University of Caloocan City
College of Liberal Arts & Sciences
Mathematics Department
Learning Module in Number Theory (LM1)
Topic PRELIMINARIES
Sub-Topic Real Number System, Properties of Integers, Summation and
Product Notation.
Duration 3 hours
Introduction A real number is a number that can be found on the number line. It consists of
numbers from negative infinity to positive infinity. Below is a diagram of a real
number system.
Real Numbers (ℝ)
Rational Numbers (ℚ) Irrational Numbers (ℚ′ )
Non-Integers (Fraction) (ℤ′ ) Integers (ℤ )
Negative Integers (ℤ− ) Whole Numbers (𝕨)
Zero (𝕆) Positive Integers or
Natural Numbers (ℤ+ )
The set of integers ℤ = { . . . . −2, −1, 0, 1, 2, 3, . . . . } has the following laws
1. Closure Laws
For any integers 𝑎 𝑎𝑛𝑑 𝑏, 𝑡ℎ𝑒𝑛
𝑎+𝑏 ∈ ℤ, 𝑎−𝑏 ∈ ℤ, 𝑎𝑛𝑑 𝑎∙𝑏 ∈ ℤ
However, ℤ is NOT closed under division.
2. Commutative Laws
For any 𝑎, 𝑏 ∈ ℤ , 𝑡ℎ𝑒𝑛
𝑎 + 𝑏 = 𝑏 + 𝑎, 𝑎𝑛𝑑 𝑎∙𝑏 = 𝑏∙𝑎
3. Associative Laws
For any 𝑎, 𝑏, 𝑐 ∈ ℤ
𝑎 + (𝑏 + 𝑐) = (𝑎 + 𝑏) + 𝑐 𝑎𝑛𝑑
𝑎 ∙ (𝑏 ∙ 𝑐) = (𝑎 ∙ 𝑏) ∙ 𝑐
4. Distributive Laws
For any 𝑎, 𝑏, 𝑐 ∈ ℤ
(𝑎 + 𝑏)𝑐 = 𝑎𝑐 + 𝑏𝑐 𝑅𝑖𝑔ℎ𝑡 ℎ𝑎𝑛𝑑 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑣𝑒
𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐 𝐿𝑒𝑓𝑡 ℎ𝑎𝑛𝑑 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑣𝑒
5. Identity Laws
𝑎+0=𝑎 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑎
𝑎∙1=𝑎 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑎
6. Inverse Laws
1
For any 𝑎 ∈ ℤ , there exists inverses – 𝑎 𝑎𝑛𝑑 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡
𝑎
1
𝑎 + (−𝑎) = 0 𝑎𝑛𝑑 𝑎 (𝑎 ) = 1 , 𝑎 ≠ 0
7. The integers 𝑎 𝑎𝑛𝑑 𝑎 + 1 are consecutive integers and there exists NO
integer x such that for any integer 𝑎
𝑎 <𝑥 <𝑎+1
8. 𝐼𝑓 𝑎 ∙ 𝑏 = 0, 𝑡ℎ𝑒𝑛 𝑒𝑖𝑡ℎ𝑒𝑟 𝑎 = 0 𝑜𝑟 𝑏 = 0 𝑜𝑟 𝑎 = 𝑏 = 0
9. 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑎
𝑎 <𝑎+1 𝑎𝑛𝑑 𝑎−1<𝑎
10. 𝑎 > 0 𝑚𝑒𝑎𝑛𝑠 𝒂 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟, 𝑎𝑛𝑑
𝑏 > 0 𝑚𝑒𝑎𝑛𝑠 𝒃 𝑖𝑠 𝑎 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
11. 𝑇ℎ𝑒 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎𝑛 𝑒𝑣𝑒𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑖𝑠 2𝑘 ; 𝑎𝑛𝑑 𝑡ℎ𝑎𝑡 𝑓𝑜𝑟 𝑜𝑑𝑑
𝑖𝑠 2𝑘 + 1 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑘.
Summation The summation notation also known as Sigma Notation is a simple form of
Notations shorthand used to give a concise expression for a sum of the values of a
variable.
The above symbol is equivalent to
∑𝑛𝑖=1 𝑥𝑖 = 𝑥1 + 𝑥2 + 𝑥3 + ∙ ∙ ∙ ∙ + 𝑥𝑛
This x indicates that you will find the sum of all the values of 𝑥 starting from 𝑥1
up to 𝑥𝑛 .
