Complex Numbers Bernd Schr oder logo1 Bernd Schr oder Louisiana - - PowerPoint PPT Presentation

logo1 Introduction Field Absolute Value

Complex Numbers

Bernd Schr¨

• der

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

Introduction

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

Introduction

• 1. We still cannot solve x2 +1 = 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

Introduction

• 1. We still cannot solve x2 +1 = 0 (in R).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

Introduction

• 1. We still cannot solve x2 +1 = 0 (in R).
• 2. But “square roots of negative numbers” are very useful in

solving equations.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

Introduction

• 1. We still cannot solve x2 +1 = 0 (in R).
• 2. But “square roots of negative numbers” are very useful in

solving equations.

• 3. So we want to have a field in which we can solve

x2 +1 = 0 and similar equations.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

Introduction

• 1. We still cannot solve x2 +1 = 0 (in R).
• 2. But “square roots of negative numbers” are very useful in

solving equations.

• 3. So we want to have a field in which we can solve

x2 +1 = 0 and similar equations.

• 4. Plus, we don’t want to lose any of the real numbers.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

Introduction

• 1. We still cannot solve x2 +1 = 0 (in R).
• 2. But “square roots of negative numbers” are very useful in

solving equations.

• 3. So we want to have a field in which we can solve

x2 +1 = 0 and similar equations.

• 4. Plus, we don’t want to lose any of the real numbers.
• 5. But that means we will lose some properties, because the

real numbers are unique.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

Introduction

• 1. We still cannot solve x2 +1 = 0 (in R).
• 2. But “square roots of negative numbers” are very useful in

solving equations.

• 3. So we want to have a field in which we can solve

x2 +1 = 0 and similar equations.

• 4. Plus, we don’t want to lose any of the real numbers.
• 5. But that means we will lose some properties, because the

real numbers are unique.

• 6. There is no way to turn C into a totally ordered field.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

Introduction

• 1. We still cannot solve x2 +1 = 0 (in R).
• 2. But “square roots of negative numbers” are very useful in

solving equations.

• 3. So we want to have a field in which we can solve

x2 +1 = 0 and similar equations.

• 4. Plus, we don’t want to lose any of the real numbers.
• 5. But that means we will lose some properties, because the

real numbers are unique.

• 6. There is no way to turn C into a totally ordered field.

(Good exercise.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

Definition.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. The complex numbers C are the set R×R

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. The complex numbers C are the set R×R

equipped with addition and multiplication defined as follows.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. The complex numbers C are the set R×R

equipped with addition and multiplication defined as follows. For all complex numbers (a,b),(c,d) ∈ C, we set (a,b)+(c,d) := (a+c,b+d)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. The complex numbers C are the set R×R

equipped with addition and multiplication defined as follows. For all complex numbers (a,b),(c,d) ∈ C, we set (a,b)+(c,d) := (a+c,b+d) and (a,b)·(c,d) := (ac−bd,ad +bc).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. The complex numbers C are the set R×R

equipped with addition and multiplication defined as follows. For all complex numbers (a,b),(c,d) ∈ C, we set (a,b)+(c,d) := (a+c,b+d) and (a,b)·(c,d) := (ac−bd,ad +bc). We define 0 := (0R,0R)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. The complex numbers C are the set R×R

equipped with addition and multiplication defined as follows. For all complex numbers (a,b),(c,d) ∈ C, we set (a,b)+(c,d) := (a+c,b+d) and (a,b)·(c,d) := (ac−bd,ad +bc). We define 0 := (0R,0R), 1 := (1R,0R)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. The complex numbers C are the set R×R

equipped with addition and multiplication defined as follows. For all complex numbers (a,b),(c,d) ∈ C, we set (a,b)+(c,d) := (a+c,b+d) and (a,b)·(c,d) := (ac−bd,ad +bc). We define 0 := (0R,0R), 1 := (1R,0R) and i := (0R,1R).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. The complex numbers C are the set R×R

equipped with addition and multiplication defined as follows. For all complex numbers (a,b),(c,d) ∈ C, we set (a,b)+(c,d) := (a+c,b+d) and (a,b)·(c,d) := (ac−bd,ad +bc). We define 0 := (0R,0R), 1 := (1R,0R) and i := (0R,1R). Complex numbers are also written in the form (a,b) = a·1+b·i = a+ib.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. The complex numbers C are the set R×R

equipped with addition and multiplication defined as follows. For all complex numbers (a,b),(c,d) ∈ C, we set (a,b)+(c,d) := (a+c,b+d) and (a,b)·(c,d) := (ac−bd,ad +bc). We define 0 := (0R,0R), 1 := (1R,0R) and i := (0R,1R). Complex numbers are also written in the form (a,b) = a·1+b·i = a+ib. For z = a+ib ∈ C, the number a is also called the real part of z

