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Unformatted text preview: 18.03 Lecture #5 Sept. 18, 2009: notes Topic for today is complex numbers. More or less the assumption is that you have seen complex numbers at least a bit before, so the emphasis is on the particular ideas that are most helpful for understanding differential equations. First, definition: a complex number is a symbol a + bi , with a and b real numbers. So 3 + 11 i , 2 − πi , and 11 (thought of as 11+0 i are complex numbers. A complex number is very often written as z (just as a real number is often written x ). If z = a + bi , then we define z = a − bi, the complex conjugate of z, (prounounced “zee bar”). Furthermore Re z = a = ( z + z ) / 2 , the real part of z, and Im z = b = ( z − z ) / 2 i, the imaginary part of z. (Notice that the “imaginary part” is the real number b , not the imaginary number bi .) A complex number z = a + bi is often identified with the point ( a,b ) in the plane; when we’re using the plane in this way it’s called the complex plane . The distance of the point to the origin turns out to be important for complex numbers, and it’s called the modulus of z : | z | = radicalbig a 2 + b 2 . Complex numbers can be added, subtracted, multiplied, and divided (except that you’re not allowed to divide by zero). These operations satisfy the basic laws of arithmetic: associative laws for addition and multiplication, and the distributive law. Addition and subtraction are easy: ( a + bi ) + ( c + di ) = ( a + c ) + ( b + d ) i, ( a + bi ) − ( c + di ) = ( a − c ) + ( b − d ) i. The rule for multiplication is determined by the distributive law and the definition i 2 = − 1. This gives ( a + bi )( c + di ) = ( ac − bd ) + ( ad + bc ) i. What you should remember is just i 2 = − 1 and how to use the distributive law to compute with that. Division is a little trickier. It’s based on the important identity w w = ( c + di )( c − di ) = c 2 + d 2 = | w | 2 ....
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This note was uploaded on 05/06/2010 for the course 18 18.03 taught by Professor Unknown during the Fall '09 term at MIT.
- Fall '09