Question about complex numbers

In summary, complex numbers are numbers that have both a real and an imaginary part, and are written in the form a + bi. They are useful in many areas of mathematics, science, and engineering, and can be used to solve certain types of equations that cannot be solved with only real numbers. To add or subtract complex numbers, you simply add or subtract the real parts and separately add or subtract the imaginary parts. To multiply complex numbers, you use the FOIL method, and to divide complex numbers, you use the complex conjugate. Complex numbers cannot be graphed on a number line, but are instead graphed on a two-dimensional coordinate plane called the complex plane. The fundamental theorem of algebra states that every polynomial equation with complex coefficients
  • #1
atwong713
20
0
when indicating the region of the complex plane that corresponds to all points z that satisfy the condition
|z|=11
this region would be the circle of radius 11 around the origin because the |z-w| is the distance between z and w, where z and w are both complex numbers
in general |z-w|=r would be a circle around the pt w on the complex plane with radius r

but how would you indicate a region that satisfies |2z+1| = 3 or |i-2z|=5 for all z?
what do these represent geometrically?
 
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  • #2
|2z+1|= |2(z+1/2)|= 2|z+ 1/2|= 3 so |z+ 1/2|= 3/2.
 

FAQ: Question about complex numbers

What are complex numbers?

Complex numbers are numbers that have both a real and an imaginary part. They are written in the form a + bi, where a is the real part and bi is the imaginary part, with i being the imaginary unit (the square root of -1).

Why do we need complex numbers?

Complex numbers are useful in many areas of mathematics, science, and engineering. They are used to solve certain types of equations that cannot be solved with only real numbers, and they have applications in fields such as electrical engineering, quantum mechanics, and signal processing.

How do you add, subtract, multiply, and divide complex numbers?

To add or subtract complex numbers, you simply add or subtract the real parts and separately add or subtract the imaginary parts. To multiply complex numbers, you use the FOIL method (multiply the first terms, then the outer terms, then the inner terms, and finally the last terms). To divide complex numbers, you use the complex conjugate (multiply the numerator and denominator by the conjugate of the denominator).

Can complex numbers be graphed on a number line?

No, complex numbers cannot be graphed on a number line. Real numbers can be represented on a number line, but complex numbers have both a real and an imaginary part, so they cannot be represented on a one-dimensional number line. Instead, they are graphed on a two-dimensional coordinate plane called the complex plane.

What is the relationship between complex numbers and the fundamental theorem of algebra?

The fundamental theorem of algebra states that every polynomial equation with complex coefficients has at least one complex root. This means that complex numbers are necessary for solving certain polynomial equations, and they play a crucial role in the fundamental theorem of algebra.

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