Types of Numbers in Mathematics

Number Types

• Natural Numbers: Positive integers starting from 1 (1, 2, 3, 4, 5, etc.).
• Whole Numbers: Natural numbers including zero (0, 1, 2, 3, 4, 5, etc.).
• Integers: Whole numbers that include both positive and negative values, as well as zero.
• Rational Numbers: Numbers expressible as a fraction where both numerator and denominator are integers; their decimal forms either terminate or repeat.
• Irrational Numbers: Numbers that cannot be expressed as fractions, including non-repeating and non-terminating decimals.

Key Mathematical Concepts

• Absolute Value: Represents a number's distance from zero on a number line; denoted as |a|, always a non-negative value (e.g., |4| = 4, |-4| = 4).
• Proper Fraction: A fraction where the numerator is less than the denominator (e.g., 3/5).
• Improper Fraction: A fraction where the numerator is greater than or equal to the denominator (e.g., 5/3).
• Mixed Numbers: A combination of a whole number and a fraction (e.g., 1 1/2).
• Reciprocals: The reciprocal of a fraction (a/b) is obtained by flipping the fraction to b/a.

Factors and Multiples

• Greatest Common Factor (GCF): The largest factor common to two or more numbers.
• Least Common Multiple (LCM): The smallest multiple that two or more numbers share, not including zero.
• Least Common Denominator (LCD): The LCM of the denominators of fractions used to find a common base for addition or subtraction.
• Clearing Fractions: To eliminate fractions from an equation by multiplying all terms by the LCM or LCD.

Operations and Properties

• Ways to Express Multiplication: Written in various forms: 3x4, 3*4, 3(4), (3)(4).
• PEMDAS (Order of Operations): Follow the sequence of Parentheses, Exponents, Multiplication/Division (from left to right), and Addition/Subtraction (from left to right).
• Associative Property: Changing the grouping in addition or multiplication does not affect the result (e.g., (7 + 4) + 8 = 7 + (4 + 8)).
• Additive Property: Adding zero to a number does not change the number (a + 0 = a).
• Multiplicative Property: Multiplying a number by one does not change its value (a * 1 = a).
• Distributive Property: Distributing a number across a sum (a(b + c) = ab + ac).

Algebraic Terminology

• Term: Represents a constant, product, or quotient of constants and variables.
• Variable: A symbol that represents an unknown value (e.g., x, y, z).
• Constant: A fixed value that does not change in an expression (e.g., 5 in 5x).
• Coefficient: The numerical factor multiplied by a variable in an algebraic expression (e.g., 3 in 3a).
• Equation: A mathematical statement asserting the equality of two expressions (e.g., x + 1 = 1).
• Solution: The value for the variable that satisfies the equation (e.g., x = 0).
• Solution Set: All values of the variable that make the equation true.
• Linear Equation: An equation of the form ax + b = c that represents a straight line when graphed.
• Identity: An equation that holds true for all values of the variable.

Problem-Solving Techniques

• Application Problems: A stepwise approach to solving real-world problems, which includes reading the problem, labeling unknowns, building a verbal model, writing equations, solving them, and providing final answers in words.
• Percent: Expresses a ratio per hundred, used to represent portions of a whole.
• Amount Calculation: The relationship between amount, percent, and base is described as amount = percent * base.