Core Concepts of Fractions, Decimals, and Exponent Rules

Fundamental Number Concepts

Fraction Definition

A fraction is an expression a / b, where a (the numerator) and b (the denominator) are integers, and b must not be zero.

Equivalent Fractions

Two fractions, a / b and c / d, are considered equivalent, written as a / b = c / d, if the cross-product holds true: a · d = b · c.

Understanding Decimal Numbers

Decimal numbers express quantities that include incomplete units. A decimal number has two parts separated by a comma (or decimal point): the integer part (located to the left) and the fractional part (located to the right).

Types of Decimal Numbers

• Terminating Decimal: A decimal number is terminating (or accurate) when it has a finite number of decimal places.
• Periodic Decimal (Repeating): A decimal number is periodic if its decimal places contain one or more figures that repeat indefinitely. The repeating figure or group of numbers is called the period.
  • Pure Periodic Decimal: If the period begins immediately after the comma.
  • Mixed Periodic Decimal: If the period does not begin immediately after the comma. The non-repeating decimal places are called the anti-period.
• Non-Terminating and Non-Periodic Decimal (Irrational): A decimal number that is neither terminating nor periodic, having infinite decimal figures where none of them repeat periodically.

Rational Numbers (Q)

The set of Rational Numbers (Q) is the set of all numbers that can be expressed as a fraction a / b, where a and b are integers and b ≠ 0.

Rules of Exponents and Powers

Powers of Positive Exponents

A power is a short form of multiplication used to express repeated multiplication where all factors are equal (e.g., aⁿ).

Key Exponent Rules

• Power of a Product: To raise a product to a power, raise each factor to that power: (a · b)ⁿ = aⁿ · bⁿ.
• Power of a Quotient: To raise a quotient to a power:
  • If the exponent is positive, raise both the numerator and denominator to that power: (a / b)ⁿ = aⁿ / bⁿ.
  • If the exponent is negative, invert the base elements and raise the new quotient to the positive power: (a / b)⁻ⁿ = (b / a)ⁿ.
• Multiplication of Powers (Same Base): To multiply powers of the same base, keep the base and add the exponents: aⁿ · aⁿ = aⁿ⁺ⁿ.
• Division of Powers (Same Base): To divide powers of the same base, keep the base and subtract the exponents: aⁿ / aⁿ = aⁿ⁻ⁿ.
• Power of a Power: To raise a power to another power, keep the base and multiply the exponents: (aⁿ)ⁿ = aⁿ·ⁿ.

Scientific Notation and Real Numbers

Understanding Powers of 10

A power of 10 with a positive exponent (10ⁿ) is equal to 1 followed by as many zeros as the exponent indicates. A power of 10 with a negative exponent (10⁻ⁿ) is the reciprocal of the positive power (1 / 10ⁿ).

Scientific Notation Definition

Scientific notation is a method of expressing numbers using the product of a number (a) such that 1 ≤ a < 10, multiplied by a power of 10 (10ⁿ). The exponent of the power of 10 (n) is called the order of magnitude.

Real Numbers (R)

The set of Real Numbers (R) is formed by the union of all rational numbers and all irrational numbers.

Intervals on the Real Line

An Interval (sometimes referred to as a Range) is a set of real numbers corresponding to the points of a segment on the real number line.