Chapter: Number Systems
1. Introduction to Number Systems
Brief overview of different types of numbers we encounter in mathematics. Importance of understanding number systems for further mathematical studies.
2. Natural Numbers (N)
Definition: Counting numbers starting from 1 (1, 2, 3, ...).
Properties:
Closed under addition: If you add two natural numbers, you always get another natural number (e.g., 2 + 3 = 5).
Closed under multiplication: If you multiply two natural numbers, you always get another natural number (e.g., 2 × 3 = 6).
Not closed under subtraction: If you subtract two natural numbers, you might not get a natural number (e.g., 2 - 5 = -3).
Not closed under division: If you divide two natural numbers, you might not get a natural number (e.g., 5 ÷ 2 = 2.5).
Examples: Counting your fingers (10), the number of pages in a book, the number of students in a class.
Questions and Solutions:
1. Is 7 a natural number?
Solution: Yes, 7 is a counting number starting from 1, so it is a natural number.
2. Are natural numbers closed under addition? Give an example.
Solution: Yes, natural numbers are closed under addition. For example, 4 + 9 = 13, which is also a natural number.
3. Are natural numbers closed under subtraction? Give an example to support your answer.
Solution: No, natural numbers are not closed under subtraction. For example, 2 - 6 = -4, which is not a natural number.
4. Give two real-life examples where you use natural numbers.
Solution: Counting the number of chairs in a room; Counting the number of cars passing by on a road.
3. Whole Numbers (W)
Definition: Natural numbers including zero (0, 1, 2, 3, ...).
Relationship with natural numbers: All natural numbers are also whole numbers, but 0 is a whole number that is not a natural number.
Properties:
Closed under addition: (e.g., 0 + 5 = 5, 3 + 7 = 10)
Closed under multiplication: (e.g., 0 × 8 = 0, 4 × 6 = 24)
Not closed under subtraction: (e.g., 3 - 5 = -2)
Not closed under division: (e.g., 7 ÷ 0 is undefined, 5 ÷ 2 = 2.5)
Examples: The number of siblings you have (could be zero), the score in a game (could start at zero).
Questions and Solutions:
1. Is 0 a natural number? Is it a whole number?
Solution: 0 is not a natural number; 0 is a whole number.
2. Are whole numbers closed under multiplication? Give an example.
Solution: Yes, whole numbers are closed under multiplication. For example, 0 × 12 = 0; 5 × 8 = 40.
3. Give an example of subtracting two whole numbers where the result is not a whole number.
Solution: 4 - 9 = -5.
4. Can you always divide two whole numbers and get a whole number? Explain with an example.
Solution: No, for example, 7 ÷ 2 = 3.5, which is not a whole number; dividing by 0 is undefined.
4. Integers (Z)
Definition: Whole numbers and their negatives (... -3, -2, -1, 0, 1, 2, 3, ...).
Representation on the number line: Imagine a line with 0 in the middle. Positive integers are to the right, and negative integers are to the left, equally spaced.
Properties:
Closed under addition: (e.g., -3 + 5 = 2, -2 + (-4) = -6)
Closed under subtraction: (e.g., 3 - 7 = -4, -5 - 2 = -7, -1 - (-3) = 2)
Closed under multiplication: (e.g., -2 × 4 = -8, -3 × (-5) = 15)
Not closed under division: (e.g., 7 ÷ 3 is not an integer)
Absolute value of an integer: The absolute value of an integer is its distance from zero on the number line. It's always non-negative. We write it with two
vertical bars: $|-5| = 5$, $|3| = 3$, $|0| = 0$.
Examples: Temperature below zero (-5°C), owing someone money (-$10), altitude above and below sea level.
Questions and Solutions:
1. Is -8 an integer? Is it a whole number? Is it a natural number?
Solution: -8 is an integer; -8 is not a whole number; -8 is not a natural number.
2. Are integers closed under subtraction? Give an example.
Solution: Yes, integers are closed under subtraction. For example, 5 - (-3) = 8; 7 - (-2) = 9.
3. What is the absolute value of -12? What is the absolute value of 15?
Solution: $|-12| = 12$; $|15| = 15$.
4. Give two real-life examples where you might use negative integers.
Solution: Representing a loss in business (e.g., -$50); Describing depth below sea level (e.g., -200 meters).
