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**Unformatted text preview: **An introduction to complex numbers Solving quadratic equations and the imaginary unit The quadratic equation = - - x 2 4 x 5 0 can be solved by factoring. = - - x 2 4 x 5 0 exactly when = ( ) + x 1 ( )- x 5 0, so that = x- 1 or = x 5. Alternatively, the method of " completing the square " can be used. = - - x 2 4 x 5 0 exactly when = - x 2 4 x 5 ------- (i). A perfect square has the form = ( ) + x p 2 + + x 2 2 p x p 2 . Comparing the expression + x 2 2 p x with - x 2 4 x we see that taking = p- 2, which is " half the coefficient of x ", enables us to complete the square for the expression - x 2 4 x by adding = p 2 ( )- 2 2 = 4 to form the perfect square = - + x 2 4 x 4 ( )- x 2 2 . Hence by adding 4 to each side of (i) we obtain = - + x 2 4 x 4 + 5 4, which is equivalent to = ( )- x 2 2 9. Then = - x 2 + 3, so that = x 2 + 3 that is, = x- 1 or = x 5, as before. These two solutions can be interpreted graphically as the x-intercepts of the graph of the equation = y- - x 2 4 x 5. On the other hand, the quadratic equation = - + x 2 4 x 5 0 ------- (ii), in which the - 5 in the previous equation is replaced by + 5, has no real number solutions . Writing the equation (ii) in the form = - x 2 4 x- 5, we can complete the square on the left side of the equation by adding 4 to the left side as before. We must also add 4 to the right to obtain an equation which is equivalent to (ii). This gives = - + x 2 4 x 4- + 5 4 that is, = ( )- x 2 2- 1. We cannot proceed any further using only real numbers, since there is no real number with a square equal to - 1. This is consistent with the fact that the graph of = y- + x 2 4 x 5 has no x intercepts. Indeed, since this equation can be written in the form = y + ( )- x 2 2 1, we can see that y has a minimum value of 1 which is obtained when = x 0. Consequently it is impossible find a real value for x to give the y value 0, which is less than the minimum value 1. Returning to the equation = ( )- x 2 2- 1, we now introduce the imaginary unit denoted by i or j in mathematical notation, which is a new number (that is, it is not a real number) with precisely the property we are looking for, namely = i 2- 1 or = j 2- 1. The symbol i is preferred by pure mathematicians, while j is commonly used in applied mathematics....

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