To see what the polar form of the number is, it helps to draw it on a graph, where the horizontal axis is the imaginary part and the vertical axis the real part. This is called the complex plane.

To find the angle , we can find its supplementary angle  and subtract it from  radians, so .

Using trigonometric ratios,    and  .

Then .

To find the distance , we need to find the distance from the origin to the point . Using the Pythagorean Theorem to find the hypotenuse ,  or .

 Imaginary numbers do not fall on the number line by definition, since they are not real numbers. However, although i is an imaginary number equal to the square root of -1,  is a real number since . Therefore, . Negative numbers fall to the left of 0 on a number line.

In complex numbers of the form , a represents the real portion of the number and b represents the imaginary portion of the number. To graph  on a plane in which real numbers are represented by the x-axis and imaginary numbers are represented by the y-axis, place a point a units right of the origin and b units above the origin. The graph shows a point 2 units right and 3 units above the origin, so the complex number represented is .

In complex numbers of the form , a represents the real portion of the number and b represents the imaginary portion of the number. To graph  on a plane in which real numbers are represented by the x-axis and imaginary numbers are represented by the y-axis, place a point a units right of the origin and b units above the origin. The graph shows a point 5 units left and 2 units below the origin, so the complex number represented is .

 In complex numbers of the form , a represents the real portion of the number and b represents the imaginary portion of the number. In the complex number ,  and .

 In complex numbers of the form ,  represents the real portion of the number and  represents the imaginary portion of the number. In the complex number ,  and 

 In complex numbers of the form , a represents the real portion of the number and b represents the imaginary portion of the number. To graph  on a plane in which real numbers are represented by the x-axis and imaginary numbers are represented by the y-axis, place a point a units right of the origin and b units above the origin. The graph shows a point 4 units right and 7 units below the origin, so the complex number represented is .

In complex numbers of the form , a represents the real portion of the number and b represents the imaginary portion of the number. To graph  on a plane in which real numbers are represented by the x-axis and imaginary numbers are represented by the y-axis, place a point a units right of the origin and b units above the origin. The graph shows a point 8 units left and 1 unit above the origin, so the complex number represented is .

To solve this inequality, it is best to break it up into two separate inequalities to eliminate the absolute value function:

 or .

Then, solve each one separately:

Combining these solutions gives: 

The inequality compares an absolute value function with a negative integer. Since the absolute value of any real number is greater than or equal to 0, it can never be less than a negative number. Therefore,  can never happen. There is no solution.