# Complex numbers on the SAT test

### SAT Subscore: Additional topics in math

## Studying complex numbers

**On the SAT test complex numbers topic** is the first topic of additional topics in math that include 7 advanced topics (see the full topics list on the top menu).

**Complex numbers topic is divided into sections** from easy to difficult (the list of the sections appears on the left menu). Each section includes detailed explanations of the required material with examples followed by a variety of self-practice questions with solutions.

**Finish studying** heart of algebra subscore topics before you study this topic or any other additional topic in math. (Heart of algebra subscore includes basic algebra topics which knowledge is required for understanding additional topics in math).

### Complex numbers- summary

**A complex number** has a real component and an imaginary component, it is written in a form of a + bi, where a and b are real numbers, and i is an imaginary number satisfying i^{2} = −1.

**An imaginary number **is a number that, when squared, has a negative result (i^{2} = −1). Since an imaginary number i is equal to a square root of a negative number -1 it does not have a tangible value (negative numbers don’t have real square roots since a square is either positive or zero).

**The unit imaginary number, i**, equals the square root of minus 1, so that **i=√-1**. As said above, when squared imaginary number has a negative result, so that **i ^{2}=(√-1)^{2}=-1**.

__For example:__ 3i is an imaginary number, and its square is (3√-1)^{2}=9*-1=-9.

**To add or subtract complex numbers,** combine like terms (real terms with real terms and imaginary terms with imaginary terms).

**To multiply complex numbers,** multiply the numbers with foil formula, replace i^{2 }with -1 and combine like terms.

**To divide complex numbers, **you need to cancel the denominator by turning the imaginary component in the denominator to a real number. This is done by multiplying the numerator and the denominator by the conjugate of the denominator. The next steps are multiplying the numbers with foil formula, replacing i^{2 }with -1 and combining like terms.

__Continue reading this page for detailed explanations and examples.__

### Adding and subtracting complex numbers

**Combine like terms**: real terms with real terms and imaginary terms with imaginary terms and write the result as a+bi.

Consider the following example:

If a=6+4i and b=2i-4, that are the values of a-b and a+b?

### Multiplying complex numbers

Remember that since i=√-1, the value of i^{2} is ** i^{2}=-1**.

If after multiplying we get i^{2}, we can write it as -1 and continue solving.

**Steps for multiplying complex numbers:**

Step 1: Multiply the numbers with foil formula.

The FOIL formula is y=(x+c)(x+d)= x^{2}+dx+cx+cd= x^{2}+(c+d)x+cd.

Step 2: Replace i^{2 }with -1.

Step 3: Combine like terms (real terms with real terms and imaginary terms with imaginary terms) and write the result as a+bi.

Consider the following example:

If a=6+4i and b=2i-4, that is the value of a*b?

__Step 1: Multiplying the numbers with foil formula:__

a=6+4i

b=2i-4

a*b=(6+4i)(2i-4)=12i-24+8i^{2}-16i

__Step 2: Replacing i__^{2 }__with -1:__

a*b=12i-24+8i^{2}-16i

a*b=12i-24+8*-1-16i

__Step 3: Combining like terms:__

a*b=-32-4i

### Dividing complex numbers

We have a numerator and a denominator as 2 complex numbers in a form of a+bi and we need to simplify the result to a form of one complex number in a form of a+bi (staying without the denominator).

To cancel the denominator, we need to turn the imaginary component in the denominator to a real number,** this is done by multiplying the numerator and the denominator by the conjugate of the denominator.**

** ****For example: **

We learned in the quadratic equations topic that (a+b)(a-b)=a^{2}-b^{2}.

If we multiply the complex number a+bi by a conjugate of a-bi we get (a+bi)(a-bi)=a^{2}-b^{2}i^{2}.

Since we know that i^{2}=-1. The expression a^{2}-b^{2}i^{2} is equal to a^{2}+b^{2}. This outcome is a real number.

**Steps for dividing complex numbers:**

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator (conjugate divided by itself is equal to 1 and we can multiply by 1 without changing the original value).

Step 2: Multiply the numbers in the numerator and the denominator with foil formula.

Step 3: Replace i^{2 }in the numerator and the denominator with -1. In the denominator you will be left with real terms without imaginary terms.

Step 4: Combine like terms and write the answer as a complex number in the numerator in a form of a+bi divided by a real number in the denominator.

Consider the following example:

If a=6+4i and b=2i-4, that is the value of a/b?

Step 1: Multiplying the numerator and the denominator by the conjugate of the denominator:

a=6+4i

b=2i-4

a = 6+4i

__ _____

b 2i-4

The conjugate of the denominator is 2i+4.

a = 6+4i = (6+4i)(2i+4)

__ _____ ____________

b 2i-4 (2i-4) (2i+4)

Step 2: Multiplying the numbers in the numerator and the denominator with foil formula:

a = 12i+24+8i^{2}+16i

__ _______________

b 4i^{2}-16

Step 3: Replacing i^{2 }in the numerator and the denominator with -1:

a = 12i+24+8i^{2}+16i = 12i+24+8*-1+16i

__ ________________ ________________

b 4i^{2}-16 4*-1-16

Step 4: Combining like terms and writing the answer as a complex number in the numerator divided by a real number in the denominator:

a = 12i+24+8*-1+16i = 28i+16 = 28i + 16 = -7i – 4

__ _________________ _______ ____ ____ ___ ___

b 4*-1-16 -20 -20 -20 5 5

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