Name:_______________________________

PreCalculus MidWinter Worksheet A 1) Consider the set of odd numbers, give a few examples of operations that are closed for the set of odd numbers, and give some examples of operations that are open for the set of odd numbers.

2) Repeat for even numbers.

3) Repeat for pure imaginary numbers (square roots of negative numbers).

4) Repeat for irrational numbers.

5) Even functions have the property f (−x)= f (x) , these functions are symmetric by reflection 2 across the y-axis. Examples are y= x and y=cos( x) . Odd functions have the property f (− x)=− f ( x) , these functions are symmetric by reflection through the origin. Examples are 3 y= x and y=sin ( x) . Consider basic operations: adding, subtracting, multiplying, dividing. The set of even functions is closed under which of these operations? The set of odd functions is closed under which of these operations?

Name:_______________________________

PreCalculus MidWinter Worksheet B 1) What does it mean for two integers to be relatively prime?

2) What does it mean for two polynomials to be relatively prime?

3) What is the fundamental theorem of arithmetic?

4) What is the fundamental theorem of algebra?

5) Give four examples of transcendental numbers.

6) What makes a number transcendental?

Name:_______________________________

PreCalculus MidWinter Worksheet C 1) Plotting points on the complex plane: On a sheet of graph-paper, make the x-axis the real numbers, make the y-axis the imaginary numbers. Plot the complex number 3 + 2i on the complex plane. 2) How far is the point 3 + 2i from the origin?

3) How far does 3 + 2i extend in the x-direction (real axis)? 4) How far does 3 + 2i extend in the y-direction (imaginary axis)? 5) What angle does a line connecting the origin to the point 3 + 2i make with the x-axis? 6) Write 3 + 2i in "trigonometric form" (otherwise known as "polar form")

7) Use De Moivre's Theorem to find the two square-roots of 3 + 2i. Check your answers by squaring each of your answers.

8) Plot your square-roots on the same graph as 3 + 2i. Bonus: The number a + bi can be plotted on the complex plane. Where would it's complex conjugate be plotted? Do they make the same angle with the x-axis? Write the general form for a complex number in polar form. Now write the conjugate of that number in polar form. Multiply the two and see what happens.

Name:_______________________________

PreCalculus MidWinter Worksheet D 1) Write out the first five terms of the Taylor Series approximation of

2) Write out the Taylor Series for

y=e

x

y=e

x

.

in "sigma notation." (S)

3) Write out the first five terms of the Taylor Series approximation of

y=sin x .

4) Write out the Taylor Series for i y=sin x n "sigma notation." (S)

5) Write out the first five terms of the Taylor Series approximation of . y= cos x

6) Write out the Taylor Series for i y= cos x n "sigma notation." (S)

7) Write out Euler's Formula.

8) Write the complex number 4 + 3i in polar form. (Hint: the angle should be in radians!)

9) Write the complex number 4 + 3i as a complex exponential. (Hint: the angle should be in radians!)

10) Write out the complex number 2 e

0.6i

11) Write the complex conjugate of 2 e

0.6i

in rectangular form.

as a complex exponential.

Name:_______________________________

PreCalculus MidWinter Worksheet E ⃗b a ⃗

1) Vector's ⃗ a +⃗b . a and ⃗b are given in the diagram. Make a diagram showing how to add: ⃗ 2) Make a diagram showing how to subtract: a⃗ −⃗b . 3) Make a diagram of 2 ⃗ b−3 ⃗ a . 4)

a +⃗b and 5 ⃗b−2 ⃗ a . a =〈3,1〉 and ⃗b=〈−1,2〉 . Find ⃗ ⃗

5) Find the lengths and angles of all four vectors in the previous problem. 6)

a −⃗b and 3 ⃗a+4 ⃗b a =2 ̂i +5 ̂j−3 k̂ and ⃗b=−3 ̂i + ̂j+4 k̂ . Find ⃗ ⃗

Bonus: write all four vectors in the last problem in spherical coordinates.

Name:_______________________________

PreCalculus MidWinter Worksheet F d =〈−3,4,1〉 , Use the following vectors in this worksheet: ⃗ a =〈3,1〉 , ⃗b=〈−1,2〉 , ⃗c=〈0,5〉 , ⃗ ̂ ⃗ ̂ ̂ ̂ ̂ ⃗e =〈1,1,1〉 , f =〈−4,0,1,2〉 , ⃗g =〈1,2,−1,3〉 , m ⃗ =2 i+5 j−3 k , ⃗n =−3 i + j+4 k̂

Find the following scalar products: 1)

a⋅⃗b ⃗

2)

a⋅⃗c ⃗

3)

⃗b⋅⃗c

4)

a⋅⃗ d ⃗

5)

⃗ d⋅⃗e

6)

⃗f⋅⃗ g

7) m ⃗ ⋅⃗n Find the following vector products: 8)

⃗ d × ⃗e

9)

d ⃗e × ⃗

10) m ⃗ ×⃗ n

Name:_______________________________

PreCalculus MidWinter Worksheet G d =〈−3,4,1〉 , Use the following vectors in this worksheet: ⃗ a =〈3,1〉 , ⃗b=〈−1,2〉 , ⃗c=〈0,5〉 , ⃗ ̂ ⃗ ̂ ̂ ̂ ̂ ⃗e =〈1,1,1〉 , f =〈−4,0,1,2〉 , ⃗g =〈1,2,−1,3〉 , m ⃗ =2 i+5 j−3 k , ⃗n =−3 i + j+4 k̂

Find the angle between the following vectors: 1) ⃗ a and ⃗b 2) m ⃗ and ⃗n 3) ⃗f and ⃗g 4) ⃗c and ⃗g 7) Find a vector that is perpendicular to both ⃗ d and ⃗e . 6) Find the angle between the two planes:

2x+3y− z=10 and − x+2y+2z=3

Name:_______________________________

PreCalculus MidWinter Worksheet H (Many equations taken from http://en.wikipedia.org/wiki/Spinors_in_three_dimensions) The spinor is an important mathematical object in Quantum Mechanics. The wikipedia page can be pretty scary, but you can understand more of this than you realize. d =〈−3,4,1〉 and ⃗e =〈1,1,1〉 . Convert them into spinors (D and E) by 1) Consider the vectors ⃗ the formula:

d and ⃗e . 2) Calculate the lengths of ⃗

3) Calculate the determinants of D and E. (hint: 2x2 is easy: (upper-left times lower right) minus (upper-right times lower left)). How are these related to your answers in 2? 4) Multiply D times itself to get D2. What do you notice? 5) Verify that this formula works for our examples:

6) Verify that this formula works for our examples: Z is the matrix you get by converting d⃗ × ⃗e into a spinor. 7) The spinors form a group that is closed. So, DE (E multiplied by D) should also be a spinor. Show that it follows the same rules as D and E in problems 4, 5 and 6.