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. The complex numbers C are the set R×R

equipped with addition and multiplication defined as follows. For all complex numbers (a,b),(c,d) ∈ C, we set (a,b)+(c,d) := (a+c,b+d) and (a,b)·(c,d) := (ac−bd,ad +bc). We define 0 := (0R,0R), 1 := (1R,0R) and i := (0R,1R). Complex numbers are also written in the form (a,b) = a·1+b·i = a+ib. For z = a+ib ∈ C, the number a is also called the real part of z, denoted ℜ(z).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. The complex numbers C are the set R×R

equipped with addition and multiplication defined as follows. For all complex numbers (a,b),(c,d) ∈ C, we set (a,b)+(c,d) := (a+c,b+d) and (a,b)·(c,d) := (ac−bd,ad +bc). We define 0 := (0R,0R), 1 := (1R,0R) and i := (0R,1R). Complex numbers are also written in the form (a,b) = a·1+b·i = a+ib. For z = a+ib ∈ C, the number a is also called the real part of z, denoted ℜ(z). The number b is also called the imaginary part

• f z

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. The complex numbers C are the set R×R

equipped with addition and multiplication defined as follows. For all complex numbers (a,b),(c,d) ∈ C, we set (a,b)+(c,d) := (a+c,b+d) and (a,b)·(c,d) := (ac−bd,ad +bc). We define 0 := (0R,0R), 1 := (1R,0R) and i := (0R,1R). Complex numbers are also written in the form (a,b) = a·1+b·i = a+ib. For z = a+ib ∈ C, the number a is also called the real part of z, denoted ℜ(z). The number b is also called the imaginary part

• f z, denoted ℑ(z).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

Theorem.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Theorem. The complex numbers C with addition,

multiplication, 0 and 1 as defined above are a field.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Theorem. The complex numbers C with addition,

multiplication, 0 and 1 as defined above are a field. Proof.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Theorem. The complex numbers C with addition,

multiplication, 0 and 1 as defined above are a field.

• Proof. Good exercise.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Theorem. The complex numbers C with addition,

multiplication, 0 and 1 as defined above are a field.

• Proof. Good exercise. For a+ib ∈ C

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Theorem. The complex numbers C with addition,

multiplication, 0 and 1 as defined above are a field.

• Proof. Good exercise. For a+ib ∈ C, the additive inverse is

−(a+bi) = (−a)+(−b)i

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Theorem. The complex numbers C with addition,

multiplication, 0 and 1 as defined above are a field.

• Proof. Good exercise. For a+ib ∈ C, the additive inverse is

−(a+bi) = (−a)+(−b)i and for a+ib ∈ C\{0}

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Theorem. The complex numbers C with addition,

multiplication, 0 and 1 as defined above are a field.

• Proof. Good exercise. For a+ib ∈ C, the additive inverse is

−(a+bi) = (−a)+(−b)i and for a+ib ∈ C\{0}, the multiplicative inverse is (a+ib)−1 = a a2 +b2 − b a2 +b2i.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Theorem. The complex numbers C with addition,

multiplication, 0 and 1 as defined above are a field.

• Proof. Good exercise. For a+ib ∈ C, the additive inverse is

−(a+bi) = (−a)+(−b)i and for a+ib ∈ C\{0}, the multiplicative inverse is (a+ib)−1 = a a2 +b2 − b a2 +b2i.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Theorem. The complex numbers C with addition,

multiplication, 0 and 1 as defined above are a field.

• Proof. Good exercise. For a+ib ∈ C, the additive inverse is

−(a+bi) = (−a)+(−b)i and for a+ib ∈ C\{0}, the multiplicative inverse is (a+ib)−1 = a a2 +b2 − b a2 +b2i. Theorem.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Theorem. The complex numbers C with addition,

multiplication, 0 and 1 as defined above are a field.

• Proof. Good exercise. For a+ib ∈ C, the additive inverse is

−(a+bi) = (−a)+(−b)i and for a+ib ∈ C\{0}, the multiplicative inverse is (a+ib)−1 = a a2 +b2 − b a2 +b2i.

