Imaginary Complex Numbers Daily Check For each equation
Imaginary & Complex Numbers
Daily Check For each equation find the discriminant and the number of solutions.
Launched Object h(t) = -16 t 2 + 64 t + 80 a) How many seconds until 2 sec. the max height is reached? 144 ft. b) What will be the max height? c) How many seconds 5 sec. until the object hits the ground?
Today’s Question: How do we take the square root of negative numbers?
2 2 2
i
-In the set of real numbers, negative numbers do not have square roots. -Imaginary numbers were invented so that negative numbers would have square roots and certain equations would have solutions. -These numbers were devised using an imaginary unit named i.
-The imaginary numbers consist of all numbers bi, where b is a real number and i is the imaginary unit, with the property that i² = -1. -The first four powers of i establish an important pattern and should be memorized. Powers of i
Examples of how we use
Examples of how we use
Complex Numbers A complex number has a real part and imaginary part. Standard form is: Real part Example: 5+4 i Imaginary part
The Complex Plane Real Axis Imaginary Axis
Graphing in the complex plane
Adding and Subtracting (add or subtract the real parts, then add or subtract the imaginary parts) Ex:
Absolute Value of a Complex Number The distance the complex number is from the origin on the complex plane. If you have a complex number the absolute value can be found using:
Graphing in the complex plane 5 2
Examples 1. 2. Which of these 2 complex numbers is closest to the origin? -2+5 i
Try These!!! 1. 2. Which of these 2 complex numbers is closest to the origin? 3 i
Powers of i 1. ) Find i 23 2. ) Find i 2006 3. ) Find i 37 4. ) Find i 828
Simplify. 3. ) 4. ) 5. ) -Express these numbers in terms of i.
You try… 6. 7. 8.
To multiply imaginary numbers or an imaginary number by a real number, it is important first to express the imaginary numbers in terms of i.
Multiplying 9. 10. 11.
Add or Subtract 12. 13. 14.
Multiplying & Dividing Complex Numbers Part of 7. 9 in your book
REMEMBER: i² = -1 Multiply 1) 2)
You try… 3) 4)
Multiply 5)
You try… 6)
You try… 7)
Conjugate -The conjugate of a + bi is a – bi -The conjugate of a – bi is a + bi
Find the conjugate of each number… 8) 9) 10) 11)
Divide… 12)
You try… 13)
Ex: Solve a= b= c= x 2+ 1 s 6 x +10 = 0 t 2 nd
- Slides: 39