Complex Numbers

Complex NumbersBecause the square of any real number can't be negative, the equation x2 = –1 does not have any solution.

Complex NumbersBecause the square of any real number can't be negative, the equation x2 = –1 does not have any solution. We make up a new imaginary number –1to be a solution of this equation

Complex NumbersBecause the square of any real number can't be negative, the equation x2 = –1 does not have any solution. We make up a new imaginary number –1 ↔ ito be a solution of this equation and we name it as “ i ”,

Complex NumbersBecause the square of any real number can't be negative, the equation x2 = –1 does not have any solution. We make up a new imaginary number –1 ↔ ito be a solution of this equation and we name it as “ i ”, i.e.

(±i)2 = –1

Complex Numbers

Using i, the “solutions” of the equations of the form

Because the square of any real number can't be negative, the equation x2 = –1 does not have any solution. We make up a new imaginary number –1 ↔ ito be a solution of this equation and we name it as “ i ”, i.e.

(±i)2 = –1

x2 = –rare x = ± ir

Complex Numbers

Using i, the “solutions” of the equations of the form

Example A. Solve x2 + 49 = 0 using imaginary numbers.

Because the square of any real number can't be negative, the equation x2 = –1 does not have any solution. We make up a new imaginary number –1 ↔ ito be a solution of this equation and we name it as “ i ”, i.e.

(±i)2 = –1

x2 = –rare x = ± ir

Complex Numbers

Using i, the “solutions” of the equations of the form

Example A. Solve x2 + 49 = 0 using imaginary numbers. Using the square-root method:x2 + 49 = 0 → x2 = –49

Because the square of any real number can't be negative, the equation x2 = –1 does not have any solution. We make up a new imaginary number –1 ↔ ito be a solution of this equation and we name it as “ i ”, i.e.

(±i)2 = –1

x2 = –rare x = ± ir

Complex Numbers

Using i, the “solutions” of the equations of the form

Example A. Solve x2 + 49 = 0 using imaginary numbers. Using the square-root method:x2 + 49 = 0 → x2 = –49 sox = ±–49

Because the square of any real number can't be negative, the equation x2 = –1 does not have any solution. We make up a new imaginary number –1 ↔ ito be a solution of this equation and we name it as “ i ”, i.e.

(±i)2 = –1

x2 = –rare x = ± ir

Complex Numbers

Using i, the “solutions” of the equations of the form

Example A. Solve x2 + 49 = 0 using imaginary numbers. Using the square-root method:x2 + 49 = 0 → x2 = –49 sox = ±–49 x = ±49–1 x = ±7i

Because the square of any real number can't be negative, the equation x2 = –1 does not have any solution. We make up a new imaginary number –1 ↔ ito be a solution of this equation and we name it as “ i ”, i.e.

(±i)2 = –1

x2 = –rare x = ± ir

A complex number is a number of the form a + biwhere a and b are real numbers,

Complex Numbers

A complex number is a number of the form a + biwhere a and b are real numbers, a is called the real part

Complex Numbers

A complex number is a number of the form a + biwhere a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number.

Complex Numbers

A complex number is a number of the form a + biwhere a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number.

Complex Numbers

Example B. 5 – 3i, 6i, –17 are complex numbers.

A complex number is a number of the form a + biwhere a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number.

Complex Numbers

Example B. 5 – 3i, 6i, –17 are complex numbers. The imaginary part of 5 – 3i is –3i.

A complex number is a number of the form a + biwhere a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number.

Complex Numbers

Example B. 5 – 3i, 6i, –17 are complex numbers. The imaginary part of 5 – 3i is –3i. The real part of 6i is 0.

A complex number is a number of the form a + biwhere a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number.

Complex Numbers

Example B. 5 – 3i, 6i, –17 are complex numbers. The imaginary part of 5 – 3i is –3i. The real part of 6i is 0.Any real number a is also a complex because a = a + 0i hence –17 = –17 + 0i.

A complex number is a number of the form a + biwhere a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number

Example B. 5 – 3i, 6i, –17 are complex numbers. The imaginary part of 5 – 3i is –3i. The real part of 6i is 0.Any real number a is also a complex because a = a + 0i hence –17 = –17 + 0i.

Complex Numbers

(Addition and subtraction of complex numbers)Treat the "i" as a variable when adding or subtracting complex numbers.

