Is z + z¯ and z × z¯ Real for Any Complex Number z?

In summary, a complex number is a number that consists of a real part and an imaginary part, typically written in the form a + bi. To check for a complex number, look for the presence of the imaginary unit, i, in the number. One example of a complex number is 3 + 2i, where 3 is the real part and 2i is the imaginary part. Complex numbers can undergo addition, subtraction, multiplication, and division, with special considerations for multiplying or dividing the imaginary parts. In science, complex numbers are used in various fields such as physics, engineering, and mathematics, particularly in dealing with quantities that have both magnitude and direction. They are also applied in signal processing, control systems, and quantum
  • #1
Yordana
3
0
I apologize in advance for my English.
I want to know if my solution is correct. :)

To verify that for every complex number z, the numbers z + z¯ and z × z¯ are real.

My solution:
z = a + bi
z¯ = a - bi
z + z¯ = a + bi + a - bi = 2a ∈ R
z × z¯ = (a + bi) × (a - bi) = a^2 + b^2 ∈ R
 
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  • #2
Yep. All correct.
 

FAQ: Is z + z¯ and z × z¯ Real for Any Complex Number z?

What is a complex number?

A complex number is a number that has both a real part and an imaginary part. It is usually written in the form a + bi, where a is the real part and bi is the imaginary part. The imaginary part is represented by the letter i, which is equal to the square root of -1.

How do you check for a complex number?

To check for a complex number, you need to look for the presence of the imaginary unit i. If the number contains i, then it is a complex number. Additionally, a number can also be considered complex if it has a non-zero imaginary part, even if it does not explicitly contain i.

What is the purpose of checking for complex numbers?

Checking for complex numbers is important in mathematics and other scientific fields, as complex numbers have many applications in solving equations and problems that involve both real and imaginary quantities. They are also used in fields such as engineering, physics, and economics.

How do you perform calculations with complex numbers?

To perform calculations with complex numbers, you can use the basic operations of addition, subtraction, multiplication, and division. To add or subtract complex numbers, you simply add or subtract the real and imaginary parts separately. To multiply complex numbers, you can use the FOIL method, and to divide complex numbers, you can use the conjugate method.

Can complex numbers be graphed?

Yes, complex numbers can be graphed on a coordinate plane known as the complex plane. The real part of the complex number is plotted on the x-axis, while the imaginary part is plotted on the y-axis. This allows for visual representation of complex numbers and their relationships with other complex numbers.

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