5. Are integers closed under division? Explain with an example.
1. Introduction to Number Systems
Brief overview of different types of numbers we encounter in mathematics. Importance of understanding number systems for further mathematical studies.
2. Natural Numbers (N)
Definition: Counting numbers starting from 1 (1, 2, 3, ...).
Properties:
Closed under addition: If you add two natural numbers, you always get another natural number (e.g., 2 + 3 = 5).
Closed under multiplication: If you multiply two natural numbers, you always get another natural number (e.g., 2 × 3 = 6).
Not closed under subtraction: If you subtract two natural numbers, you might not get a natural number (e.g., 2 - 5 = -3).
Not closed under division: If you divide two natural numbers, you might not get a natural number (e.g., 5 ÷ 2 = 2.5).
Examples: Counting your fingers (10), the number of pages in a book, the number of students in a class.
Questions and Solutions:
1. Is 7 a natural number?
Solution: Yes, 7 is a counting number starting from 1, so it is a natural number.
2. Are natural numbers closed under addition? Give an example.
Solution: Yes, natural numbers are closed under addition. For example, 4 + 9 = 13, which is also a natural number.
3. Are natural numbers closed under subtraction? Give an example to support your answer.
Solution: No, natural numbers are not closed under subtraction. For example, 2 - 6 = -4, which is not a natural number.
4. Give two real-life examples where you use natural numbers.
Solution: Counting the number of chairs in a room; Counting the number of cars passing by on a road.
3. Whole Numbers (W)
Definition: Natural numbers including zero (0, 1, 2, 3, ...).
Relationship with natural numbers: All natural numbers are also whole numbers, but 0 is a whole number that is not a natural number.
Properties:
Closed under addition: (e.g., 0 + 5 = 5, 3 + 7 = 10)
Closed under multiplication: (e.g., 0 × 8 = 0, 4 × 6 = 24)
Not closed under subtraction: (e.g., 3 - 5 = -2)
Not closed under division: (e.g., 7 ÷ 0 is undefined, 5 ÷ 2 = 2.5)
Examples: The number of siblings you have (could be zero), the score in a game (could start at zero).
Questions and Solutions:
1. Is 0 a natural number? Is it a whole number?
Solution: 0 is not a natural number; 0 is a whole number.
2. Are whole numbers closed under multiplication? Give an example.
Solution: Yes, whole numbers are closed under multiplication. For example, 0 × 12 = 0; 5 × 8 = 40.
3. Give an example of subtracting two whole numbers where the result is not a whole number.
Solution: 4 - 9 = -5.
4. Can you always divide two whole numbers and get a whole number? Explain with an example.
Solution: No, for example, 7 ÷ 2 = 3.5, which is not a whole number; dividing by 0 is undefined.
4. Integers (Z)
Definition: Whole numbers and their negatives (... -3, -2, -1, 0, 1, 2, 3, ...).
Representation on the number line: Imagine a line with 0 in the middle. Positive integers are to the right, and negative integers are to the left, equally spaced.
Properties:
Closed under addition: (e.g., -3 + 5 = 2, -2 + (-4) = -6)
Closed under subtraction: (e.g., 3 - 7 = -4, -5 - 2 = -7, -1 - (-3) = 2)
Closed under multiplication: (e.g., -2 × 4 = -8, -3 × (-5) = 15)
Not closed under division: (e.g., 7 ÷ 3 is not an integer)
Absolute value of an integer: The absolute value of an integer is its distance from zero on the number line. It's always non-negative. We write it with two
vertical bars: $|-5| = 5$, $|3| = 3$, $|0| = 0$.
Examples: Temperature below zero (-5°C), owing someone money (-$10), altitude above and below sea level.
Questions and Solutions:
1. Is -8 an integer? Is it a whole number? Is it a natural number?
Solution: -8 is an integer; -8 is not a whole number; -8 is not a natural number.
2. Are integers closed under subtraction? Give an example.
Solution: Yes, integers are closed under subtraction. For example, 5 - (-3) = 8; 7 - (-2) = 9.
3. What is the absolute value of -12? What is the absolute value of 15?
Solution: $|-12| = 12$; $|15| = 15$.
4. Give two real-life examples where you might use negative integers.
Solution: Representing a loss in business (e.g., -$50); Describing depth below sea level (e.g., -200 meters).
5. Are integers closed under division? Explain with an example.
