Math Operations with Complex Numbers College Algebra 1.3 day 2 Notes

Learning Target: Students will be able to… Add, Subtract, Multiply and Divide Complex Numbers

Adding Complex Numbers looks very similar to Adding Polynomials 𝐴𝑁𝑆: 1+2𝑥 𝐴𝑁𝑆: 1+2𝑖 Ex 3) 4−5𝑖 +(−5+8𝑖) Ex 4) 2 −𝑖 +(−3 2 +9𝑖) 𝐴𝑁𝑆: −1+3𝑖 𝐴𝑁𝑆: −2 2 +8𝑖

Subtracting Complex Numbers looks very similar to Subtracting Polynomials −4−3𝑥−6+7𝑥 −4−3𝑖−6+7𝑖 𝐴𝑁𝑆: −10+4𝑥 ANS: −10+4𝑖 Ex 3) −10+7𝑖 −(5−3𝑖) Ex 4) 7 3 −4𝑖 −(−3 3 +9𝑖) −15+10𝑖 10 3 −13𝑖

Multiplying Complex Numbers looks very similar to Multiplying Polynomials….with one exception! 4−3𝑥 (6−7𝑥) 4−3𝑖 (6−7𝑖) 24−28𝑖−18𝑖+21 𝑖 2 24−28𝑥−18𝑥+21 𝑥 2 24−46𝑖−21 𝑖 2 24−46𝑥+21 𝑥 2 24−46𝑖−21(−1) 45−46𝑖

You try it! (5+3𝑖)(2−7𝑖) 4−5𝑖 2 31−29𝑖 −9−40𝑖 (7−𝑖 2 )(5+𝑖) Multiply each complex number. (5+3𝑖)(2−7𝑖) 4−5𝑖 2 31−29𝑖 −9−40𝑖 (7−𝑖 2 )(5+𝑖) 35+7𝑖−5𝑖 2 + 2 35+ 2 +7𝑖−5𝑖 2 𝑜𝑟 35+ 2 + 7−5 2 𝑖

Division with Complex Numbers…

Division with Complex Numbers…

Practice finding conjugates… The conjugate of a binomial is the result of reversing the sign between the two terms. Example: (a+bi) and (a-bi) are conjugates Note: Complex conjugates have the same REAL parts, but OPPOSITE imaginary parts Find the conjugates for each of the following. Then multiply them together. (5+2𝑖) 29 (1− 6 ) −5 5−2𝑖 ∗____________=______________ 1+ 6 ∗____________=______________ 3+4𝑖 ∗____________=______________ (3−4𝑖) 25 4+𝑖 5 4−𝑖 5 ∗____________=______________ 21 36 −6𝑖 ∗____________=______________ (6𝑖)

Dividing by Complex Numbers….

You try it! 3+4𝑖 2−𝑖 15 −𝑖 9+4𝑖 1+𝑖 2+11𝑖 5 15𝑖 13−5𝑖 2 2 5 + 11 5 𝑖 or or 2 5 + 11 5 𝑖 13 2 − 5 2 𝑖

Modeling Alternating Current with Ohm’s Law Complex numbers are used to describe current (I), voltage (E), and impedance (Z). Impedance is the opposition to current. The relation between these three is demonstrated by the formula for Ohm’s Law, E = (I)(Z) 𝐸= 5+7𝑖 6+4𝑖 𝐸=2+62𝑖 Ex 1) Given I = (5+7i) and Z = (6+4i), find E Ex 2) Given I = (10+4i) and E = (88+128i), find Z 𝑍= 𝐸 𝐼 = 88+128𝑖 10+4𝑖 𝑍=12+8𝑖