Rational Approximations and Irrational Numbers

The video tutorial explains rational approximations, focusing on how rational numbers can closely estimate irrational numbers. It uses examples like pi and square roots to demonstrate the concept. The tutorial provides step-by-step calculations for approximating irrational numbers such as pi, 2 pi, √6, and √42 with rational numbers. It concludes with homework instructions for students to practice finding rational approximations using given methods.

What is a rational approximation?

• A number that cannot be expressed as a fraction

• An irrational number that approximates a rational number

• A rational number that is a close estimate of an irrational number

• A rational number that exactly equals an irrational number

Why is pi considered an irrational number?

• It can be expressed as a fraction

• It is a whole number

• It goes on forever without repeating

• It has a finite decimal expansion

What is the rational approximation of pi commonly used in calculations?

• 3.15

• 3.13

• 3.14

• 3.16

How do you approximate 2 pi using a rational number?

• Multiply 3.14 by 3

• Multiply 3.14 by 2

• Divide 3.14 by 2

• Add 3.14 to 2

What is the rational approximation of pi/2?

• 1.56

• 1.59

• 1.57

• 1.58

Why is the square root of 6 considered irrational?

• 6 is a rational number

• 6 is an integer

• 6 is not a perfect square

• 6 is a perfect square

What is the rational approximation of the square root of 6?

• 2.6

• 2.3

• 2.4

• 2.5

Between which two perfect squares does the square root of 42 lie?

• 25 and 36

• 36 and 49

• 49 and 64

• 16 and 25

What is the rational approximation of the square root of 42?

• 6.6

• 6.4

• 6.5

• 6.3

How do you determine which perfect square a number is closer to?

• By guessing

• By finding the exact square root

• By comparing the number to the square roots

• By using a calculator