Complex Numbers - ACT Math

Complex numbers take the form a + bi, where a is the real term in the complex number and bi is the nonreal (imaginary) term in the complex number. Taking this, we can see that for the real number 8, we can rewrite the number as , where represents the (zero-sum) non-real portion of the complex number.

Thus, any real number can be added to any complex number simply by considering the nonreal portion of the number to be .

Complex numbers take the form , where is the real term in the complex number and is the nonreal (imaginary) term in the complex number.

Thus, to balance the equation, add like terms on the left side.

Substituting for and , we have

This simplifies to

which equals

A complex number is a combination of a real and imaginary number. To add complex numbers, add each element separately.

In equation , is the real component and is the imaginary component (designated by ).

In equation , is the real component and is the imaginary component.

When added,

A complex number is a combination of a real and imaginary number. To add complex numbers, add each element separately.

First, distribute:

Then, group the real and imaginary components:

Solve to get:

When adding or subtracting complex numbers, the real terms are additive/subtractive, and so are the nonreal terms.

This problem can be solved in a way similar to other kinds of division problems (with binomials, for example). We need to get the imaginary number out of the denominator, so we will multiply the denominator by its conjugate and multiply the top by it as well to preserve the number's value.

Then, recall by definition, so we can simplify this further:

This is as far as we can simplify, so it is our final answer.

Multiply both numberator and denominator by :

First, divide 100 by as follows:

Now dvide this result by :

First, divide 100 by as follows:

Now, divide this by :

Square both sides of the equation:

Then Solve for x:

Therefore,

First, evaluate :

Now divide this into :

First, evaluate using the square pattern:

Divide this into :

This problem can be solved very similarly to a binomial such as . In this case, both the real and nonreal terms in the complex number are eligible to be divided by the real divisor.

, so

Solving this problem using a conjugate is just like conjugating a binomial to simplify a denominator.

Multiply both terms by the denominator's conjugate.

Simplify. Note .

Combine and simplify.

Simplify the numerator.

The prime denominator prevents further simplifying.

Thus, .

Multiplying complex numbers is like multiplying binomials, you have to use foil. The only difference is, when you multiply the two terms that have in the them you can simplify the to negative 1. Foil is first, outside, inside, last

First

Outside:

Inside

Last

Add them all up and you get

This equation can be solved very similarly to a binomial like . Distribution takes place into both the real and nonreal terms inside the complex number, where applicable.

This problem can be solved very similarly to a binomial like .

When dealing with complex numbers, remember that .

If we square , we thus get .

Yet another exponent gives us OR .

But when we hit , we discover that

Thus, we have a repeating pattern with powers of , with every 4 exponents repeating the pattern. This means any power of evenly divisible by 4 will equal 1, any power of divisible by 4 with a remainder of 1 will equal , and so on.

Thus,

Since the remainder is 3, we know that .

Solving this problem requires eliminating the nonreal term of the denominator. Our best bet for this is to cancel the nonreal term out by using the conjugate of the denominator.

Remember that for all binomials , there exists a conjugate such that .

This can also be applied to complex conjugates, which will eliminate the nonreal portion entirely (since )!

Multiply both terms by the denominator's conjugate.

Simplify. Note .

FOIL the numerator.

Combine and simplify.

Simplify the fraction.

Thus, .