A complex number is a number of the form a + biwhere a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number

Complex Numbers

(Addition and subtraction of complex numbers)Treat the "i" as a variable when adding or subtracting complex numbers. Example C.(7 + 4i) + (5 – 3i)

Example B. 5 – 3i, 6i, –17 are complex numbers. The imaginary part of 5 – 3i is –3i. The real part of 6i is 0.Any real number a is also a complex because a = a + 0i hence –17 = –17 + 0i.

A complex number is a number of the form a + biwhere a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number

Complex Numbers

(Addition and subtraction of complex numbers)Treat the "i" as a variable when adding or subtracting complex numbers. Example C.(7 + 4i) + (5 – 3i) = 7 + 4i + 5 – 3i

Example B. 5 – 3i, 6i, –17 are complex numbers. The imaginary part of 5 – 3i is –3i. The real part of 6i is 0.Any real number a is also a complex because a = a + 0i hence –17 = –17 + 0i.

A complex number is a number of the form a + biwhere a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number

Complex Numbers

(Addition and subtraction of complex numbers)Treat the "i" as a variable when adding or subtracting complex numbers. Example C.(7 + 4i) + (5 – 3i) = 7 + 4i + 5 – 3i = 12 + i

Example B. 5 – 3i, 6i, –17 are complex numbers. The imaginary part of 5 – 3i is –3i. The real part of 6i is 0.Any real number a is also a complex because a = a + 0i hence –17 = –17 + 0i.

A complex number is a number of the form a + biwhere a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number

Complex Numbers

(Addition and subtraction of complex numbers)Treat the "i" as a variable when adding or subtracting complex numbers. Example C.(7 + 4i) + (5 – 3i) = 7 + 4i + 5 – 3i = 12 + i(7 + 4i) – (5 – 3i)

Example B. 5 – 3i, 6i, –17 are complex numbers. The imaginary part of 5 – 3i is –3i. The real part of 6i is 0.Any real number a is also a complex because a = a + 0i hence –17 = –17 + 0i.

A complex number is a number of the form a + biwhere a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number

Complex Numbers

(Addition and subtraction of complex numbers)Treat the "i" as a variable when adding or subtracting complex numbers. Example C.(7 + 4i) + (5 – 3i) = 7 + 4i + 5 – 3i = 12 + i(7 + 4i) – (5 – 3i) = 7 + 4i – 5 + 3i

Example B. 5 – 3i, 6i, –17 are complex numbers. The imaginary part of 5 – 3i is –3i. The real part of 6i is 0.Any real number a is also a complex because a = a + 0i hence –17 = –17 + 0i.

A complex number is a number of the form a + biwhere a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number

Complex Numbers

(Addition and subtraction of complex numbers)Treat the "i" as a variable when adding or subtracting complex numbers. Example C.(7 + 4i) + (5 – 3i) = 7 + 4i + 5 – 3i = 12 + i(7 + 4i) – (5 – 3i) = 7 + 4i – 5 + 3i = 2 + 7i

Example B. 5 – 3i, 6i, –17 are complex numbers. The imaginary part of 5 – 3i is –3i. The real part of 6i is 0.Any real number a is also a complex because a = a + 0i hence –17 = –17 + 0i.

(Multiplication of complex numbers)Complex Numbers

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i)

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i The conjugate of (a + bi) is (a – bi) and vice–versa.

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i The conjugate of (a + bi) is (a – bi) and vice–versa.The most important complex number multiplication formula is the product of a pair of conjugate numbers.

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i

(Conjugate Multiplication)

The conjugate of (a + bi) is (a – bi) and vice–versa.The most important complex number multiplication formula is the product of a pair of conjugate numbers.

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i

(Conjugate Multiplication) The nonzero conjugate product is(a + bi)(a – bi) = a2 + b2 which is always positive.

The conjugate of (a + bi) is (a – bi) and vice–versa.The most important complex number multiplication formula is the product of a pair of conjugate numbers.

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i

Example E.

(4 – 3i)(4 + 3i)

The conjugate of (a + bi) is (a – bi) and vice–versa.The most important complex number multiplication formula is the product of a pair of conjugate numbers. (Conjugate Multiplication) The nonzero conjugate product is (a + bi)(a – bi) = a2 + b2 which is always positive.

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i

Example E.

(4 – 3i)(4 + 3i) = 42 + 32 = 25

The conjugate of (a + bi) is (a – bi) and vice–versa.The most important complex number multiplication formula is the product of a pair of conjugate numbers. (Conjugate Multiplication) The nonzero conjugate product is(a + bi)(a – bi) = a2 + b2 which is always positive.

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i

Example E.

