π― Learning Objectives
- Define and distinguish between sets β, π, and β€.
- Understand the Closure Property under basic operations.
- Perform arithmetic involving negative integers and absolute values.
- Apply Euclidβs Division Lemma to find quotients and remainders.
π Detailed Educational Notes
A. Defining the Sets
- β (Natural Numbers): The set used for counting.
β = {1, 2, 3, 4, ...}
- π (Whole Numbers): Natural numbers plus zero.
π = {0, 1, 2, 3, ...}
- β€ (Integers): Positive and negative whole numbers.
β€ = {..., -2, -1, 0, 1, 2, ...}
Relationship: All natural numbers are whole numbers, and all whole numbers are integers (β β π β β€).
B. Absolute Value |x|
This represents the distance from zero. Because it is a distance, the result is never negative.
| -5 | = 5 and | 5 | = 5
C. Euclidβs Division Lemma
For any positive integers a and b, there exist unique integers q (quotient) and r (remainder) such that:
a = bq + r, where 0 β€ r < b
π‘ Worked Examples
1. Integer Arithmetic: Simplify -10 - (-4).
Step: Subtracting a negative is adding a positive: -10 + 4.
Result: -6
2. Rules of Signs: Evaluate (-2) Γ (-3) Γ (-4).
Step: 6 Γ (-4) = -24 (Odd number of negatives results in negative).
Result: -24
3. Euclid's Lemma: If a = 14 and b = 3, find q and r.
Step: 14 Γ· 3 = 4 with remainder 2.
Result: q = 4, r = 2
βοΈ Practice Exercises (20 Questions)
1. Which set includes zero but no negative numbers?
Whole Numbers (π).
2. Solve: | -20 | + | 5 |
25. (20 + 5).
3. True or False: Every integer is a natural number.
False. -5 is an integer but not a natural number.
4. Simplify: -8 + (-12)
-20.
5. What is the smallest positive integer?
1. (Zero is neither positive nor negative).
6. Simplify: -36 Γ· (-9)
4. Negative Γ· Negative = Positive.
7. In a = bq + r, what is the maximum value of r if b = 4?
3. The remainder must be less than the divisor.
8. What is the value of (-1) to the power of 20?
1. (Even power makes it positive).
9. Compare: -5 ___ -10
> (Greater than). -5 is to the right of -10 on the number line.
10. What is the distance of -15 from zero?
15. (Distance is absolute value).
11. Solve: 0 - 7
-7.
12. Is the set β€ closed under division?
No. 5 Γ· 2 = 2.5, which is not an integer.
13. Additive inverse of 10?
-10.
14. Evaluate: (-2)Β³
-8. (-2 Γ -2 Γ -2).
15. Is 0 an even number?
Yes. It is divisible by 2 without a remainder.
16. Multiply: 15 Γ (-1)
-15.
17. Define | x | if x is negative.
-x (This makes the negative number positive).
18. Simplify: -100 + 100
0.
19. In a = bq + r, if a=10 and b=3, what is q?
3. (10 = 3Γ3 + 1).
20. What is 5 - (-5)?
10.