Understanding Real Numbers

Real Numbers

• Real numbers consist of rational numbers and irrational numbers.

Classification of Real Numbers

• Counting numbers are the most basic numbers, used in algebra to count objects (1, 2, 3, 4, 5,...), also called natural numbers.
• Counting or natural numbers start with 1 and continue: N = {1, 2, 3, 4, 5,...}.
• Whole numbers include the counting numbers and zero: W = {0, 1, 2, 3, 4, 5,...}.
• A negative number is less than 0 and represented on a number line.
• It is customary to omit the plus sign (+) before a positive number.
• Integers are the set of counting numbers, their opposites, and 0: Z = {..., -3, -2, -1, 0, 1, 2, 3,...}.
• Rational numbers can be written in the form 𝑝/𝑞 where p and q are integers but 𝑞 ≠ 0.
• Integers are rational numbers because they can be written as a ratio of two integers.
• A rational number can be written as a ratio of integers or as a decimal that either stops (e.g., 4.275) or repeats (e.g., 2.757575...).
• Irrational numbers are numbers that cannot be written as a ratio of two integers, with decimal forms that do not stop or repeat (e.g.,√5 = 2.236067978...).
• Pi (π) is an irrational number crucial for describing circles and has a decimal form that neither stops nor repeats (π = 3.141592654...).

Properties of Real Numbers

• Mastering mathematics requires constant attention.
• Reviewing basic topics is helpful, especially after a long break.
• Algebra uses numbers and symbols to represent ideas and words.

Closure Properties

• If a and b are real numbers, then a + b is also a real number (Closure Property of Addition).
• If a and b are real numbers, then a * b is also a real number (Closure Property of Multiplication).

Commutative Properties

• If a and b are real numbers, then a + b = b + a (Commutative Property of Addition).
• If a and b are real numbers, then a * b =b * a (Commutative Property of Multiplication).

Associative Properties

• If a, b, and c are real numbers, then (a + b) + c = a + (b + c) (Associative Property of Addition).
• If a, b, and c are real numbers, then (a * b) * c = a * (b * c) (Associative Property of Multiplication).

Distributive Property

• If a, b, and c are real numbers, then a * (b + c) = a * b + a *c (Distributive Property).
• The Distributive Property is used to remove parentheses when simplifying expressions.

Identity Properties

• For any real number a, a + 0 = a; 0 is called the additive identity (Identity Property of Addition).
• For any real number a, a * 1 = a; 1 is called the multiplicative identity (Identity Property of Multiplication).

Inverse Properties

• For any real number a, a + (-a) = 0; -a is the additive inverse of a (Inverse Property of Addition).
• For any real number 𝑎 ≠ 0, 𝑎 ⋅1/ 𝑎 = 1; 1/𝑎 is the multiplicative inverse of a (Inverse Property of Multiplication).

Properties of Zero

• Zero is the additive identity because adding zero to any number doesn't change the number's identity.
• a * 0 = 0 for any real number a (Multiplication by Zero).
• 0 / a = 0 for any real number a ≠ 0 (Division with Zero).
• a/0 is undefined for any real number a (Division with Zero).

Fractions

• A fraction represents parts of a whole.
• The denominator (b) indicates the number of equal parts, and the numerator (a) indicates included parts.
• A fraction is written as a/b, where a and b are integers, and b ≠ 0.
• A proper fraction has a < b, and an improper fraction has a ≥ b.
• Improper fractions can be converted into mixed numbers and vice versa.
• A multiple of a number is the product of that number and a counting number.
• A number is a multiple of n if it is the product of a counting number and n and is divisible by n.

Factors

• 𝑓 𝑎 * 𝑏 = 𝑚, then a and b are factors of m, and m is the product of a and b.
• Divide the number by each counting number and determine if the divisor and quotient are a pair of factors.
• A number with only two factors (1 and itself) is prime.
• A number with more than two factors is composite.
• The number 1 is neither prime nor composite.
• Equivalent fractions have the same value.
• A fraction is simplified if there are no common factors in the numerator and denominator.
• Multiplying fractions involves multiplying across the numerators and denominators: 𝘢/𝘣 * 𝘤/𝘥 = 𝘢𝘤/𝘣𝘥 (Fraction Multiplication).

Reciprocals

• The reciprocal of the fraction 𝘢/𝑏 is 𝑏/𝑎, where 𝑎 ≠ 0 and 𝑏 ≠ 0.
• The product of a number and its reciprocal is 1.
• Dividing fractions involves multiplying by the reciprocal of the second fraction 𝑎/𝑏 ÷ 𝑐/𝑑 =𝑎/𝑏 * 𝑑/𝑐 (Fraction Division).

Adding and Subtracting Fractions

• 𝑎/𝑐 + 𝑏/𝑐 = (𝑎+𝑏)/𝑐 and 𝑎/𝑐 − 𝑏/𝑐 = (𝑎−𝑏)/𝑐 if a, b, and c are numbers where 𝑐≠ 0.
• When adding fractions with different denominators you must convert them to equivalent fractions with a common denominator.
• The least common denominator (LCD) is the least common multiple (LCM) of their denominators.
• Factor each denominator into its primes, then list the primes lining up matching primes in columns when possible.
• Multiply the factors to compute the LCD of different fractions.

Decimals

• Adding and subtracting decimals requires writing numbers vertically with decimal points lined up and adding zeros as placeholders.
• Add (or subtract) the numbers as if they were whole numbers, placing the decimal in the answer beneath the decimal points in the given numbers.

Ratios

• A ratio compares two numbers or quantities measured with the same unit.
• The ratio of a to b can be written as a to b, 𝑎/𝑏, or a:b.
• A percent is a ratio with a denominator of 100, indicated by the percent symbol (%).
• To convert a percent to a fraction, write the percent as a ratio with the denominator as 100.
• To convert a percent to a decimal you must move the decimal point two places to the left and remove the % sign.