Number Systems & Algebra Notes (Conceptual)

Number Systems

a) Real Numbers

• Include all numbers that can be placed on a number line.

• Divided into:

  • Rational numbers – numbers that can be expressed as fractions; decimals terminate or repeat.

  • Irrational numbers – decimals that never terminate or repeat.

b) Rational Numbers

• Fractions and integers are rational.

• Properties:

  • Closed under addition, subtraction, multiplication, and division (except by zero).

  • Any rational number can be expressed as a decimal (terminating or repeating).

c) Irrational Numbers

• Cannot be expressed as fractions.

• Examples: Numbers like the square root of a non-perfect square, pi.

• Adding or multiplying two irrational numbers may give either a rational or irrational number.

d) Laws of Exponents & Number Properties

• Real numbers follow commutative, associative, and distributive laws.

• Closure property ensures the result of operations on real numbers stays within real numbers.

• Identity elements:

  • Additive identity → 0

  • Multiplicative identity → 1

e) HCF & LCM

• HCF (Highest Common Factor): Largest number dividing two or more numbers.

• LCM (Lowest Common Multiple): Smallest number divisible by two or more numbers.

• Prime factorization is the easiest way to find HCF and LCM.

2. Algebra

a) Introduction

• Algebra is the branch of mathematics that uses symbols (like letters) to represent numbers.

• Symbols allow us to generalize patterns and solve problems without specific numbers.

b) Expressions

• An expression is a combination of numbers, variables, and operations.

• Expressions can be simplified by combining like terms.

c) Polynomials

• Polynomials are algebraic expressions with variables and coefficients.

• Degree of a polynomial: The highest power of the variable in the polynomial.

• Types of polynomials:

  • Monomial – one term

  • Binomial – two terms

  • Trinomial – three terms

  • Polynomial – more than three terms

d) Factorization

• Breaking an expression into products of simpler expressions is factorization.

• Methods include:

  • Taking out a common factor

  • Grouping terms

  • Using special formulas for squares and cubes

e) Algebraic Identities (Conceptual)

• Algebraic identities help to simplify expressions quickly.

• They show patterns for squares, cubes, and differences.

• Useful in factorization, expansion, and solving equations.

f) Applications of Algebra

• Solving equations

• Expressing word problems mathematically

• Simplifying complex expressions

3. Important Points to Remember

• Numbers are classified into natural, whole, integers, rational, irrational, and real numbers.

• Algebra helps to generalize patterns and solve unknowns.

• Factorization and identities are tools to simplify and solve problems quickly.

• HCF and LCM connect number theory with algebra.

• Always check whether operations lead to rational or irrational results.