University:
Johns Hopkins University
Course:
AS.110.601 | Algebra I
Academic year:

2024
Views:

298

Pages:

4
Author:

Cristian Santos

Question 1 What is the absolute value of 0 2 2 0 ? Answer 1 Absolute value measures the distance from zero. 0 2 2 0 simplifies to 0 0. So, ∣ 0 ∣ ∣0∣ is just 0 0. Answer 2 Absolute value is defined as the distance of a number from 0 0. 0 0 divided by anything (except zero) is zero. The absolute value of 0 0 is 0 0. Question 2 How do you find the principal square root of 8 8? Answer 1 The principal square root of 8 8 is the positive solution to 𝑥 2 = 8 x 2 =8. The square root of 8 8 can be simplified: 8 = 4 ⋅ 2 = 4 ⋅ 2 = 2 2 . 8 = 4⋅2 = 4 ⋅ 2 =2 2 . Answer 2 To approximate 8 8 : Start with 3 3 because 3 2 = 9 3 2 =9 (close to 8). 8 ÷ 3 = 2.6667. 8÷3=2.6667. Average 3 3 and 2.6667 2.6667: 3 + 2.6667 2 = 2.8334 ( rounded ) . 2 3+2.6667 =2.8334(rounded). Repeat with 2.8334 2.8334: 8 ÷ 2.8334 = 2.8235 , average = 2.8284. 8÷2.8334=2.8235,average=2.8284. Final result: 8 ≈ 2.8284 8 ≈2.8284. Question 3 Is 12 + 31 12 + 31 irrational? Answer 1 The theory of field extensions shows that 𝑄 ( 12 ) Q( 12 ) and 𝑄 ( 31 ) Q( 31 ) are not equal to 𝑄 Q (the rational numbers). If 12 + 31 12 + 31 were rational, then 12 12 and 31 31 would also need to be rational, but they are not. Thus, 12 + 31 12 + 31 is irrational. Answer 2 Algebraic integers like 12 12 and 31 31 are closed under addition. Since 12 + 31 12 + 31 does not simplify to an integer, it must be irrational. Also, notice: 2 < 12 + 31 < 3. 2< 12 + 31 <3. Thus, 12 + 31 12 + 31 is irrational. Question 4 True or False: If 𝑥 2 − 5 𝑥 = 𝑦 2 − 5 𝑦 x 2 −5x=y 2 −5y and 𝑥 ≠ 𝑦 x  =y, then 𝑥 + 𝑦 = 5 x+y=5. 𝜋 π can be expressed as a repeating decimal. Dividing an irrational number by a nonzero integer gives an irrational result. Answer True: Rearrange 𝑥 2 − 5 𝑥 = 𝑦 2 − 5 𝑦 x 2 −5x=y 2 −5y: 𝑥 2 − 𝑦 2 = 5 𝑥 − 5 𝑦 . x 2 −y 2 =5x−5y. Factor both sides: ( 𝑥 − 𝑦 ) ( 𝑥 + 𝑦 ) = 5 ( 𝑥 − 𝑦 ) . (x−y)(x+y)=5(x−y). Divide by 𝑥 − 𝑦 x−y (since 𝑥 ≠ 𝑦 x  =y): 𝑥 + 𝑦 = 5. x+y=5. False: 𝜋 π is irrational. It cannot be expressed as a terminating or repeating decimal. True: Proof by contradiction: Assume dividing 𝑥 x (irrational) by 𝑎 a (integer, 𝑎 ≠ 0 a  =0) gives a rational result. Then 𝑥 / 𝑎 x/a is rational, which contradicts 𝑥 x being irrational. Question 5 (a) Is the sum of a rational number 𝑟 r and an irrational number 𝑡 t irrational? (b) Is the product of a rational number 𝑟 r and an irrational number 𝑡 t irrational? Answer (a) Yes, the sum is irrational. Proof by contradiction: Assume 𝑟 + 𝑡 r+t is rational. Then 𝑡 = ( 𝑟 + 𝑡 ) − 𝑟 t=(r+t)−r, which would make 𝑡 t rational. This contradicts 𝑡 t being irrational. (b) Yes, the product is irrational (if 𝑟 ≠ 0 r  =0). Proof by contradiction: Assume 𝑟 ⋅ 𝑡 r⋅t is rational. Then 𝑡 = rational 𝑟 t= r rational , which makes 𝑡 t rational. This contradicts 𝑡 t being irrational. Thus, both statements are true.