[Func-Num] 6.1 Real Numbers, Order, and Absolute Value

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

Objectives

1. A number on a number line
2. A number as positive, negative, or zero
3. A number as belonging to one or more sets of numbers
4. Given 2 numbers A and B, determine whether A = B, A < B, or A > B
5. Given a number A, determine its additive inverse and absolute value
6. Signed numbers in tables of economic and occupations data

Sets of Real Numbers

1. Natural Numbers

{1, 2, 3, 4, …} is the set of natural numbers
→ Numbers which we can count

2. Whole Numbers

{0, 1, 2, 3, 4, …} is the set of whole numbers
→ Including 0

3. Number Line

Number Line Image Link

• Origin: “0”
• The right of “0”: Positive numbers {1, 2, 3, …}
• The left of “0”: Negative numbers {-1, -2, -3, …}

• Signed numbers: Positive numbers and Negative numbers
• “0” itself: Neither positive nor negative
• Practical applications of negative numbers: Temperatures, Altitude below sea level

Signed and Unsigned Numbers
Signed vs Unsigned

Difference between Signed and Unsigned in MySQL

• Signed value - Variables with signed numerals can store 0, positive, and negative numbers.
• Unsigned value - Variables with unsigned numerals can store only 0 and positive numbers.

4. Integers

{…, -2, -1, 0, 1, 2, …} is the set of integers
→ The natural numbers, negative numbers, and zero make up the set of integers.

• An integer, such as 2, is also a rational number → 2 = 2/1

5. Rational Numbers

{x/x is a quotient of two integers, with denominator not equal to 0} is the set of rational numbers

• The set symbolism used in the definition of rational numbers: {x/x has a certain property} → Set-builder notation
• Set-builder notation
• Set-builder notation(KR)

6. Irrational Numbers

Irrational numbers Image Link

{x/x is a number on the number line that is not rational} is the set of irrational numbers
→ Cannot be expressed as a quotient of integers (a fraction of integer over integer)

Pi π
Pi π Image Link

7. Real Numbers

{x/x is a number that can be represented by a point on a number line} is the set of real numbers
→ Can be written as decimal numbers

• Rational numbers: Can be written as a decimal that will either come to end(terminate) or repeat in a fixed “block” of digits
  • Terminating decimals: 2/5 = 0.4
  • Repeating decimals: 3/11 = 0.2727272727……
• Irrational numbers: Neither terminates nor repeats

Real number representation 1 Image Link
Real number representation 2 Image Link

Order in the Real Numbers

1. Law of Trichotomy

For any real numbers A and B, one and only one of the following holds true
{A = B, A < B, A > B}

Inequality (Mathematics)

2. Additive Inverse and Absolute Value

Additive Inverse

For any non-zero real number x, there is exactly one number on the number line the same distance from 0 as x but on the opposite side of 0
→ {3, -3} is called additive inverses, negatives, or opposites of each other

• The additive inverse of the number 0 is 0 itself
• Double negative rule: For any number x, the following holds true → {-(-x) = x}

Absolute Value

Absolute Value Image Link

• The absolute value of a real number can be defined as the undirected distance between 0 and the number on a number line
• Read “the absolute value of x”
• The absolute value of a number is never negative
• The absolute value of 0 → 0

3. Applications of Real Numbers

• Interpreting Signed Numbers in a Table: Consumer Price Index (CPI)
• Comparing Occupational Rates of Change

Exercise

Section 6.1 → No. 1 ~ 11

Concept Check. Give a number that satisfies the given condition.

1. An integer between 4.5 and 5.5 → 5
2. A rational number between 2.8 and 2.9 → 2.85
3. A whole number that is not positive and is less than 1 → 0
4. A whole number greater than 4.5 → 5
5. An irrational number that is between Root 17 and Root 19 → Root 18
6. A real number that is neither negative nor positive → 0

Concept Check. Write true or false for each statement.

1. Every natural number is positive. → T
2. Every whole number is positive. → F
3. Every integer is a rational number. → T
4. Every rational number is a real number. → T

List all numbers from each set that are (a)natural numbers; (b)whole numbers; (c)integers; (d)rational numbers; (e)irrational numbers; (f)real numbers.

1. {-9, -Root 7, -1 and 1/4, -3/5, 0, Root 5, 3, 5.9, 7}
   • (a)natural numbers: 3, 7
   • (b)whole numbers: 0, 3, 7
   • (c)integers: 0, 3, 7, -9
   • (d)rational numbers: 0, 3, 7, -9, -1 and 1/4, -3/5, 5.9
   • (e)irrational numbers: -Root 7, Root 5
   • (f)real numbers: -9, -Root 7, -1 and 1/4, -3/5, 0, Root 5, 3, 5.9, 7

Section 6.1 → No. 31 ~ 39 (Part b)

Concept Check.

1. Match each expression in Column I with its value in Column II. Some choices in Column II may not be used.
   → Answer. (a)-A, (b)-A, (c)-B, (d)-B

Concept Check.

1. Fill in the blanks with the correct values:
   → Answer. The opposite of -2 is [2], while the absolute value of -2 is [2]. The additive inverse of -2 is [2], while the additive inverse of the absolute value of -2 is [-2].

Find (a) the additive inverse (or opposite) of each number and (b) the absolute value of each number.

1. -2 → (a)2, (b)2
2. -8 → (a)8, (b)8
3. 6 → (a)-6, (b)6
4. 11 → (a)-11, (b)11
5. 7 - 4 → (a)-3, (b)3
6. 8 - 3 → (a)-5, (b)5
7. 7 - 7 → (a)0, (b)0

Section 6.1 → No. 53 ~ 63 (odd)

Write true or false for each statement.

1. T
2. T
3. T
4. F
5. T
6. F

---

Back to Top

See other articles in Category Func-Num