Rational, Irrational, Complex, Transcendental

1. From the Bible, I Kings 7:23-26

He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and ﬁve cubits high. It took a line of thirty cubits to measure around it.

What is the biblical value of π?

1 2. Rational or Irrational? 1 √ √ √ √ 5 5 22 √ √ √ √ 1, , 2, e, 9, 10, 3 4, , , π, , 3 8, 2 + 7, 2 + 7, 3 π 91823218223 7 √ √ 3 2 √ 4 − 91 2 − 3 0 1 , √ , √ , 1.44, 983711 + 2 2 2

3. Rational or irrational? √ 2, π, e, π + e

2 a 4. If possible, write the following decimals as a rational number , where a and b are integers. b (a) 1.5 (b) 1.63 (c) 1.333333 ... = 1.3 (d) 1.555555 ... = 1.5 (e) 0.101010 ... = 0.10 (f) 0.252525 ... = 0.25 (g) π = 3.1415926 ... (h) 0.1234567891011121314 ...

What is the relationship between decimals, rational numbers and irrational numbers?

3 5. The Real Number Game We specify a mathematical world, one of:

Natural Numbers, Integers, Rational Numbers, Irrational Numbers, Real Numbers

Then, we roll the dice and we form as many numbers of the given category as possible. Here are some “rules”:

Numbers can be put together: 9 2 would represent the number 92. √ √ Roots are represented with indexes: 2 9 would represent the square root of 9: 9. √ 3 8 is the cube root of 8 (or 2). Parenthesis can be added anywhere needed. Exponents are represented with ∗

4 6. Practice the following. Write you answer in “standard” form for a complex number (a + bi).

(a) (1 + 13i)(4 − 7i) (b) (1 + i)4 (c) (1 + 2i)4 1 (d) i 1 (e) 1 + i 1 − i (f) 1 + i Do you feel like you can comfortably multiply any two complex numbers? What is the general formula for multiplying (a + bi)(c + di)? How about dividing two complex numbers? What is the general formula for dividing: a + bi c + di

5 7. Remember our “Mathematical Worlds” that we were investigating. We had Integer, Rational Num- bers, Real Numbers, Complex Numbers and Gaussian Integers.

(a) Describe the diﬀerences between these worlds. (b) Find 5 rational numbers that are not integers. (c) Find 5 real numbers that are not rational. (d) Find 5 complex numbers that are not real numbers. (e) Find 5 complex numbers that are real numbers. (f) Find 5 gaussian integers that are not integers.

8. Property Hunt: Multiplicative Inverses (This is much more diﬃcult than the previous problems.)

(a) What is a multiplicative inverse? (b) Do additive multiplicative inverses exist in our mathematical worlds? Explore this for each mathematical world.

6 9. Property Hunt: Factoring Numbers This is more diﬃcult and we need to understand what we mean by factoring. Certainly we know that we factor something if we write the starting object as a product of two (or more) other objects. Discuss the issues involved in the following examples of these products:

(a) 6 = 2 × 3 (b) 48 = 2 × 2 × 12 (c) 48 = 2 × 2 × 2 × 3 (d) 6 = 1 × 6 (e) 6 = (−1) × (−6) (f) 6 = (−2) × (−3) (g) −6 = (−1) × (6) (h) 7 = 1 × 7 7 (i) 7 = 3 × 3 2 35 (j) 7 = 5 × 2

10. Property Hunt: Factoring Integers

(a) What does it mean to factor an integer? (b) What does it mean to factor an integer? (c) What does it mean to completely factor an integer?

7 11. Property Hunt: Factoring Polynomials Big picture hint: I am trying to have you look at familiar properties of numbers and generalize them to objects that you probably are not as comfortable with. Hopefully, this will help us all understand the mathematical worlds that we work with every day a bit better. Check to make sure the multiplications/factorizations below are correct.

(a) x2 − 4 = (x − 2)(x + 2) √ √ (b) x2 − 3 = (x − 3)(x + 3) (c) x2 + 4 = (x − 2i)(x + 2i) √ √ (d) x2 + 3 = (x − 3 i)(x + 3 i)

I am sure you like some of the above factorizations and you don’t like others (maybe you hate them all?). What is it about the factorizations you don’t like? (Or, if you love them all, what are the diﬀerences between the factorizations?)

12. What is a prime? We usually talk about prime numbers living in integer land. For the other mathematical worlds, we might call the analog to prime numbers irreducible. For each of our mathematical worlds:

(a) Write down several irreducibles. (So, when you are working with integers, write down several primes.) 2 (b) In integer land, is 7 a prime number? How about −7? How about 3 ? How about in rational land? (c) Explain what it means to be irreducible or prime.

8 13. Size of numbers In each of our mathematical worlds, there is a list of number living in that world. Order them from smallest to biggest.

Integers: 100, −2, 8, 32, 94, −100 3 902 3 Rationals: − 1, , , 98, − 8 32√ 8 √ π 2 1 Reals: 0, 2, π, , 100, , −π 6 2 i Complex: 0, 4, 4i, −4i, 1 + i, 100 − i, , π + i 9 Gaussian: 0, 4, −4, 4i, 1 + i, 1 − i, 2 + i, 1 − 2i Polynomialss: 0, 4, x, x2 − 4, 3x2 − 8x + 2, x − 1

14. Plotting Complex Numbers (and Gaussian Integers) Given a complex number, a + bi, we can plot it on the plane as (a, b). Plot the following:

(a) 2 − 3i (b) 1 + i (c) 4 (d) 5i (e) −5i (f) 5 (g) 100 + 13i

Can you use this idea to sort of order the complex numbers in Problem 13?