Algebra 1

• Numbers are classified into two types: cardinal numbers and ordinal numbers
• Cardinal numbers allow us to count the objects or ideas in a given collection
• Ordinal numbers state the position of individual objects in a sequence
• Numerals are symbols or combination of symbols that describe a number
• The most widely used numerals are the Arabic numerals and the Roman numerals
• Arabic numerals were modified from Hindu-Arabic number signs and are written in Arabic digits
• Roman numerals are numbers written in the Latin alphabet
• Roman numerals and their equivalent Arabic numbers:

  • I = 1
  • V = 5
  • X = 10
  • L = 50
  • C = 100
  • D = 500
  • M = 1000
• The Romans used brackets to multiply numbers by 100 times, vinculum to multiply by 1000 times, and doorframe to multiply by 1000000 times
• A digit is a specific symbol or symbols used alone or in combination to denote a number
• In Roman numerals, the number 9 is denoted as IX, where I and X are used together to represent 9
• A system of numbers using cardinal numbers is established and widely used in mathematical computations or engineering applications
• Imaginary numbers are denoted as i and are equal to the square root of -1
• Rational numbers can be expressed as a quotient of two integers
• Irrational numbers cannot be expressed as a quotient of two integers
• A complex number is an expression of both real and imaginary numbers combined, taking the form of a + bi
• If a = 0, a pure imaginary number is produced; if b = 0, a real number is obtained
• Integers include all natural numbers, the negative of natural numbers, and the number zero
• Natural numbers are considered as the "counting numbers" (e.g., 1, 2, 3, ...)
• For non-terminating decimals:

  • Repeating decimals like 0.3333... are rational numbers
  • Non-repeating decimals like pi = 3.14159... are irrational numbers
• Properties of INTEGERS:

  • ADDITION PROPERTIES:
  • Closure property: a + b = integer
  • Commutative property: a + b = b + a
  • Associative property: (a + b) + c = a + (b + c)
  • Identity property: a + 0 = a
  • Inverse property: a + (-a) = 0
  • Distributive property: a(b + c) = ab + ac
  • MULTIPLICATION PROPERTIES:
  • Closure property: ab = integer
  • Commutative property: ab = ba
  • Associative property: (ab)c = a(bc)
  • Identity property: a * 1 = a
  • Inverse property: 1/a, where a is the multiplicative inverse
  • Distributive property: a(b + c) = ab + ac
• • Multiplication property of zero: a(0) = 0
  • EQUALITY PROPERTIES:
  • Reflexive property: a = a
  • Symmetric property: If a = b, then b = a
  • Transitive property: If a = b and b = c, then a = c
  • Substitution property: If a = b, then a can be replaced by b in any expression
  • Addition/Subtraction property: If a = b, then a + c = b + c and a - c = b - c
  • Multiplication/Division property: If a = b, then ac = bc and a/b = c, where c is not equal to 0
  • Cancellation property: If a + c = b + c, then a = b and if ac = bc and c is not equal to 0, then a = b