• Theorem. i2 = −1.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Theorem. The complex numbers C with addition,

multiplication, 0 and 1 as defined above are a field.

• Proof. Good exercise. For a+ib ∈ C, the additive inverse is

−(a+bi) = (−a)+(−b)i and for a+ib ∈ C\{0}, the multiplicative inverse is (a+ib)−1 = a a2 +b2 − b a2 +b2i.

• Theorem. i2 = −1.

Proof.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Theorem. The complex numbers C with addition,

multiplication, 0 and 1 as defined above are a field.

• Proof. Good exercise. For a+ib ∈ C, the additive inverse is

−(a+bi) = (−a)+(−b)i and for a+ib ∈ C\{0}, the multiplicative inverse is (a+ib)−1 = a a2 +b2 − b a2 +b2i.

• Theorem. i2 = −1.

Proof. i2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Theorem. The complex numbers C with addition,

multiplication, 0 and 1 as defined above are a field.

• Proof. Good exercise. For a+ib ∈ C, the additive inverse is

−(a+bi) = (−a)+(−b)i and for a+ib ∈ C\{0}, the multiplicative inverse is (a+ib)−1 = a a2 +b2 − b a2 +b2i.

• Theorem. i2 = −1.

Proof. i2 = (0+1i)·(0+1i)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Theorem. The complex numbers C with addition,

multiplication, 0 and 1 as defined above are a field.

• Proof. Good exercise. For a+ib ∈ C, the additive inverse is

−(a+bi) = (−a)+(−b)i and for a+ib ∈ C\{0}, the multiplicative inverse is (a+ib)−1 = a a2 +b2 − b a2 +b2i.

• Theorem. i2 = −1.

Proof. i2 = (0+1i)·(0+1i) = (0·0−1·1)+(0·1+1·0)i

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Theorem. The complex numbers C with addition,

multiplication, 0 and 1 as defined above are a field.

• Proof. Good exercise. For a+ib ∈ C, the additive inverse is

−(a+bi) = (−a)+(−b)i and for a+ib ∈ C\{0}, the multiplicative inverse is (a+ib)−1 = a a2 +b2 − b a2 +b2i.

• Theorem. i2 = −1.

Proof. i2 = (0+1i)·(0+1i) = (0·0−1·1)+(0·1+1·0)i = −1.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Theorem. The complex numbers C with addition,

multiplication, 0 and 1 as defined above are a field.

• Proof. Good exercise. For a+ib ∈ C, the additive inverse is

−(a+bi) = (−a)+(−b)i and for a+ib ∈ C\{0}, the multiplicative inverse is (a+ib)−1 = a a2 +b2 − b a2 +b2i.

• Theorem. i2 = −1.

Proof. i2 = (0+1i)·(0+1i) = (0·0−1·1)+(0·1+1·0)i = −1.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

Definition.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. For a+ib ∈ C we define |z| :=
• a2 +b2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. For a+ib ∈ C we define |z| :=
• a2 +b2 and we

call it the absolute value of z.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. For a+ib ∈ C we define |z| :=
• a2 +b2 and we

call it the absolute value of z. Theorem.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. For a+ib ∈ C we define |z| :=
• a2 +b2 and we

call it the absolute value of z.

• Theorem. Properties of the absolute value.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. For a+ib ∈ C we define |z| :=
• a2 +b2 and we

call it the absolute value of z.

• Theorem. Properties of the absolute value.
• 0. For all z ∈ C, we have |z| ≥ 0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. For a+ib ∈ C we define |z| :=
• a2 +b2 and we

call it the absolute value of z.

• Theorem. Properties of the absolute value.
• 0. For all z ∈ C, we have |z| ≥ 0. (Because |z| ∈ R, it is

permissible to use inequalities here.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. For a+ib ∈ C we define |z| :=
• a2 +b2 and we

call it the absolute value of z.

• Theorem. Properties of the absolute value.
• 0. For all z ∈ C, we have |z| ≥ 0. (Because |z| ∈ R, it is

permissible to use inequalities here.)

• 1. For all z ∈ C, we have |z| = 0 iff z = 0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. For a+ib ∈ C we define |z| :=
• a2 +b2 and we

call it the absolute value of z.

• Theorem. Properties of the absolute value.
• 0. For all z ∈ C, we have |z| ≥ 0. (Because |z| ∈ R, it is

permissible to use inequalities here.)

