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Types of Numbers
Natural Numbers
Notation: ℕ
Natural numbers are the counting numbers you first learn when you start math.
They include: 1, 2, 3, 4, 5, …
They do not include:
0
Negative numbers
Fractions or decimals
Example: Is 5 a natural number? Yes.
Practice Problem
Write down the first five natural numbers.
Answer: 1, 2, 3, 4, 5
Whole Numbers
Notation: W, sometimes also written as ℕ₀ or just Whole Numbers (varies by syllabus).
Whole numbers are like natural numbers but include 0.
They include: 0, 1, 2, 3, 4, 5, …
Example: Is 0 a whole number? Yes.
Practice Problem
Which of these is not a whole number?
A. 3
B. 0
C. –2
D. 8
Answer: C
Integers
Notation: ℤ
Integers are all whole numbers and their negatives.
They include: …, –3, –2, –1, 0, 1, 2, 3, …
They do not include:
Fractions
Decimals
Example: Is –7 an integer? Yes.
Practice Problem
List all integers between –3 and 2.
Answer: -2, -1, 0, and 1
Prime Numbers
Prime numbers are natural numbers greater than 1 with only two factors: 1 and itself.
Examples of prime numbers: 2, 3, 5, 7, 11, 13, 17…
Examples of numbers that are not prime:
1 (only one factor)
4 (factors: 1, 2, 4)
Example: is 13 a prime number? Yes, factors are only 1 and 13.
Practice Problem
Circle all prime numbers: 2, 4, 5, 9, 11.
Answer: 5, 11
Square Numbers
Square numbers are the result of multiplying a number by itself.
Notation: n²
Examples:
1² = 1
2² = 4
3² = 9
4² = 16
5² = 25
Example: what is 7²? 49.
Practice Problem
Find 6².
Answer: 6x6=36
Cube Numbers
Cube numbers are the result of multiplying a number by itself twice (e.g. 8x8x8)
Notation: n³
Examples:
1³ = 1 (1x1x1)
2³ = 8 (2x2x2)
3³ = 27
4³ = 64
5³ = 125
Example: what is 3³? 3x3x3= 27.
Practice Problem
Calculate 4³.
Answer: 4x4x4= 64
Factors
A factor is a number that divides exactly into another number without a remainder.
Example: factors of 12 = 1, 2, 3, 4, 6, 12 (because 12 ÷ 3 = 4 exactly)
Practice Problem
List all factors of 18.
Answer: 1, 2, 3, 6, 18
Multiples
Multiples of a number are what you get when you multiply it by whole numbers. (the number given multiplied by 1, 2, 3, 4, etc.)
Example: multiples of 4 = 4, 8, 12, 16, 20, … (because 4 × 1, 4 × 2, 4 × 3, 4 × 4)
Practice Problem
Write the first four multiples of 6.
Answer: 6, 12, 18, 24
Common Factors
A common factor is a number that is a factor of two or more numbers.
Example:
Factors of 12 = 1, 2, 3, 4, 6, 12
Factors of 18 = 1, 2, 3, 6, 9, 18
Common factors = 1, 2, 3, 6
Practice Problem
Find the common factors of 20 and 30.
Answer:
Factors of 20 = 1, 2, 4, 5, 10, 20
Factors of 30 = 1, 2, 3, 5, 6, 10, 30
Common factors = 1, 2, 5, 10
Common Multiples
A common multiple is a number that is a multiple of two or more numbers.
Example:
Multiples of 4 = 4, 8, 12, 16, 20, 24…
Multiples of 6 = 6, 12, 18, 24, 30…
Common multiples = 12, 24, 36…
Practice Problem
What is the lowest common multiple (LCM) of 3 and 4?
Answer:
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, ....
Multiples of 4: 4, 8, 12, 16, 20, 24, ....
Common multiples: 12, 24, ... but the lowest is 12
Rational Numbers
Notation: ℚ
Rational numbers can be written as fractions (a/b, where a and b are integers, and b ≠ 0).
Examples: 1/2 , –3/4, 0.75 (which is 3/4), 5 (which is 5/1).
Example: is 0.6 rational? Yes, because 0.6 = 3/5.
Practice Problem
Write 0.25 as a fraction.
Answer: 1/4
Irrational Numbers
Numbers that cannot be written as exact fractions.
Their decimal forms never end and never repeat.
Examples:
π ≈ 3.14159…
√2 ≈ 1.414213…
Example: is √2 rational? No, it’s irrational.
Practice Problem
Is 22/7 rational or irrational?
Answer: rational as it can be written in the form a/b even though it's an approximation for pi
Reciprocals
The reciprocal of a number is 1 divided by that number.
Notation: Reciprocal of x = 1/x.
Examples:
Reciprocal of 5 = 1/5 (since 5 is technically 5/1)
Reciprocal of 2/3 = 3/2.
Reciprocal of –4 = –1/4. (since -4 is technically -4/1)
Example: what is the reciprocal of 7? 1/7.
Practice Problem
Find the reciprocal of 3/4.
Answer: 4/3
NOTES DONE BY FARIDA SABET
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