Presentation on theme: "COMPLEX NUMBERS Objectives"— Presentation transcript:
1COMPLEX NUMBERS Objectives Use the imaginary unit i to write complex numbers.Add, subtract, and multiply complex numbers.Use complex conjugates to write the quotient of two complex numbers in standard form.Perform operations with square roots of negative numbersSolve quadratic equations with complex imaginary solutions
2Complex Numbers C R Real Numbers R Irrational Numbers Q -bar Integers ZImaginaryNumbers iWhole numbers WNaturalNumbers NRational Numbers Q
3What is an imaginary number? It is a tool to solve an equation and was invented to solve quadratic equations of the form 𝒂 𝒙 𝟐 +𝒃𝒙+𝒄..It has been used to solve equations for the last 200 years or so.“Imaginary” is just a name, imaginary do indeed exist; they are numbers.
4The Imaginary Unit i 𝒊= −𝟏 𝒊 𝟐 =−𝟏 Previously, when we encountered square roots of negative numbers in solving equations, we would say “no real solution” or “not a real number”.𝒊= −𝟏𝒊 𝟐 =−𝟏
5Complex Numbers & Imaginary Numbers a + bi represents the set of complex numbers, where a and b are real numbers and i is the imaginary part.a + bi is the standard form of a complex number. The real number a is written first, followed by a real number b multiplied by i. The imaginary unit i always follows the real number b, unless b is a radical. Example: 2+3𝑖If b is a radical, then write i before the radical.𝑖 2
6Adding and Subtracting Complex Numbers (5 − 11i) + (7 + 4i)Simplify and treat the i like a variable.= 5 − 11i i= (5 + 7) + (− 11i + 4i)= 12 − 7iStandard form
7Adding and Subtracting Complex Numbers (− 5 + i) − (− 11 − 6i)= − 5 + i i= − i + 6i= i
14Complex ConjugatesThe complex conjugate of the number a + bi is a − bi.Example: the complex conjugate of 𝟐+𝟒𝒊 is 𝟐−𝟒𝒊The complex conjugate of the number a − bi is a + bi.Example: the complex conjugate of 3−𝟓𝒊 is 3+5𝒊
15Complex ConjugatesWhen we multiply the complex conjugates together, we get a real number.(a + bi) (a − bi) = a² + b²Example: 2+3𝑖 2−3𝑖 =4−9 𝑖 24−9 𝑖 2 =4−9 −1=4+9=13
16Complex ConjugatesWhen we multiply the complex conjugates together, we get a real number.(a − bi) (a + bi) = a² + b²Example: 2−5𝑖 2+5𝑖 =4−25 𝑖 24−25 𝑖 2 =4−25(−1)=4+25=29
17Using Complex Conjugates to Divide Complex Numbers Divide and express the result in standard form:7 + 4i2 − 5iThe complex conjugate of the denominator is 2 + 5i.Multiply both the numerator and the denominator by the complex conjugate.