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Transcript
Chapter 2. Section 2 Page 1 of 2 Section 2.2 – Complex Numbers • • • • • • What does it mean to have 9 ? Why is 9 = 3 ? The inverse of the square function is the square root function. And 9 = 3 because 32 = 9 In the Real Number system, we know that a negative times a negative is always positive. So how can we ever achieve squaring a number and getting a negative value? This translates into the square root function as well, which is why we can't take the square root of a negative value (like −9 ) in the Real Number system But sometimes we want to investigate this possibility, so we have what is known as the Complex Number system The Number i is defined so that i 2 = −1 or i = −1 , so we can express square roots of negative numbers according to this value − p = i2 p = i2 p =i p Complex Numbers: • A complex number is a number of the form a +bi, where a and b are real number values. The number a is said to be the real part and the number b is said to be the imaginary part • What are the restrictions on a and b? • Are complex numbers real numbers? Are real numbers complex numbers? • If b is nonzero, then the complex number a + bi is also called an imaginary number • Let's take a look at how the entire system maps out in what is known as a Venn diagram • The real number system we investigated in Chapter R • We now add complex numbers, imaginary numbers and pure imaginary numbers C. Bellomo, revised 26-Sep-07 Chapter 2. Section 2 Page 2 of 2 Operations of Imaginary Numbers: • The imaginary numbers obey the same laws as the real numbers, like commutative, associative and distributive • So we operate on them in much the same way (like we do binomials). We collect the real and imaginary parts • The only difference is we always replace i2 with −1 • Example. Simplify ( −6 − 5i ) + (9 + 2i ) ( −6 + 9) + (2i − 5i ) = 3 − 3i • Example. Simplify ( −3 − 4i ) − (8 − i ) ( −3 − 4i ) + ( −8 + i ) = ( −3 − 8) + ( −4i + i ) = −11 − 3i • Example. Simplify 3i (6 + 4i ) 3i (6 + 4i ) = 3i (6) + 3i (4i ) = 18i + 12i 2 = −12 + 18i • Example. Simplify (2 + 3i )(2 + 5i ) (2 + 3i )(2 + 5i ) = 2 ⋅ 2 + 2 ⋅ 5i + 3i ⋅ 2 + 3i ⋅ 5i = 4 + 16i − 15 = −11 + 16i Complex Conjugates: • The conjugate of a complex number a + bi is a − bi • The product of a complex number and its' conjugate is a real number, i.e. (a + bi )(a − bi ) = a 2 + bi − bi − b 2i 2 = a 2 + b 2 • The conjugate is used to simplify an expression. Somewhere along the way, it was decided that one shouldn't have a complex number in the denominator of an expression (much the same way of not having a square root in the denominator) i • Example. Simplify 2+i i ⎛ 2 − i ⎞ 2i − i 2 1 = (1 + 2i ) ⎜ ⎟= 2 + i ⎝ 2 − i ⎠ 4 − i2 5 1− i • Example. Simplify (1 + i ) 2 1− i 1 − i −2i −2 − 2i −1 = = (1 + i ) = 2 (1 + i ) 2i −2i 4 2 FG IJ H K Powers of i: • Note the cycle of powers of i: i 1 = i i 2 = −1 i 3 = −i i 4 = 1 i 5 = i... • We can simplify every power of i into one of the values above • Example. Simplify i24 i 24 = (i 2 )12 = ( −1)12 = 1 • Example. Simplify (2i)5 (2i ) 5 = 25 i 5 = 32 ⋅ i ⋅ i 4 = 32i (i 2 ) 2 = 32i C. Bellomo, revised 26-Sep-07