Rational and Irrational Numbers

Rational Numbers

• A rational number is a real number that can be expressed as the ratio of two integers, i.e., p/q, where p and q are integers and q ≠ 0.
• Rational numbers can be expressed as finite decimals or recurring decimals.
• Examples: 3/4, 22/7, 0.5, 0.333...

Irrational Numbers

• An irrational number is a real number that cannot be expressed as the ratio of two integers.
• Irrational numbers have infinite non-repeating decimals.
• Examples: π, e, √2, √3...

Square Roots

• A square root of a number is a value that, when multiplied by itself, gives the original number.
• Square root of a number x is denoted by √x.
• A square root can be positive or negative, e.g., √16 = ±4.
• Square roots of non-perfect squares are irrational numbers.

Indices

• An index (or exponent) is a small number that indicates the power to which a base number should be raised.
• Indices are used to simplify expressions and equations.
• Examples: 2^3, 5^2, x^4...

Surds

• A surd is a square root of a number that cannot be simplified to a whole number or a rational number.
• Surds are irrational numbers that can be expressed using square roots.
• Examples: √2, √3, 2√5...
• Surds can be simplified by combining like terms, e.g., 2√2 + 3√2 = 5√2.

Numbers

• A rational number can be expressed as the ratio of two integers (p/q), where p and q are integers and q ≠ 0.
• Rational numbers can be expressed as finite decimals or recurring decimals.
• Examples of rational numbers include 3/4, 22/7, 0.5, and 0.333...

Irrational Numbers

• An irrational number cannot be expressed as the ratio of two integers.
• Irrational numbers have infinite non-repeating decimals.
• Examples of irrational numbers include π, e, √2, and √3.

Square Roots

• A square root of a number is a value that, when multiplied by itself, gives the original number.
• The square root of a number x is denoted by √x.
• A square root can be positive or negative, for example, √16 = ±4.
• Square roots of non-perfect squares are irrational numbers.

Indices

• An index (or exponent) is a small number that indicates the power to which a base number should be raised.
• Indices are used to simplify expressions and equations.
• Examples of indices include 2^3, 5^2, and x^4.

Surds

• A surd is a square root of a number that cannot be simplified to a whole number or a rational number.
• Surds are irrational numbers that can be expressed using square roots.
• Examples of surds include √2, √3, and 2√5.
• Surds can be simplified by combining like terms, for example, 2√2 + 3√2 = 5√2.

Life Processes

Photosynthesis

• Defined as the process where green plants, algae, and some bacteria convert light energy into chemical energy in the form of glucose
• Occurs in the presence of light, CO2, and H2O, producing glucose and oxygen as byproducts
• Equation: 6CO2 + 6H2O + light energy → C6H12O6 (glucose) + 6O2
• Crucial for producing oxygen, supporting life on Earth, and providing energy and organic compounds for plants and animals
• Factors influencing photosynthesis include light intensity, temperature, water availability, and CO2 concentration

Respiration

• Defined as the process where cells generate energy from glucose, releasing CO2 and H2O as byproducts
• Equation: C6H12O6 (glucose) + 6O2 → 6CO2 + 6H2O + energy (ATP)
• Types: aerobic respiration (with oxygen) and anaerobic respiration (without oxygen)
• Essential for providing energy for cellular activities and maintaining life
• Occurs in the mitochondria of cells

Transportation

• Defined as the movement of substances within an organism, including nutrients, oxygen, and waste products
• Mechanisms: diffusion (passive), osmosis (passive), and active transport (energy-dependent)
• Crucial for maintaining cellular homeostasis and enabling cells to respond to environmental changes
• Occurs through various transport mechanisms, including blood vessels, nerves, and cellular membranes

Nutrition

• Defined as the process where an organism obtains and utilizes nutrients for energy, growth, and maintenance
• Modes: autotrophic nutrition (self-nutrition, e.g., photosynthesis) and heterotrophic nutrition (other-organism nutrition, e.g., ingestion)
• Essential for growth, development, and maintenance of life
• Influences health and well-being, as optimal nutrition supports overall health

Excretion

• Defined as the process where an organism removes waste products from its body
• Mechanisms: excretory organs (e.g., kidneys, liver) and excretory systems (e.g., urinary, digestive)
• Crucial for maintaining cellular homeostasis and preventing toxic buildup of waste products
• Occurs through various excretory systems, including kidneys, liver, and skin

Understand the difference between rational and irrational numbers, including definitions, examples, and properties.