[Func-Num] 6.4 Irrational Numbers and Decimal Representation

Date: 2023.01.23 Updated: 2023.01.23

Categories: Func-Num

📋 This is my note-taking from what I learned in the class “Math175-002 Functions & Number Systems”

Objectives

Illustrate how irrational numbers differ from rational numbers in their decimal representations

Follow the proof that sqrt 2 is an irrational number

Use a calculator to find square roots

Apply the product and quotient rules for square roots

Rationalize a denominator

Explain the relevance of the irrational numbers such as the Golden Ratio, the ratio of the circumference of a circle to its diameter, and a constant approximately equal to 2.71828

Definition and Basic Concepts

Every rational number has a decimal form that terminates or repeats, and every repeating or terminating decimal represents a rational number
However, there are numbers that do not terminate and do not repeat → We call this “Irrational Numbers”

Irrational Numbers

Irrational numbers = { \({x \over x}\) is a number represented by a nonrepeating, non-terminating decimal}
→ As the name implies, an irrational number cannot be represented as a quotient of integers

E.g. root, the Golden Ratio, the ratio of the circumference of a circle to its diameter, and a constant approximately equal to 2.71828 …

Irrationality of root 2 and Proof by Contradiction

The proof that root 2 is irrational is a classic example of a proof by contradiction. We begin by assuming that root 2 is rational, which leads to a contradiction, or absurdity. The method is also called reductio ad absurdum (Latin for “reduce to the absurd”). In order to understand the proof, we consider three preliminary facts:

When a rational number is written in lowest terms, the greatest common factor of the numerator and denominator is 1.
If an integer is even, then it has 2 as a factor and may be written in the form 2k, where k is an integer.
If a perfect square is even, then its square root is even.

contradiction

Operations with Square Roots

Recall that \(\sqrt{a}\) , for a >= 0, is the nonnegative number whose square is a
That is, {(\(\sqrt{a}\))}^2 = a

Examples of square roots that are irrational: \(\sqrt{2}\), \(\sqrt{3}\), and \(\sqrt{13}\)

Examples of square roots that are rational: \(\sqrt{4}\) = 2, \(\sqrt{36}\) = 6, and \(\sqrt{100}\) = 10

If n is a positive integer that is not the square of an integer, then 2n is an irrational number.
A calculator with a square root key can give approximations of square roots of numbers that are not perfect squares.
We use the ≈ symbol to indicate “is approximately equal to.”
Sometimes, for convenience, the = symbol is used even if the statement is actually one of approximation, such as pi = 3.14.

Product Rule for Square Roots

For nonnegative real numbers a and b, the following holds true

\(\sqrt{a}\) * \(\sqrt{b}\) = \(\sqrt{a*b}\)

Conditions for a Simplified Square Root Radical

Just as every rational number \({a \over b}\) can be written in lowest terms (by using the fundamental property of rational numbers), every square root radical has a simplified form

A square root radical is in simplified form if the following three conditions are met:

The number under the radical (radicand) has no factor (except 1) that is a perfect square

The radicand has no fractions

No denominator contains a radical

Quotient Rule for Square Roots

For nonnegative real numbers a and positive real numbers b, the following holds true

\({\sqrt{a} \over \sqrt{b}}\) = \({\sqrt{\) {a \over b} \(}}\)

Adding and Subtracting Square Root Radicals

Square root radicals may be combined, however, if they have the same radicand. Such radicals are called like radicals. We add (and subtract) like radicals using the distributive property.
Like radicals may be added or subtracted by adding or subtracting their coef-ficients (the numbers by which they are multiplied) and keeping the same radical.

Exercise

Section 6.4: 7 ~ 19 (odd)

Identify each number as rational or irrational

7: 4/9 → rational
9: root 10 → irrational
11: 1.618 → rational
13: 0.41 (repeating 41) → rational
15: Pi → irrational
17: 3.14159 → rational
19: 0.878778777877778… → irrational

Section 6.4: 23 ~ 29 (odd)

Use a calculator to find a rational decimal approximation for each irrational number

23: root 39 → 6.244 997 998 4
25: root 15.1 → 3.885 871 845 55
27: root 884 → 29.732 137 494 6
29: root 9/8 → 1.060 660 171 78

Section 6.4: 49 ~ 67 (odd)

Use the methods of Examples 3 and 4 to simplify each expression. Then, use a calculator to approximate both the given expression and the simplified expression. (Both should be the same.)

49: root 50 → 5(root 2) → 7.071 067 811 87
51: root 75 → 5(root 3) → 8.660 254 037 84
53: root 288 → 12(root 2) → 16.970 562 748 5
55: 5/(root 6) → 5/6(root 6) → 2.041 241 452 32
57: root (7/4) → (root 7)/2 → 1.322 875 655 53
59: root (7/3) → (root 21)/3 → 1.527 525 231 65

Use the method of Example 5 to perform the indicated operations

61: root 17 + 2(root 17) → 3(root 17)
63: 5(root 7) - root 7 → 4(root 7)
65: 3(root 18) + root 2 → 10(root 2)
67: -(root 12) + root 75 → 3(root 3)

Seyeon