Graphing Functions with Complex Numbers - Algebra 2

Use the change of base formula for logarithmic functions and incorporate the fact that and

Or

can be solved using

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 .

Imaginary numbers do not fall on the number line-- they are by definition not real numbers.

** If the question asked where falls on the number line, the answer would be to the left of 0, because .

Use the change of base formula for logarithmic functions and incorporate the fact that and

Or

can be solved using

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 .