Rational and Irrational Numbers

Study Notes

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...

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...

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.

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...

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.

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...

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.

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.

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.

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

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

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

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

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