A complex number is a variety of the type a + b ns , where a and b are actual numbers and also ns is the imagine unit , the square source of − 1 .

In a complex number z = a + b i , a is called the "real part" the z and b is dubbed the "imaginary part." If b = 0 , the complicated number is a genuine number; if a = 0 , climate the complex number is "purely imaginary."

We deserve to graph a facility number ~ above the Cartesian plane , utilizing the horizontal axis as the genuine axis and also the upright axis as the imaginary axis. Once we usage the Cartesian plane this way, we speak to it the complicated plane .

The facility number a + b i deserve to be plotted as the notified pair ( a , b ) on the complicated plane. The absolute value or modulus of a complex number z = a + b i have the right to be construed as the street of the point ( a , b ) from the origin on a complex plane.

utilizing the street Formula,

| z | = | a + b i |           = ( a − 0 ) 2 + ( b − 0 ) 2           = a 2 + b 2 You are watching: Which complex number has a distance of from the origin on the complex plane?

instance 1:

Plot the number − 5 + 6 ns top top a complicated plane.

The real component of the facility number is − 5 and also the imaginary component is 6 .

begin at the origin. Move 5 devices to the left on the actual axis to reach the point ( − 5 , 0 ) . Now, move 6 systems upward to reach the suggest ( − 5 , 6 ) . If the real component of a complicated number is zero, the number lies top top the imaginary axis. Similarly, if the imaginary component of a facility number is zero, the number lies ~ above the real axis.

See more: Which Of The Following Functions Has A Rate Of Change That Stays The Same ?

instance 2:

Plot the number 6 top top the facility plane.

The real component of the complex number is 6 and the imaginary component is 0 . So, the number will certainly lie on the real axis.

start at the origin. Relocate 6 systems to the appropriate on the actual axis to reach the point ( 6 , 0 ) . example 3:

Plot the number − 4 ns on the facility plane.

The real part of the complicated number − 4 ns is zero and also the imaginary part is − 4 .

start at the origin. Relocate 4 systems down follow me the imagine axis to with the point ( 0 , − 4 ) .