In this course, natural numbers INCLUDE 0.
Natural Numbers: ℕ = {0, 1, 2, 3, 4, ...}
Natural numbers are used for counting and represent abstract quantities.
The number 0 is of Indian origin and is crucial for our place value numbering system.
Numbers like 7 represent what is common between different collections (7 balls, 7 pencils).
Zero represents "nothing to count" but is essential for mathematics.
Some textbooks exclude 0 from natural numbers, but THIS COURSE INCLUDES IT. Always remember this convention for exams and assignments.
Integers: ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...}
Integers extend natural numbers with negative numbers
All natural numbers are integers: ℕ ⊆ ℤ
Integers are discrete: every integer (except boundaries) has a unique "next" and "previous" integer
Combining quantities
Finding difference
Repeated addition
Splitting into equal parts
Repeated multiplication
For integers a and b (where b ≠ 0):
where:
Divide 17 by 5:
a mod b = remainder when a is divided by b
17 mod 5 = 2
Since 17 = 5 × 3 + 220 mod 6 = 2
Since 20 = 6 × 3 + 215 mod 3 = 0
Since 15 = 3 × 5 + 0We say a | b (a divides b) if a mod b = 0
This means b is divisible by a with no remainder.
Factors of a number n are all numbers that divide n evenly (with remainder 0).
Let's find all numbers that divide 24:
Factors of 24: {1, 2, 3, 4, 6, 8, 12, 24}
Find the quotient and remainder when 47 is divided by 6.
Calculate: (a) 23 mod 7, (b) 100 mod 9, (c) 50 mod 10
Find all factors of 36.