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10.7 Complex Numbers Up to this point, we have not been able to take the square root of a negative number. If the radicand is negative, we have to stop and say the expression is โnot a real numberโ or it is โundefined.โ This problem with the set of real numbers can be overcome by coming up with (inventing) a new number that does allow us to take the square root of negative numbers. We define the new number, called the imaginary unit, ๐, to be the principal square root of โ1. Definition of ๐ Thus, ๐ 2 = (โโ1) ๐ 2 = โ1 ๐ = โโ1 2 With this definition and the product rule for square roots, we can find the square root of any negative number in terms of ๐. โโ5 = โโ1 โ 5 = โโ1 โ โ5 = ๐โ5 โโ36 = โโ1 โ 36 = โโ1 โ โ36 = ๐ โ36 = 6๐ Square Root of Any Negative Number โโ๐ = โโ1 โ ๐ = ๐โ๐ where ๐ is any positive real number Examples: โโ9 = ๐โ9 = 3๐ โโ 9 16 = ๐โ 9 16 3 = ๐ 4 โโ6 = ๐ โ6 โโ8 = ๐ โ8 = 2๐โ2 โโ80 = ๐ โ80 = ๐โ16 โ 5 = 4๐โ5 Complex Numbers The numbers in our examples are not real numbers. They are a new type of number called imaginary numbers or just imaginaries. We now form a new set of numbers, the set of complex numbers, that consists of two mutually exclusive sets: the set of real numbers and the set of imaginary numbers. See the diagram at the top of page 720. Standard Form of a Complex Number A complex number is written in the standard form of ๐ + ๐๐ where ๐ and ๐ are real numbers ๐ is the ๐ซ๐๐๐ฅ ๐ฉ๐๐ซ๐ญ and ๐ is the ๐ข๐ฆ๐๐ ๐ข๐ง๐๐ซ๐ฒ ๐ฉ๐๐ซ๐ญ of the complex number ๏ท if ๐ = 0 then ๐ + 0๐ = ๐ is a ๐ซ๐๐๐ฅ ๐ง๐ฎ๐ฆ๐๐๐ซ 2 2 6 + 0๐ = 6 + 0๐ = โ5 + 0๐ = โ5 3 3 ๏ท if ๐ โ 0 and ๐ โ 0 then ๐ + ๐๐ is an ๐ข๐ฆ๐๐ ๐ข๐ง๐๐ซ๐ฒ ๐ง๐ฎ๐ฆ๐๐๐ซ 1 3 3 + 2๐ 4 โ 7๐ + ๐ 2 4 ๏ท if ๐ = 0 and ๐ โ 0 then 0 + ๐๐ = ๐๐ is an imaginary number called a ๐ฉ๐ฎ๐ซ๐ ๐ข๐ฆ๐๐ ๐ข๐ง๐๐ซ๐ฒ 0 + 2๐ = 2๐ 0 โ 8๐ = โ8๐ 0 + 0.7๐ = 0.7๐ --------------------------------------------------------------------------------------------Operations with Complex Numbers Adding and Subtracting We add and subtract complex numbers by combining the real parts and combining the imaginary parts. (2 + 3๐) + (โ4 + 5๐) = (2 + (โ4)) + (3๐ + 5๐) = โ2 + 8๐ --------------------------------------------------------------------------------------------(5 โ 11๐) + (7 + 4๐) = (5 + 7) + (โ11๐ + 4๐) = 12 โ 7๐ --------------------------------------------------------------------------------------------(โ5 + 7๐)โ (11 โ 6๐) = โ5 + 7๐ โ 11 + 6๐ = (โ5 โ 11) + (7๐ + 6๐) = โ16 + 13๐ --------------------------------------------------------------------------------------------6 โ (2 + ๐) = 6 โ 2 โ ๐ = (6 โ 2) โ ๐ = 4 โ ๐ --------------------------------------------------------------------------------------------โ2๐ โ (3 โ 9๐) = โ2๐ โ 3 + 9๐ = โ3 + (โ2๐ + 9๐) = โ3 + 7๐ Multiplying When multiplying complex numbers, you may get a factor of ๐ 2 . When this occurs, remember to replace ๐ 2 with โ 1 and continue simplifying. 2(3 โ 4๐) = 2 โ 3 โ 2 โ 4๐ = 6 โ 8๐ --------------------------------------------------------------------------------------------3๐(6 + ๐) = 3๐ โ 6 + 3๐ โ ๐ = 18๐ + 3๐ 2 = 18๐ + 3(โ1) = 18๐ โ 3 = โ3 + 18๐ --------------------------------------------------------------------------------------------(โ5 + 2๐)(7 โ 6๐) = โ5 โ 7 + 5 โ 6๐ + 2๐ โ 7 โ 2๐ โ 6๐ = โ35 + 30๐ + 14๐ โ 12๐ 2 = โ35 + 44๐ โ 12(โ1) = โ35 + 44๐ + 12 = โ23 + 44๐ Dividing Divisor is a Monomial and Real Use the distributive property and simplify the division in each term. 3+9๐ 3 3 =3+ 9๐ 3 = 1 + 3๐ Divisor is a Monomial and Imaginary Multiply the numerator and denominator by i then use the distributive property and simplify the division in each term. โ2+4๐ 5๐ = (โ2+4๐)๐ 5๐โ๐ = โ2๐+4๐ 2 5๐ 2 = โ4โ2๐ โ5 = โ4 โ5 + โ2 โ5 4 2 5 5 ๐= + ๐ Divisor is a Binomial of the form a + bi Multiply the numerator and denominator by the complex conjugate of the denominator. ๐ + ๐๐ and ๐ โ ๐๐ are complex conjugates. When complex conjugates are multiplied together, the result is a real number. (๐ + ๐๐)(๐ โ ๐๐) = ๐2 โ (๐๐)2 = ๐2 โ ๐ 2 ๐ 2 = ๐2 โ ๐ 2 (โ1) = ๐2 + ๐ 2 2โ3๐ 4+๐ = (2โ3๐)(4โ๐) (4+๐)(4โ๐) = 8โ2๐โ12๐+3๐ 2 4 2 +12 = 5โ14๐ 17 = 5 17 โ 14 17 ๐ --------------------------------------------------------------------------------------------Powers of i The powers of i are cyclical. They repeat in a cycle of 4. ๐1 = ๐ ๐ 2 = โ1 ๐ 3 = ๐ 2 โ ๐ = โ1 โ ๐ = โ๐ ๐ 4 = ๐ 2 โ ๐ 2 = (โ1)(โ1) = 1 ๐ 5 = ๐ 4 โ ๐ = 1๐ = ๐ ๐ 7 = ๐ 4 โ ๐ 3 = 1 โ โ๐ = โ๐ ๐ 6 = ๐ 4 โ ๐ 2 = 1 โ โ1 = โ1 ๐8 = ๐4 โ ๐4 = 1 โ 1 = 1 To simplify powers of i, divide the exponent by 4. ๏ท If the remainder is 1, the value of the expression is ๐. ๏ท If the remainder is 2, the value of the expression is โ1. ๏ท If the remainder is 3, the value of the expression is โ๐. ๏ท If the remainder is 0, the value of the expression is 1. ๐ 23 23 ÷ 4 = 5 remainder 3 ๐ 23 = โ๐ ๐ 81 81 ÷ 4 = 20 remainder 1 ๐ 81 = ๐ NOTE: On your calculator, 1โ a remainder of 1 will show as the decimal .25 4 2โ = 1โ a remainder of 2 will show as the decimal .5 4 2 3 a remainder of 3 will show as the decimal .75 โ4 a remainder of 0 will show a whole number answer