• 1. For all z ∈ C, we have |z| = 0 iff z = 0.
• 2. For all z1,z2 ∈ C, we have |z1z2| = |z1||z2|.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. For a+ib ∈ C we define |z| :=
• a2 +b2 and we

call it the absolute value of z.

• Theorem. Properties of the absolute value.
• 0. For all z ∈ C, we have |z| ≥ 0. (Because |z| ∈ R, it is

permissible to use inequalities here.)

• 1. For all z ∈ C, we have |z| = 0 iff z = 0.
• 2. For all z1,z2 ∈ C, we have |z1z2| = |z1||z2|.
• 3. The triangular inequality holds.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. For a+ib ∈ C we define |z| :=
• a2 +b2 and we

call it the absolute value of z.

• Theorem. Properties of the absolute value.
• 0. For all z ∈ C, we have |z| ≥ 0. (Because |z| ∈ R, it is

permissible to use inequalities here.)

• 1. For all z ∈ C, we have |z| = 0 iff z = 0.
• 2. For all z1,z2 ∈ C, we have |z1z2| = |z1||z2|.
• 3. The triangular inequality holds. That is, for all z1,z2 ∈ C

we have |z1 +z2| ≤ |z1|+|z2|.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

Proof.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Proof. Parts 0 to 2 are good exercises.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Proof. Parts 0 to 2 are good exercises. For part 3, let z1 = a+ib

and z2 = c+id.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Proof. Parts 0 to 2 are good exercises. For part 3, let z1 = a+ib

and z2 = c+id. Then 0 ≤ (ad −bc)2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Proof. Parts 0 to 2 are good exercises. For part 3, let z1 = a+ib

and z2 = c+id. Then 0 ≤ (ad −bc)2 = a2d2 −2abcd +b2c2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Proof. Parts 0 to 2 are good exercises. For part 3, let z1 = a+ib

and z2 = c+id. Then 0 ≤ (ad −bc)2 = a2d2 −2abcd +b2c2, so 2abcd ≤ a2d2 +b2c2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Proof. Parts 0 to 2 are good exercises. For part 3, let z1 = a+ib

and z2 = c+id. Then 0 ≤ (ad −bc)2 = a2d2 −2abcd +b2c2, so 2abcd ≤ a2d2 +b2c2 a2c2 +2abcd +b2d2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Proof. Parts 0 to 2 are good exercises. For part 3, let z1 = a+ib

and z2 = c+id. Then 0 ≤ (ad −bc)2 = a2d2 −2abcd +b2c2, so 2abcd ≤ a2d2 +b2c2 a2c2 +2abcd +b2d2 ≤ a2c2 +a2d2 +b2c2 +b2d2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Proof. Parts 0 to 2 are good exercises. For part 3, let z1 = a+ib

and z2 = c+id. Then 0 ≤ (ad −bc)2 = a2d2 −2abcd +b2c2, so 2abcd ≤ a2d2 +b2c2 a2c2 +2abcd +b2d2 ≤ a2c2 +a2d2 +b2c2 +b2d2 (ac+bd)2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Proof. Parts 0 to 2 are good exercises. For part 3, let z1 = a+ib

and z2 = c+id. Then 0 ≤ (ad −bc)2 = a2d2 −2abcd +b2c2, so 2abcd ≤ a2d2 +b2c2 a2c2 +2abcd +b2d2 ≤ a2c2 +a2d2 +b2c2 +b2d2 (ac+bd)2 ≤

• a2 +b2

c2 +d2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Proof. Parts 0 to 2 are good exercises. For part 3, let z1 = a+ib

and z2 = c+id. Then 0 ≤ (ad −bc)2 = a2d2 −2abcd +b2c2, so 2abcd ≤ a2d2 +b2c2 a2c2 +2abcd +b2d2 ≤ a2c2 +a2d2 +b2c2 +b2d2 (ac+bd)2 ≤

• a2 +b2

c2 +d2 2ac+2bd

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Proof. Parts 0 to 2 are good exercises. For part 3, let z1 = a+ib

and z2 = c+id. Then 0 ≤ (ad −bc)2 = a2d2 −2abcd +b2c2, so 2abcd ≤ a2d2 +b2c2 a2c2 +2abcd +b2d2 ≤ a2c2 +a2d2 +b2c2 +b2d2 (ac+bd)2 ≤