(4 – 3i)(4 + 3i) = 42 + 32 = 25(5 – 7i)(5 + 7i)

The conjugate of (a + bi) is (a – bi) and vice–versa.The most important complex number multiplication formula is the product of a pair of conjugate numbers. (Conjugate Multiplication) The nonzero conjugate product is(a + bi)(a – bi) = a2 + b2 which is always positive.

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i

Example E.

(4 – 3i)(4 + 3i) = 42 + 32 = 25(5 – 7i)(5 + 7i) = (5)2 + 72

The conjugate of (a + bi) is (a – bi) and vice–versa.The most important complex number multiplication formula is the product of a pair of conjugate numbers. (Conjugate Multiplication) The nonzero conjugate product is(a + bi)(a – bi) = a2 + b2 which is always positive.

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i

Example E.

(4 – 3i)(4 + 3i) = 42 + 32 = 25(5 – 7i)(5 + 7i) = (5)2 + 72 = 54

The conjugate of (a + bi) is (a – bi) and vice–versa.The most important complex number multiplication formula is the product of a pair of conjugate numbers. (Conjugate Multiplication) The nonzero conjugate product is(a + bi)(a – bi) = a2 + b2 which is always positive.

Complex Numbers(Division of Complex Numbers)

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

*

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i Example F. SimplifyMultiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 25

4 + 3i

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 2512 – 8i – 9i + 6i2

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 2512 – 8i – 9i + 6i2

–6

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 2512 – 8i – 9i + 6i2

–6

6 – 17i = 256

2517i –

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 2512 – 8i – 9i + 6i2

–6

6 – 17i = 256

2517i –

Using the quadratic formula, we can solve all 2nd degree equations and obtain their complex number solutions.

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 2512 – 8i – 9i + 6i2

–6

6 – 17i = 256

2517i –

Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers.

Using the quadratic formula, we can solve all 2nd degree equations and obtain their complex number solutions.

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 2512 – 8i – 9i + 6i2

–6

6 – 17i = 256

2517i –

Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers.To find b2 – 4ac first: a = 2, b = –2, c = 3,

Using the quadratic formula, we can solve all 2nd degree equations and obtain their complex number solutions.

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 2512 – 8i – 9i + 6i2

–6

6 – 17i = 256

2517i –

Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers.To find b2 – 4ac first: a = 2, b = –2, c = 3, so b2 – 4ac = –20.

Using the quadratic formula, we can solve all 2nd degree equations and obtain their complex number solutions.

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 2512 – 8i – 9i + 6i2

–6

6 – 17i = 256

2517i –

Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers.To find b2 – 4ac first: a = 2, b = –2, c = 3, so b2 – 4ac = –20.

x = 2 ± –204

Using the quadratic formula, we can solve all 2nd degree equations and obtain their complex number solutions.

Hence

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 2512 – 8i – 9i + 6i2

–6

6 – 17i = 256

2517i –

Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers.To find b2 – 4ac first: a = 2, b = –2, c = 3, so b2 – 4ac = –20.

x = 2 ± –204 = 2 ± 2–5

4

Using the quadratic formula, we can solve all 2nd degree equations and obtain their complex number solutions.

Hence

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 2512 – 8i – 9i + 6i2

–6

6 – 17i = 256

2517i –

Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers.To find b2 – 4ac first: a = 2, b = –2, c = 3, so b2 – 4ac = –20.

x = 2 ± –204 = 2 ± 2–5

4 = 2(1 ± i5)4 = 1 ± i5

2

Using the quadratic formula, we can solve all 2nd degree equations and obtain their complex number solutions.

Hence

Powers of i

The powers of i go in a cycle as shown below:

The powers of i go in a cycle as shown below:

i

-1 = i2

Powers of i

The powers of i go in a cycle as shown below:

i

-1 = i2

-i = i3

Powers of i

The powers of i go in a cycle as shown below:

i

-1 = i2

-i = i3

1 = i4

Powers of i

The powers of i go in a cycle as shown below:

i = i5

-1 = i2

-i = i3

1 = i4

Powers of i

The powers of i go in a cycle as shown below:

i = i5

-1 = i2 = i6 ..

-i = i3

1 = i4

Powers of i

The powers of i go in a cycle as shown below:

i = i5

-1 = i2 = i6 ..

-i = i3 = i7 ..

1 = i4

Powers of i

The powers of i go in a cycle as shown below:

i = i5

-1 = i2 = i6 ..

-i = i3 = i7 ..

1 = i4 = i8 ..

Powers of i

The powers of i go in a cycle as shown below:

i = i5 = i9 ..

-1 = i2 = i6 ..

