imaginary number

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Complex numbers

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 i2 = −1.

An imaginary number is a number that, when squared, has a negative result (i2 = −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 i2=(√-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 i2 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 i2 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?

a=6+4i b=2i-4 a-b=6+4i-(2i-4)= 6+4i-2i+4=10+2i a+b=6+4i+(2i-4)= 6+4i+2i-4=2+6i

Multiplying complex numbers

Remember that since i=√-1, the value of i2 is i2=-1.

If after multiplying we get i2, 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)= x2+dx+cx+cd= x2+(c+d)x+cd.

Step 2: Replace i2 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+8i2-16i

Step 2: Replacing i2 with -1:

a*b=12i-24+8i2-16i

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)=a2-b2.
If we multiply the complex number a+bi by a conjugate of a-bi we get (a+bi)(a-bi)=a2-b2i2.
Since we know that i2=-1. The expression a2-b2i2 is equal to a2+b2. 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 i2 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+8i2+16i
__      _______________
b              4i2-16

Step 3: Replacing i2 in the numerator and the denominator with -1:

a   =  12i+24+8i2+16i  =   12i+24+8*-1+16i
__     ________________       ________________
b               4i2-16                        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