# Complex numbers on the SAT test

### SAT Subscore: Additional topics in math

**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