Solve the quadratic equation x2 + 1 = 0. Solving for x , gives x2 = – 1

12 x

1x

We make the following definition:

1i

Bell Work #1

1i

Complex Numbers

12 iNote that squaring both sides yields:therefore

and

so

and

iiiii *1* 13 2

1)1(*)1(* 224 iii

iiiii *1*45

1*1* 2246 iiii

And so on…

Real NumbersImaginary Numbers

Real numbers and imaginary numbers are subsets of the set of complex numbers.

Complex Numbers

Definition of a Complex Number Definition of a Complex Number

If a and b are real numbers, the number a + bi is a complex number, and it is said to be written in standard form.

If b = 0, the number a + bi = a is a real number.

If a = 0, the number a + bi is called an imaginary number.

Write the complex number in standard form

81 81 i 241 i 221 i

Addition and Subtraction of Complex Addition and Subtraction of Complex Numbers Numbers

If a + bi and c +di are two complex numbers written in standard form

)()( dicbia

)()( dicbia

Sum:

Difference:

Perform the subtraction and write the answer in standard form.

( 3 + 2i ) – ( 6 + 13i ) 3 + 2i – 6 – 13i –3 – 11i

234188 i

234298 ii

234238 ii

4

Multiplying Complex NumbersMultiplying Complex Numbers

Multiplying complex numbers is similar to multiplying polynomials and combining like terms.

Perform the operation and write the result in standard form. ( 6 – 2i )( 2 – 3i )

F O I L12 – 18i – 4i + 6i2

12 – 22i + 6 ( -1 )6 – 22i

Consider ( 3 + 2i )( 3 – 2i )9 – 6i + 6i – 4i2

9 – 4( -1 )9 + 4 13

This is a real number. The product of two complex numbers can be a real number.