Properties of Summation
𝐼𝑓 𝑘 𝑖𝑠 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑡ℎ𝑒𝑛
1. ∑𝑛𝑖=1 𝑘 = 𝑘𝑛
Example:
∑61=1 5 = 5(6) = 30
2. ∑𝑛𝑖=1 𝑘𝑥𝑖 = 𝑘 ∑𝑛𝑖=1 𝑥𝑖
Example:
∑4𝑖=1 7𝑥𝑖 = 7 ∑4𝑖=1 𝑥𝑖 = 7(𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 )
3. ∑𝑛𝑖=1(𝑥𝑖 ± 𝑦𝑖 ) = ∑𝑛𝑖=1 𝑥𝑖 ± ∑𝑛𝑖=1 𝑦𝑖
Example
∑3𝑖=1(𝑥𝑖 + 𝑦𝑖 ) = ∑3𝑖=1 𝑥𝑖 ± ∑3𝑖=1 𝑦𝑖
= (𝑥1 + 𝑥2 + 𝑥3 ) + (𝑦1 + 𝑦2 + 𝑦3 )
NOTE: In examples 2 and 3, to evaluate for specific value of the summation the
values of 𝑥1 , 𝑥2 , 𝑥3 𝑎𝑛𝑑 𝑠𝑜 𝑜𝑛 𝑚𝑢𝑠𝑡 𝑏𝑒 𝑔𝑖𝑣𝑒𝑛 𝑠𝑜 𝑎𝑠 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑦1 , 𝑦2 ….
Example 1:
𝐼𝑓 𝑥1 = 5 , 𝑥2 = 9, 𝑥3 = −6 𝑎𝑛𝑑 𝑦1 = 7, 𝑦2 = 10 , 𝑦3 = 0
Evaluate the following
1. ∑3𝑖=1 𝑥𝑖
2. ∑3𝑖=1 6𝑦𝑖
3. ∑3𝑖=2(𝑥𝑖 + 𝑦𝑖 )
Solution.
1. ∑3𝑖=1 𝑥𝑖 = 𝑥1 + 𝑥2 + 𝑥3 = 5 + 9 + (−6) = 8
2. ∑3𝑖=1 6𝑦𝑖 = 6 ∑3𝑖=1 𝑦𝑖 = 6(𝑦1 + 𝑦2 + 𝑦3 )
= 6(7 + 10 + 0) = 6(17)
= 102
3. 𝑖=2(𝑥𝑖 + 𝑦𝑖 ) = 𝑖=2 𝑥𝑖 + ∑3𝑖=2 𝑦𝑖
∑ 3
∑ 3
= (𝑥2 + 𝑥3 ) + (𝑦2 + 𝑦3 )
= (9 + (−6)) + (10 + 0)
= (3) + (10)
= 13
Example 2:
Find the value of ∑5𝑖=1(−1)𝑖 (4𝑖 − 3)
Solution.
∑5𝑖=1(−1)𝑖 (4𝑖 − 3) = (−1)1 [4(1) − 3] + (−1)2 [4(2) − 3] +
(−1)3 [4(3) − 3] + (−1)4 [4(4) − 3] + (−1)5 [4(5) − 3]
= (−1)[1] + (1)[5] + (−1)[9] + (1)[13] + (−1)[17]
= −9
Product Notation
The product notation also known as Pi Notation is used to indicate repeated
multiplication. It is similar to Sigma symbol except that succeeding terms are
multiplied instead of added.
∏𝑛𝑖=1 𝑖 = 1 ∙ 2 ∙ 3 ∙ ∙ ∙ 𝑛 = 𝑛!
Properties of Product
𝐼𝑓 𝑘 𝑖𝑠 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟
1. ∏𝑛𝑖=1 𝑘 = 𝑘 𝑛
2. ∏𝑛𝑖=1 𝑥𝑖𝑘 = (∏𝑛𝑖=1 𝑥𝑖 )𝑘
3. ∏𝑛𝑖=1 𝑘𝑥𝑖 = 𝑘 ∏𝑛𝑖=1 𝑥𝑖
4. ∏𝑛𝑖=1 𝑥𝑖 𝑦𝑖 = (∏𝑛𝑖=1 𝑥𝑖 ) (∏𝑛𝑖=1 𝑦𝑖 )
Example 3:
Evaluate the following:
1. ∏4𝑖=1(−1)𝑖+1
2. ∏3𝑖=1(3𝑖 + 1)𝑖
Solution.
1. ∏4𝑖=1(−1)𝑖+1 = [(−1)1+1 ] [(−1)2+1 ] [(−1)3+1 ] [(−1)4+1 ]
= [(−1)2 ] [(−1)3 ] + [(−1)4 ][(−1)5 ]
= [1][−1][1 ] [−1 ]
= 1
2. ∏3𝑖=1(3𝑖 + 1)𝑖 = [3(1) + 1]1 [3(2) + 1]2 [3(3) + 1]3
= [4]1 [7]2 [10]3
= (4)(49)(1000)
= 196 000
Prepared and Submitted by:
TEODULFO T. UCHI
Faculty
