# Number Theory (part 1)

Secondary School Teacher à Tingloy National High School (TNHS)
25 May 2016
1 sur 18

### Number Theory (part 1)

• 1. MaryJane R. De Ade BATSTATEU - JPLPC Malvar Campus
• 2. Notations ℝ set of real numbers ℂ set of complex numbers ℚ set of rational numbers ℚ′ set of irrational numbers 𝕎 set of whole numbers ℤ set of integers
• 3. ℤ+ set of positive integers ℤ− set of negative integers 𝐴, 𝐵 sets 𝑎, 𝑏 elements ∧ and ∨ or ∀ for all Notations
• 4. ⊂ subset ⊃ supersubset ∩ intersection ∪ union 𝑈 universal ∴ therefore ∵ because ~ equivalent Notations
• 5. Try these: Identify the elements of the given set 𝐴 = −9, − 48 12 , − 1 3 , − 5, 0, 5 6 , 12 4 , 18, 56 a. Natural numbers b. Whole numbers c. Negative integers d. Integers e. Rational numbers f. Irrational numbers
• 6. Subtraction of real numbers is defined in terms of addition as follows: If 𝑎 and 𝑏 are real numbers, the difference of 𝑎 and 𝑏, denoted by 𝑎 − 𝑏 is a real number 𝑑 and 𝑎 − 𝑏 = 𝑑 if and only if 𝑎 = 𝑏 + 𝑑 Math Ideas to Remember
• 7. The real number system consists of the set of real numbers and two operation called addition and multiplication. Addition is denoted by the symbol + while multiplication is denoted by ∙ or ×. Math Ideas to Remember
• 8. Division of real numbers defined as follows: 𝑎 and 𝑏 are real numbers, 𝑏 ≠ 0, the quotient of 𝑎 and 𝑏 denoted by 𝑎 ÷ 𝑏 or 𝑎 𝑏 is a real number 𝑐 and 𝑎 ÷ 𝑏 = 𝑐 if and only if 𝑎 = 𝑏𝑐 Math Ideas to Remember
• 9. Basic Properties of Integers
• 10. Closure 𝑎 + 𝑏 and 𝑎 − 𝑏 are integers whenever 𝑎 and 𝑏 are integers
• 11. Commutative Law 𝑎 + 𝑏 = 𝑎 + 𝑏 and 𝑎 ∙ 𝑏 = 𝑎 ∙ 𝑏 ∀ integers 𝑎 and 𝑏
• 12. Associative Law 𝑎 + 𝑏 + 𝑐 = 𝑎 + 𝑏 + 𝑐 and 𝑎 ∙ 𝑏 𝑐 = 𝑎 𝑏 ∙ 𝑐 ∀ integers
• 13. Distributive Law 𝑎 + 𝑏 + 𝑐 = 𝑎 ∙ 𝑐 + 𝑏. 𝑐 ∀ integers 𝑎, 𝑏 and 𝑐
• 14. Identity Elements 𝑎 + 0 = 𝑎 and 𝑎 ∙ 1 = 𝑎 ∀ integers 𝑎
• 15. Additive Inverse For every integers 𝑎, there is an integer solution 𝑥 to the equation at 𝑥 = 0. This integer 𝑥 is called the additive inverse of 𝑎 and is denoted by 𝑎. By 𝑏 − 𝑎, we mean 𝑏 + (−𝑎).
• 16. Cancellation Law If 𝑎, 𝑏 and 𝑐 are integers with 𝑎 ∙ 𝑐 = 𝑏 ∙ 𝑐 with 𝑐 ≠ 0 then 𝑎 = 𝑏.
• 17. Use the axioms for the integers to prove the following statements for all integers 𝑎, 𝑏 and 𝑐. 1. 𝑎 𝑏 + 𝑐 = 𝑎𝑏 + 𝑎 2. (𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏2 3. 𝑎 + 𝑏 + 𝑐 = (𝑐 + 𝑎) + 𝑏 Exercises
• 18. Thank you for Listening… “Without mathematics, there's nothing you can do. Everything around you is mathematics. Everything around you is numbers.” -Shakuntala Devi