5.9: imaginary + complex numbers -defining i -simplifying negative radicands -powers of i -solving...
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5.9: Imaginary + Complex Numbers
-Defining i
-Simplifying negative radicands
-Powers of i
-Solving equations
-Complex numbers
-Operations with complex #s
Imaginary units
• Consider the equation 2x2 + 2 = 0
• You end up with x2 = -1
• There is no real number that, when squared, equals -1
• We define the imaginary unit, i, to be the square root of -1… and i2 = -1
• We can then simplify negative radicands (if the index is even) in terms of i
Combining terms with negative radicands
• Recall that by definition, i equals the square root of negative 1 and i2 = -1
• When combining two or more terms with negative radicands, always rewrite each radical in terms of i first!!!
Higher powers of i
• i raised to ANY power equals either 1, -1, i or –i
• For this reason, your answer should NEVER contain i raised to a power
• To simplify, rewrite as i2 raised to a power, or as i * (i2 raised to a power)
• Ex: i14 = (i2)7 = (-1)7 = -1
• Ex. i29 = i * i28 = i*(i2)14 = i* (-1)14 = i*1 = i
Solving equations with squared term
• Isolate the squared term/expression first
• Then take the square root of each side!
• REMEMBER when you take the root yourself, stick the ± in front
• Then simplify the radical, using i if necessary
Solve
Answer:
Original equation
Subtract 20 from each side.
Divide each side by 5.
Take the square root of each side.
Complex Numbers
• A complex number is a number that can be written in the form a + bi , where a and b are real numbers
• That is, a complex number contains two parts, a real part (a) and an imaginary part (bi)
• Examples: 4 + 5i, 7 – 2i • Also: 4 (can be written as 4 + 0i)• Also: -3i (can be written as 0 – 3i)
Equality of complex numbers
• Two complex numbers a + bi and c + di are equal iff a = c and b = d
• If confused, set the coefficients of the I term equal to each other and solve for the variable
• Then you can set the “real” parts equal and solve
Find the values of x and y that make the equationtrue.
Set the real parts equal to each other and the imaginary parts equal to each other.
Real parts
Divide each side by 2.
Imaginary parts
Answer:
Operations with complex #s
• Adding/subtracting – just add/subtract the “real” components and the imaginary components
• Multiplying – distribute or use FOIL.. Just remember that i2 = -1
• Rationalizing (may need to use the COMPLEX CONJUGATE)
Application
• Complex #s are used with electricity.. Except they use j instead of i (the letter i is used elsewhere)
• E = I * Z, where E is the voltage, I is the current, and Z is the impedance
• Not that important to know.. Just an example of multiplying complex #s
Answer: The voltage is volts.
Electricity In an AC circuit, the voltage E, current I, and impedance Z are related by the formulaFind the voltage in a circuit with current 1 + 4 j ampsand impedance 3 – 6 j ohms.
Electricity formula
FOIL
Multiply.
Add.
Electricity In an AC circuit, the voltage E, current I, and impedance Z are related by the formula E = I • Z. Find the voltage in a circuit with current 1 – 3 j ampsand impedance 3 + 2 j ohms.
Answer: 9 – 7 j