Number and Numeration

 Number and Numeration

1. Introduction to Numbers

Numbers are mathematical objects used to count, measure, and label. They are the foundation of mathematics and are classified into different types based on their properties.

2. Types of Numbers

1. Natural Numbers (N):

   - These are counting numbers starting from 1.  

   - Example: 1, 2, 3, 4, 5, ...

2. Whole Numbers (W):

   - These include all natural numbers and zero.  

   - Example: 0, 1, 2, 3, 4, ...

3. Integers (Z): 

   - These include whole numbers and their negative counterparts.  

   - Example: ..., -3, -2, -1, 0, 1, 2, 3, ...

4. Rational Numbers (Q): 

   - These are numbers that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \).  

   - Example: \( \frac{1}{2}, \frac{3}{4}, -0.5, 2 \).

5. Irrational Numbers:  

   - These are numbers that cannot be expressed as a fraction. They have non-terminating, non-repeating decimal expansions.  

   - Example: \( \sqrt{2}, \pi, e \).

6. Real Numbers (R): 

   - These include all rational and irrational numbers.  

   - Example: \( 1, \frac{1}{2}, \sqrt{3}, \pi \).

7. Prime Numbers:

   - These are natural numbers greater than 1 that have no positive divisors other than 1 and themselves.  

   - Example: 2, 3, 5, 7, 11, ...

8. Composite Numbers:

   - These are natural numbers greater than 1 that are not prime (i.e., they have divisors other than 1 and themselves).  

   - Example: 4, 6, 8, 9, 10, ...

3. Numeration Systems

Numeration refers to the method of representing numbers. The most common systems are:

1. Decimal System (Base 10):

   - This is the standard system using digits 0 to 9.  

   - Example: 123, 45, 6789.

2. Binary System (Base 2):

   - This system uses only two digits: 0 and 1.  

   - Example: \( 101_2 = 5_{10} \).

3. Octal System (Base 8):

   - This system uses digits 0 to 7.  

   - Example: \( 12_8 = 10_{10} \).

4. Hexadecimal System (Base 16): 

   - This system uses digits 0 to 9 and letters A to F.  

   - Example: \( 1A_{16} = 26_{10} \).

4. Place Value and Face Value

- Place Value: The value of a digit based on its position in a number.  

  Example: In 345, the place value of 4 is 40 (tens place).

5- Face Value: The actual value of a digit in a number.  

  Example: In 345, the face value of 4 is 4.

5. Rounding Numbers

Rounding is the process of approximating a number to a specific place value.  

- Rules for Rounding:

  1. Identify the place value to which you want to round.  

  2. Look at the digit to the right of that place value.  

  3. If the digit is 5 or greater, round up. If it is less than 5, round down.  

  Example: Round 123.456 to the nearest tenth:  

  - Tenths place: 4  

  - Hundredths place: 5 (round up)  

  - Rounded number: 123.5  

6. Factors and Multiples

-  Factors: Numbers that divide another number exactly without leaving a remainder.  

  Example: Factors of 12 are 1, 2, 3, 4, 6, 12.

- Multiples: The product of a number and an integer.  

  Example: Multiples of 3 are 3, 6, 9, 12, 15, ...

7. Prime Factorization

Prime factorization is the process of expressing a number as a product of its prime factors.  

- Example:  

  \( 24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3 \).

8. LCM and HCF

- LCM (Least Common Multiple): The smallest number that is a multiple of two or more numbers.  

  Example: LCM of 4 and 6 is 12.

- HCF (Highest Common Factor): The largest number that divides two or more numbers exactly.  

  Example: HCF of 12 and 18 is 6.

9. Number Patterns and Sequences

- Arithmetic Sequence: A sequence where the difference between consecutive terms is constant.  

  Example: 2, 5, 8, 11, 14, ... (common difference = 3).

- Geometric Sequence: A sequence where the ratio between consecutive terms is constant.  

  Example: 3, 6, 12, 24, 48, ... (common ratio = 2).

10. Practice Questions

1. Write the prime factors of 36.  

2. Find the LCM and HCF of 12 and 18.  

3. Round 567.893 to the nearest hundredth.  

4. Convert \( 1011_2 \) to decimal.  

5. Identify the next number in the sequence: 3, 6, 12, 24, ...