Imaginary 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 .

When adding complex numbers, add the real parts and the imaginary parts separately to get another complex number in standard form.

Adding the real parts gives , and adding the imaginary parts gives .

This is essentially the following expression after translation:

Now add the real parts together for a sum of , and add the imaginary parts for a sum of .

The first step is to distribute which gives us:

which is in standard form.

Multiply both the numerator and the denominator by the conjugate of the denominator which is which results in

The numerator after simplification give us

The denominator is equal to

Hence, the final answer in standard form =

Multiply both the numerator and the denominator by the conjugate of the denominator which is resulting in

This is equal to

Since you can make that substitution of in place of in both numerator and denominator, leaving:

When you then cancel the negatives in both numerator and denominator (remember that , simplifying each term), you're left with a denominator of and a numerator of , which equals .

The absolute value is a measure of the distance of a point from the origin. Using the pythagorean distance formula to calculate this distance.

Combine like terms. Treat as if it were any other variable.

Substitute to eliminate .

Simplify.

Multiply the numerator and denominator by the numerator's complex conjugate.

Reduce/simplify.

Recall that the definition of imaginary numbers gives that and thus that . Therefore, we can use Exponent Rules to write