Class 9 Number Systems Mathematic All Formula & Concept Notes | Formula Sheet

Formulas and concepts related to Number Systems for Class 9 Mathematics:

1. Real Numbers

• Types of Numbers:

  • Natural Numbers (N): $1, 2, 3, 4, \ldots$
  • Whole Numbers (W): $0, 1, 2, 3, \ldots$
  • Integers (Z): $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$
  • Rational Numbers (Q): Numbers that can be expressed as $\frac{p}{q}$, where $p$ and $q$ are integers, and $q \neq 0$. Examples include $\frac{1}{2}$, $\frac{-3}{4}$, etc.
  • Irrational Numbers: Numbers that cannot be expressed as a fraction $\frac{p}{q}$. 
  • Real Numbers (R): The set of all rational and irrational numbers.

• Properties of Real Numbers:

  • Closure Property: For any two real numbers $a$ and $b$, $a + b$ and $a \cdot b$ are also real numbers.
  • Commutative Property: $a + b = b + a$ and $a \cdot b = b \cdot a$.
  • Associative Property: $(a + b) + c = a + (b + c)$ and $(a \cdot b) \cdot c = a \cdot (b \cdot c)$.
  • Distributive Property: $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$.
  • Identity Elements:
    • Addition: $a + 0 = a$
    • Multiplication: $a \cdot 1 = a$
  • Inverse Elements:
    • Addition: For every $a$, there exists $-a$ such that $a + (-a) = 0$.
    • Multiplication: For every $a \neq 0$, there exists $\frac{1}{a}$ such that $a \cdot \frac{1}{a} = 1$.

2. Euclid’s Division Lemma

• Statement: For any two positive integers $a$ and $b$, there exist unique integers $q$ (quotient) and $r$ (remainder) such that: $$a = b \cdot q + r$$ where $0 \leq r < b$.

3. Fundamental Theorem of Arithmetic

• Statement: Every composite number can be expressed as a product of prime numbers in a unique way, except for the order of the factors.

4. Euclid’s Division Algorithm

• Algorithm: To find the greatest common divisor (GCD) of two positive integers $a$ and $b$:
  1. Divide $a$ by $b$ and get the remainder $r$.
  2. Replace $a$ with $b$ and $b$ with $r$.
  3. Repeat the process until $r$ becomes 0. The non-zero remainder at this step is the GCD of $a$ and $b$.

5. Properties of Irrational Numbers

• Addition/Subtraction: The sum or difference of a rational number and an irrational number is irrational.
• Multiplication: The product of a non-zero rational number and an irrational number is irrational.
• Division: The quotient of an irrational number by a non-zero rational number is irrational.

6. Representation of Real Numbers on the Number Line

• Rational Numbers: Can be represented as points on the number line by their fractional or decimal form.
• Irrational Numbers: Can also be represented on the number line, though they have non-terminating, non-repeating decimal expansions.

7. Representation of Real Numbers

• Decimal Expansions:
  • Rational Numbers: Can be expressed as terminating or repeating decimals.
  • Irrational Numbers: Have non-terminating, non-repeating decimals.

8. Comparing Real Numbers

• Order of Real Numbers: For any two real numbers $a$ and $b$, $a < b$, $a = b$, or $a > b$ can be determined by comparing their decimal representations or by using inequalities.

9. Absolute Value

• Definition: The absolute value of a real number $a$ is denoted by $|a|$ and is defined as: $$|a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases}$$

These concepts and formulas form the basis of Number Systems in Class 9 Mathematics. If you need more details or examples on any specific topic, just let me know!