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350 Imaginary and Complex Numbers
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you re more likely to go into science or engineering than pure mathematics, so you should get used to the notation they prefer
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Positive and negative j The square root of 1 can have either of two values, just as can the square root of any positive real number One of these is j The other is j, the product of j and 1 These two numbers are not the same, just as the positive and negative square roots of 1 are not the same!
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Are you confused
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Some people have trouble envisioning this unit imaginary number, also called the j operator Does the idea escape your mind s eye If so, don t worry about it Recall from Chap 3 that the natural numbers the simplest ones are built up from a set containing nothing! All numbers are abstract in the literal sense, so j isn t any more bizarre than 0, or 1, or any other number
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Here s a challenge!
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All of the laws of real-number arithmetic also apply to the unit imaginary number Based on that fact, figure out what happens as j is raised to increasing integer powers starting with the 1st power
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Solution
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Keep in mind that j is the positive square root of 1, which is ( 1)1/2 The parentheses are important in this expression If we leave them out, someone might get the idea that we re discussing the quantity (11/2), which is equal to 1 Because all the laws of the reals also apply to j, we can be sure that j 1 = j By definition, j 2 = 1 From this we can calculate j3 = j2 j = 1 j = j Now for the 4th power: j4 = j3 j = j j = 1 j j = 1 j 2 = 1 ( 1) =1 And the 5th power: j5 = j4 j =1 j =j
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The Imaginary Number Line
And the 6th: j6 = j5 j =j j = j2 = 1
Can you see what will happen if we keep going like this, increasing the integer power by 1 over and over We ll go in a four-way cycle If you grind things out, you ll see for yourself that j 7 = j, j 8 = 1, j 9 = j, j 10 = 1, and so on In general, if n is a positive integer, j n = j n+4
The Imaginary Number Line
The unit imaginary number j can be multiplied by any real number to get the positive square root of some negative real number Conversely, the positive square root of any negative real number is equal to some positive-real multiple of j If we want to multiply j by a positive real number b, we write jb If we want to multiply j by a negative real number b, we write jb, putting the minus sign in front of j rather than between j and b For example, j 5 = ( 1)1/2 251/2 = ( 1 5)1/2 = ( 5)1/2 and ( 4)1/2 = ( 1 4)1/2 = ( 1)1/2 41/2 = j2 If we take the real number line and multiply the value of every point by j, the result is the imaginary number line (Fig 21-1)
Are you confused
Why, you might ask, do we write j before the real-number numeral and not after it It s a matter of preference Engineers usually write the j before the real number If you see other notations for imaginary numbers such as 2j, 2i (the way most pure mathematicians write it), or even i 2, keep in mind that they all refer to the same quantity, which we would call j 2
Be careful!
In the challenge calculation at the end of the previous section, j was raised to integer powers If we re not careful, we can confuse expressions like these with integer multiples of j We must pay close attention to whether that real number is meant to be a multiple of j (as in j4), or a power of j (as in j 4)