Ans. We know how to multiply two polynomials and so we also know how to multiply two complex numbers. All we need to do is “foil” the two complex numbers to get,
Problems for Practice - Complex Numbers | Algebra - Mathematics
All we need to do to finish the problem is to recall that i2=−1. Upon using this fact we can finish the problem.
Problems for Practice - Complex Numbers | Algebra - Mathematics

Q.2. (−3−i)−(6−7i)
Ans. We know how to subtract two polynomials and so we also know how to subtract two complex numbers.
Problems for Practice - Complex Numbers | Algebra - Mathematics

Q.3. (1+4i)−(−16+9i)
Ans. 
We know how to subtract two polynomials and so we also know how to subtract two complex numbers.
Problems for Practice - Complex Numbers | Algebra - Mathematics

Q.4. 8i(10+2i) 
Ans. We know how to multiply two polynomials and so we also know how to multiply two complex numbers. All we need to do is distribute the 8i to get,
Problems for Practice - Complex Numbers | Algebra - Mathematics
All we need to do to finish the problem is to recall that i2=−1. Upon using this fact we can finish the problem.
Problems for Practice - Complex Numbers | Algebra - Mathematics

Q.5. (−3−9i)(1+10i)
Ans. We know how to multiply two polynomials and so we also know how to multiply two complex numbers. All we need to do is “foil” the two complex numbers to get,
Problems for Practice - Complex Numbers | Algebra - Mathematics
All we need to do to finish the problem is to recall that i2=−1. Upon using this fact we can finish the problem.
Problems for Practice - Complex Numbers | Algebra - Mathematics

Q.6. (2+7i)(8+3i)
Ans. We know how to multiply two polynomials and so we also know how to multiply two complex numbers. All we need to do is “foil” the two complex numbers to get,
Problems for Practice - Complex Numbers | Algebra - Mathematics
All we need to do to finish the problem is to recall that i2=−1. Upon using this fact we can finish the problem.
Problems for Practice - Complex Numbers | Algebra - Mathematics

Q.7. Problems for Practice - Complex Numbers | Algebra - Mathematics
Ans. Because standard form does not allow for i’s to be in the denominator we’ll need to multiply the numerator and denominator by the conjugate of the denominator, which is 2−10i.
Multiplying by the conjugate gives,
Problems for Practice - Complex Numbers | Algebra - Mathematics
Now all we need to do is do the multiplication in the numerator and denominator and put the result in standard form.
Problems for Practice - Complex Numbers | Algebra - Mathematics

Q.8. Problems for Practice - Complex Numbers | Algebra - Mathematics
Ans. Because standard form does not allow for ii’s to be in the denominator we’ll need to multiply the numerator and denominator by the conjugate of the denominator, which is 3i.
Multiplying by the conjugate gives,
Problems for Practice - Complex Numbers | Algebra - Mathematics
Now all we need to do is do the multiplication in the numerator and denominator and put the result in standard form.
Problems for Practice - Complex Numbers | Algebra - Mathematics

Q.9.Problems for Practice - Complex Numbers | Algebra - Mathematics
Ans. Because standard form does not allow for i’s to be in the denominator we’ll need to multiply the numerator and denominator by the conjugate of the denominator, which is 8+i.
Multiplying by the conjugate gives,
Problems for Practice - Complex Numbers | Algebra - Mathematics
Now all we need to do is do the multiplication in the numerator and denominator and put the result in standard form.
Problems for Practice - Complex Numbers | Algebra - Mathematics

The document Problems for Practice - Complex Numbers | Algebra - Mathematics is a part of the Mathematics Course Algebra.
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FAQs on Problems for Practice - Complex Numbers - Algebra - Mathematics

1. What are complex numbers and how are they used in mathematics?
Ans. Complex numbers are numbers that consist of a real part and an imaginary part. They are used in mathematics to solve equations that cannot be solved using real numbers alone. Complex numbers are commonly used in fields such as engineering, physics, and computer science.
2. How are complex numbers represented and written?
Ans. Complex numbers are typically written in the form a + bi, where a is the real part and bi is the imaginary part. The imaginary unit i is defined as the square root of -1. The real part and imaginary part can be any real numbers.
3. What is the geometric interpretation of complex numbers?
Ans. Geometrically, complex numbers can be represented as points in a plane called the complex plane. The real part of a complex number determines the horizontal position on the plane, while the imaginary part determines the vertical position. This allows complex numbers to be visualized and manipulated using geometric concepts.
4. How are complex numbers added and subtracted?
Ans. To add or subtract complex numbers, you simply add or subtract the real parts separately and the imaginary parts separately. For example, to add (3 + 2i) and (1 - 4i), you would add 3 + 1 for the real parts and 2i - 4i for the imaginary parts, resulting in 4 - 2i.
5. What is the complex conjugate and why is it important?
Ans. The complex conjugate of a complex number a + bi is obtained by changing the sign of the imaginary part, resulting in a - bi. The complex conjugate is important because it allows for simplification of complex numbers and plays a crucial role in operations such as division and finding the magnitude of a complex number.
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