-i = i3 = i7 ..

1 = i4 = i8 ..

Powers of i

The powers of i go in a cycle as shown below:

i = i5 = i9 ..

-1 = i2 = i6 ..

-i = i3 = i7 ..

1 = i4 = i8 ..

Example H. Simplify i59

Powers of i

The powers of i go in a cycle as shown below:

i = i5 = i9 ..

-1 = i2 = i6 ..

-i = i3 = i7 ..

1 = i4 = i8 ..

Example H. Simplify i59

59 = 4*14 + 3,

Powers of i

The powers of i go in a cycle as shown below:

i = i5 = i9 ..

-1 = i2 = i6 ..

-i = i3 = i7 ..

1 = i4 = i8 ..

Example H. Simplify i59

59 = 4*14 + 3, hence i59 = i4*14+3

Powers of i

The powers of i go in a cycle as shown below:

i = i5 = i9 ..

-1 = i2 = i6 ..

-i = i3 = i7 ..

1 = i4 = i8 ..

Example H. Simplify i59

59 = 4*14 + 3, hence i59 = i4*14+3 = i4*14+3

Powers of i

The powers of i go in a cycle as shown below:

i = i5 = i9 ..

-1 = i2 = i6 ..

-i = i3 = i7 ..

1 = i4 = i8 ..

Example H. Simplify i59

59 = 4*14 + 3, hence i59 = i4*14+3 = i4*14+3 = (i4)14 i3

Powers of i

The powers of i go in a cycle as shown below:

i = i5 = i9 ..

-1 = i2 = i6 ..

-i = i3 = i7 ..

1 = i4 = i8 ..

Example H. Simplify i59

59 = 4*14 + 3, hence i59 = i4*14+3 = i4*14+3 = (i4)14 i3 = 114 i3

Powers of i

The powers of i go in a cycle as shown below:

i = i5 = i9 ..

-1 = i2 = i6 ..

-i = i3 = i7 ..

1 = i4 = i8 ..

Example H. Simplify i59

59 = 4*14 + 3, hence i59 = i4*14+3 = i4*14+3 = (i4)14 i3 = 114 i3 = i3 = -i

Powers of i

Complex NumbersIn what sense are the complex numbers, numbers?Real numbers are physically measurable quantities (or the lack of such quantities in the case of the negative numbers).Theoretically, we can forever improve upon the construction of a stick with length exactly 2. But how do we make a stick of length 3i, or a cookie that weigh 3i oz? Well, we can’t. Imaginary numbers and complex numbers in general are not physically measurable in the traditional sense. Only the real numbers, which are a part of the complex numbers, are tangible in traditional sense.

Complex numbers are directional measurements. They keep track of measurements and directions, i.e. how much and in what direction (hence the two–component form of the complex numbers). Google the terms “complex numbers, 2D vectors” for further information.

Complex NumbersExercise A. Write the complex numbers in i’s. Combine the following expressions. 1. 2 – 2i + 3 + √–4 2. 4 – 5i – (4 – √–9) 3. 3 + 2i + (4 – i√5) 4. 4 – 2i + (–6 + i√3) 5. 4 – √–25 – (9 – √–

16) 6. 11 – 9i + (–7 + i√12) 7. ½ – (√–49)/3 – (3/4 – √–16) Exercise B. Do by inspection.8. (1 – 2i)(1 + 2i) 9. (1 + 3i)(1 – 3i) 10. (2 + 3i)(2 – 3i)11. (3 – 4i)(3 + 4i) 12. (9 + i√3)(9 – √3i) 13. (7 – i√5)(7 + i√5)14. (9 + i√3) (7 – i√5)(9 – i√3) (7 + i√5)15. (√3 + i√3) (√7 – i√5)(√3 – i√3)(√7 + i√5)Exercise C. Expand and simplify.16. (1 – 3i)(1 + 2i) 17. (2 + 3i)(1 – 3i) 18. (2 + 3i)(3 – 2i)19. (4 – 3i)(3 – 4i) 20. (5 + 3i)(5 + 3i) 21. (1 – i)2

22. (2 + 3i)2 23. (5 + 2i)2

Complex NumbersExercise D. Divide by rationalizing the denominators.

2 + 3ii24. 3 – 4i

i25. 3 – 4ii26.

1 + i1 – i27. 2 – i

3 – i28. 3 – 2i2 + i29.

2 + 3i2 – 3i30. 3 – 4i

3 – 2i31. 3 – 4i2 + 5i32.

33. Is there a difference between √4i and 2i?