Basic Arithmetic Pointers

• Number systems


  • Natural numbers (N)
  • Whole numbers (W)
  • Integers (Z)
  • Rational numbers (Q)
  • Irrational numbers
  • Real numbers (R)
• Natural numbers (N)


  Counting numbers: 1, 2, 3, 4, 5, ...
• Whole numbers (W)


  Natural numbers including 0: 0, 1, 2, 3, ...
• Integers (Z)


  Whole numbers and their opposites: ..., -3, -2, -1, 0, 1, 2, 3, ...
• Rational numbers (Q)


  Numbers that can be expressed as a ratio of two integers
• Irrational numbers


  Numbers that cannot be expressed as a ratio of two integers
• Real numbers (R)


  The set of rational and irrational numbers
• The set of natural numbers is a subset of the set of whole numbers, which is a subset of the set of integers, which is a subset of the set of rational numbers
• Adding a positive and a negative number

  1. Ignore the signs and find the positive difference between the number parts

  2. Attach the sign of the original number with the larger number part
• Subtracting numbers


  Turn the subtraction into an addition problem
• Adding/subtracting a string of positives and negatives

  1. Turn everything into addition

  2. Combine the positives and negatives so the string is reduced to the sum of a single positive and a single negative number
• Multiplying/dividing positives and negatives

  1. Treat the number parts as usual

  2. Attach a minus sign if there were originally an odd number of negatives
• Successor property


  • If x is a natural number, then x + 1 is the succeeding natural number
• Closure property


  • A set S is closed under an operation if whenever a and b are in S, a ⊕ b is in S (⊕ represents any mathematical operation)
• Commutative property for addition and multiplication


  • a + b = b + a and a × b = b × a
• Associative property for addition and multiplication

  • (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c)
• Distributive property of multiplication over addition and subtraction

  • a(b + c) = (a × b) + (a × c) and a(b - c) = (a × b) - (a × c)
• Additive and multiplicative identity properties

  • a + 0 = 0 + a = a and a × 1 = 1 × a = a
• Zero property


  • For all a in R, a × 0 = 0
• Inverse properties for addition and multiplication

  • a + (-a) = (-a) + a = 0 and a × (1/a) = (1/a) × a = 1 (where a ≠ 0)
• Density property


  • Between any two rational numbers, there is another rational number, and between any two irrational numbers, there is another irrational number. Likewise, between any two rational numbers, there is an irrational number.
• Inverse operations


  Subtraction "undoes" addition, division "undoes" multiplication, and finding roots "undoes" raising to a power
• Operations involving zero


  0 - a = 0 + (-a) = -a, a × 0 = 0 × a = 0, a / 0 is undefined
• Properties of equality


  • Reflexive, symmetric, transitive, and substitution properties
• Rounding off numbers


  If the digit to be dropped is less than 5, drop it and replace with zeros. If the digit to be dropped is 5 or more, add 1 to the digit in front of it before dropping.
• Factoring


  The process of rewriting a number as the product of its factors
• Subtraction


  If 0-a = 0+(-a) = -a
• Multiplication


  If a is any real number, then a*0=0*a=0
• Division


  If a is any real number except zero, then a/0 is undefined
• Properties of Equality


  • 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 may be replaced by b
• Rounding off numbers


  If the digit to be dropped is less than 5, just drop it and replace with zeros. If the digit to be dropped is 5 or more, add 1 to the digit in front of it before dropping.
• Factoring


  Rewriting a number as the product of its factors
• Prime factorization


  The process of getting only the prime factors of a number
• Greatest Common Factor (GCF)


  The largest integer that is a factor of each of the numbers. If GCF is 1, the numbers are relatively prime.
• Least Common Multiple (LCM)


  The smallest positive integer that is a multiple of each of the numbers
• Finding GCF and LCM

  • GCF of 108 and 240 is 12
  • LCM of 108 and 240 is 2160
• Divisibility tests for whole numbers

  • Units digit is even: divisible by 2
  • Sum of digits is multiple of 3: divisible by 3
  • Last two digits are multiple of 4: divisible by 4
  • Units digit is 0 or 5: divisible by 5
  • Divisible by 2 and 3: divisible by 6
  • Integer without units digit minus twice of units digit is 0 or divisible by 7: divisible by 7
  • Last three digits are divisible by 8: divisible by 8
  • Sum of digits is multiple of 9: divisible by 9
  • Units digit is 0: divisible by 10
  • Sum of odd place digits minus even place digits is 0 or divisible by 11: divisible by 11
  • Divisible by 3 and 4: divisible by 12
• Fraction


  The ratio of two numbers indicating a portion (numerator) taken from a whole (denominator)
• Lowest terms


  A fraction is in the lowest terms when the greatest common factor (GCF) of the numerator and denominator is 1
• Equivalent fractions


  Fractions and where b, d ≠ 0 are equivalent if and only if ad = bc