• a2 +b2

c2 +d2 2ac+2bd ≤ 2

• a2 +b2

c2 +d2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Proof. Parts 0 to 2 are good exercises. For part 3, let z1 = a+ib

and z2 = c+id. Then 0 ≤ (ad −bc)2 = a2d2 −2abcd +b2c2, so 2abcd ≤ a2d2 +b2c2 a2c2 +2abcd +b2d2 ≤ a2c2 +a2d2 +b2c2 +b2d2 (ac+bd)2 ≤

• a2 +b2

c2 +d2 2ac+2bd ≤ 2

• a2 +b2

c2 +d2 a2+2ac+c2+b2+2bd+d2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Proof. Parts 0 to 2 are good exercises. For part 3, let z1 = a+ib

and z2 = c+id. Then 0 ≤ (ad −bc)2 = a2d2 −2abcd +b2c2, so 2abcd ≤ a2d2 +b2c2 a2c2 +2abcd +b2d2 ≤ a2c2 +a2d2 +b2c2 +b2d2 (ac+bd)2 ≤

• a2 +b2

c2 +d2 2ac+2bd ≤ 2

• a2 +b2

c2 +d2 a2+2ac+c2+b2+2bd+d2 ≤ a2+b2+2

• a2+b2

c2+d2+c2+d2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Proof. Parts 0 to 2 are good exercises. For part 3, let z1 = a+ib

and z2 = c+id. Then 0 ≤ (ad −bc)2 = a2d2 −2abcd +b2c2, so 2abcd ≤ a2d2 +b2c2 a2c2 +2abcd +b2d2 ≤ a2c2 +a2d2 +b2c2 +b2d2 (ac+bd)2 ≤

• a2 +b2

c2 +d2 2ac+2bd ≤ 2

• a2 +b2

c2 +d2 a2+2ac+c2+b2+2bd+d2 ≤ a2+b2+2

• a2+b2

c2+d2+c2+d2 (a+c)2 +(b+d)2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Proof. Parts 0 to 2 are good exercises. For part 3, let z1 = a+ib

and z2 = c+id. Then 0 ≤ (ad −bc)2 = a2d2 −2abcd +b2c2, so 2abcd ≤ a2d2 +b2c2 a2c2 +2abcd +b2d2 ≤ a2c2 +a2d2 +b2c2 +b2d2 (ac+bd)2 ≤

• a2 +b2

c2 +d2 2ac+2bd ≤ 2

• a2 +b2

c2 +d2 a2+2ac+c2+b2+2bd+d2 ≤ a2+b2+2

• a2+b2

c2+d2+c2+d2 (a+c)2 +(b+d)2 ≤

• a2 +b2 +
• c2 +d2

2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Proof. Parts 0 to 2 are good exercises. For part 3, let z1 = a+ib

and z2 = c+id. Then 0 ≤ (ad −bc)2 = a2d2 −2abcd +b2c2, so 2abcd ≤ a2d2 +b2c2 a2c2 +2abcd +b2d2 ≤ a2c2 +a2d2 +b2c2 +b2d2 (ac+bd)2 ≤

• a2 +b2

c2 +d2 2ac+2bd ≤ 2

• a2 +b2

c2 +d2 a2+2ac+c2+b2+2bd+d2 ≤ a2+b2+2

• a2+b2

c2+d2+c2+d2 (a+c)2 +(b+d)2 ≤

• a2 +b2 +
• c2 +d2

2 |z1 +z2|2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Proof. Parts 0 to 2 are good exercises. For part 3, let z1 = a+ib

and z2 = c+id. Then 0 ≤ (ad −bc)2 = a2d2 −2abcd +b2c2, so 2abcd ≤ a2d2 +b2c2 a2c2 +2abcd +b2d2 ≤ a2c2 +a2d2 +b2c2 +b2d2 (ac+bd)2 ≤

• a2 +b2

c2 +d2 2ac+2bd ≤ 2

• a2 +b2

c2 +d2 a2+2ac+c2+b2+2bd+d2 ≤ a2+b2+2

• a2+b2

c2+d2+c2+d2 (a+c)2 +(b+d)2 ≤

• a2 +b2 +
• c2 +d2

2 |z1 +z2|2 ≤

• |z1|+|z2|

2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Proof. Parts 0 to 2 are good exercises. For part 3, let z1 = a+ib

and z2 = c+id. Then 0 ≤ (ad −bc)2 = a2d2 −2abcd +b2c2, so 2abcd ≤ a2d2 +b2c2 a2c2 +2abcd +b2d2 ≤ a2c2 +a2d2 +b2c2 +b2d2 (ac+bd)2 ≤

