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

Introduction Field Absolute Value Complex Numbers Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Complex Numbers

Introduction Field Absolute Value Introduction logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Complex Numbers

Introduction Field Absolute Value Introduction 1. We still cannot solve x 2 + 1 = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Complex Numbers

Introduction Field Absolute Value Introduction 1. We still cannot solve x 2 + 1 = 0 (in R ). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Complex Numbers

Introduction Field Absolute Value Introduction 1. We still cannot solve x 2 + 1 = 0 (in R ). 2. But “square roots of negative numbers” are very useful in solving equations. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Complex Numbers

Introduction Field Absolute Value Introduction 1. We still cannot solve x 2 + 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 x 2 + 1 = 0 and similar equations. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Complex Numbers

Introduction Field Absolute Value Introduction 1. We still cannot solve x 2 + 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 x 2 + 1 = 0 and similar equations. 4. Plus, we don’t want to lose any of the real numbers. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Complex Numbers

Introduction Field Absolute Value Introduction 1. We still cannot solve x 2 + 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 x 2 + 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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Complex Numbers

Introduction Field Absolute Value Introduction 1. We still cannot solve x 2 + 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 x 2 + 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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Complex Numbers

Introduction Field Absolute Value Introduction 1. We still cannot solve x 2 + 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 x 2 + 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.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Complex Numbers

Introduction Field Absolute Value Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Complex Numbers

Introduction Field Absolute Value Definition. The complex numbers C are the set R × R logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Complex Numbers

Introduction Field Absolute Value Definition. The complex numbers C are the set R × R equipped with addition and multiplication defined as follows. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Complex Numbers

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 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Complex Numbers

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 ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Complex Numbers

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 : = ( 0 R , 0 R ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Complex Numbers

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 : = ( 0 R , 0 R ) , 1 : = ( 1 R , 0 R ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Complex Numbers

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 : = ( 0 R , 0 R ) , 1 : = ( 1 R , 0 R ) and i : = ( 0 R , 1 R ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Complex Numbers

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 : = ( 0 R , 0 R ) , 1 : = ( 1 R , 0 R ) and i : = ( 0 R , 1 R ) . Complex numbers are also written in the form ( a , b ) = a · 1 + b · i = a + ib. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Complex Numbers

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 : = ( 0 R , 0 R ) , 1 : = ( 1 R , 0 R ) and i : = ( 0 R , 1 R ) . 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 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Complex Numbers

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 : = ( 0 R , 0 R ) , 1 : = ( 1 R , 0 R ) and i : = ( 0 R , 1 R ) . 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 ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Complex Numbers

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 : = ( 0 R , 0 R ) , 1 : = ( 1 R , 0 R ) and i : = ( 0 R , 1 R ) . 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 of z logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Complex Numbers