• a2 +b2

c2 +d2 2ac+2bd ≤ 2

• a2 +b2

c2 +d2 a2+2ac+c2+b2+2bd+d2 ≤ a2+b2+2

• a2+b2

c2+d2+c2+d2 (a+c)2 +(b+d)2 ≤

• a2 +b2 +
• c2 +d2

2 |z1 +z2|2 ≤

• |z1|+|z2|

2 |z1 +z2|

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Proof. Parts 0 to 2 are good exercises. For part 3, let z1 = a+ib

and z2 = c+id. Then 0 ≤ (ad −bc)2 = a2d2 −2abcd +b2c2, so 2abcd ≤ a2d2 +b2c2 a2c2 +2abcd +b2d2 ≤ a2c2 +a2d2 +b2c2 +b2d2 (ac+bd)2 ≤

• a2 +b2

c2 +d2 2ac+2bd ≤ 2

• a2 +b2

c2 +d2 a2+2ac+c2+b2+2bd+d2 ≤ a2+b2+2

• a2+b2

c2+d2+c2+d2 (a+c)2 +(b+d)2 ≤

• a2 +b2 +
• c2 +d2

2 |z1 +z2|2 ≤

• |z1|+|z2|

2 |z1 +z2| ≤ |z1|+|z2|

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Proof. Parts 0 to 2 are good exercises. For part 3, let z1 = a+ib

and z2 = c+id. Then 0 ≤ (ad −bc)2 = a2d2 −2abcd +b2c2, so 2abcd ≤ a2d2 +b2c2 a2c2 +2abcd +b2d2 ≤ a2c2 +a2d2 +b2c2 +b2d2 (ac+bd)2 ≤

• a2 +b2

c2 +d2 2ac+2bd ≤ 2

• a2 +b2

c2 +d2 a2+2ac+c2+b2+2bd+d2 ≤ a2+b2+2

• a2+b2

c2+d2+c2+d2 (a+c)2 +(b+d)2 ≤

• a2 +b2 +
• c2 +d2

2 |z1 +z2|2 ≤

• |z1|+|z2|

2 |z1 +z2| ≤ |z1|+|z2|

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

Definition.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. For z = a+ib ∈ C, the complex conjugate of z is

z := a−ib.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. For z = a+ib ∈ C, the complex conjugate of z is

z := a−ib. Proposition.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. For z = a+ib ∈ C, the complex conjugate of z is

z := a−ib.

• Proposition. For all z ∈ C, the equalities z+z = 2ℜ(z)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. For z = a+ib ∈ C, the complex conjugate of z is

z := a−ib.

• Proposition. For all z ∈ C, the equalities z+z = 2ℜ(z) and

|z|2 = zz hold.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. For z = a+ib ∈ C, the complex conjugate of z is

z := a−ib.

• Proposition. For all z ∈ C, the equalities z+z = 2ℜ(z) and

|z|2 = zz hold. Moreover, for all z ∈ C\{0} the multiplicative inverse is 1 z = z |z|2.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. For z = a+ib ∈ C, the complex conjugate of z is

z := a−ib.

• Proposition. For all z ∈ C, the equalities z+z = 2ℜ(z) and

|z|2 = zz hold. Moreover, for all z ∈ C\{0} the multiplicative inverse is 1 z = z |z|2.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. For z = a+ib ∈ C, the complex conjugate of z is

z := a−ib.

• Proposition. For all z ∈ C, the equalities z+z = 2ℜ(z) and

|z|2 = zz hold. Moreover, for all z ∈ C\{0} the multiplicative inverse is 1 z = z |z|2. Proof.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. For z = a+ib ∈ C, the complex conjugate of z is

z := a−ib.

• Proposition. For all z ∈ C, the equalities z+z = 2ℜ(z) and

|z|2 = zz hold. Moreover, for all z ∈ C\{0} the multiplicative inverse is 1 z = z |z|2.

• Proof. Good exercise.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers

logo1 Introduction Field Absolute Value

• Definition. For z = a+ib ∈ C, the complex conjugate of z is

z := a−ib.

• Proposition. For all z ∈ C, the equalities z+z = 2ℜ(z) and

|z|2 = zz hold. Moreover, for all z ∈ C\{0} the multiplicative inverse is 1 z = z |z|2.

• Proof. Good exercise.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Complex Numbers