NUMBERS
,1-Integers
Integers
Integers: whole numbers placed on a number line.
Positive: right of 0
Negative: left of 0
0: non-positive & non-negative
Conventions:
Non-positive: negative or equal to 0
Non-negative: positive or equal to 0
Ordering of integers
Moving left on the number line: numbers get smaller (e.g., -3<2)
Moving right on the number line: numbers get bigger (e.g., 3>1)
Comparing larger positive numbers: align last digits - at the first differing position, the highest digit =
biggest number.
23,150 vs. 23,148: 5>4: 23,150>23,148
Comparing larger negative numbers: align last digits - at the first differing position, the highest digit =
smallest number.
-23,150 vs. -23,148: 5>4: -23,150<-23,148
Sum, terms, product & factors
Addition & subtraction:
Addition = sum; the numbers added are terms (e.g., 2+3=5; terms are 2 & 3)
Subtraction = difference; the numbers subtracted are terms (e.g., 5-3=2; terms are 5 & 3).
Multiplication & division:
Multiplication = product; the numbers multiplied are factors (e.g., 3x8=24; factors are 3 & 8)
Division = quotient; the numbers divided are factors (e.g., 4÷2=4x1/2=2; factors: 4 & 1/2)
Divisors
Integer a is a divisor of integer b if b a yields an integer q (i.e., a × q = b). The result q is the quotient.
Divisors of 12: 1, 2, 3, 4, 6, 12
Division with remainder:
When a is not a divisor of b: b = a × q + r, where r is the remainder - a non-negative integer smaller
than a. To find q & r:
Find the largest q such that a × q ≤ b
Then r = b − a × q
,Order of operations
When multiple operations appear in a calculation, apply these rules in order:
1. Brackets first - always calculate what's inside brackets before anything else
2. Multiplication & division before addition & subtraction
3. Addition & subtraction worked left to right
Factorization
Writing a positive integer as a product of smaller positive integers is a factorization of that integer (Note:
12 × 1 = 12 is not a factorization since 12 is not smaller than 12; not every number can be factorized, e.g.,
7)
Factorizations of 12: 2 × 6, 3 × 4, 4 × 3, 6 × 2
Relation to divisors: the factors of an integer are the same as its divisors (e.g., 3 & 4 are factors of 12
because 3 × 4 = 12).
Prime numbers
A prime number is a positive integer with exactly 2 divisors: 1 & itself (Note: 1 is not prime - it has only
one divisor). Prime numbers can't be factorized. First few primes:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97...
Prime factorization
A prime factorization expresses an integer as a product of prime factors only (e.g., 60 = 2 × 2 × 3 × 5 ).
Every positive integer has a unique prime factorization.
Every positive integer that is not itself prime can be written as a prime factorization. For prime numbers,
we conventionally say their prime factorization is the number itself.
e.g., 4 = 2 × 2, 6 = 2 × 3, 8 = 2 × 2 × 2, = 3 × 3, 10 = 2 × 5, 12 = 3 × 4, 15 = 3 × 5, 16 = 2 × 2 × 2 × 2
Greatest common divisor & least common multiple
Greatest common divisor (cd): the largest integer that divides both numbers.
Common divisors of 40 & 160: 1, 2, 4, 5, 8, 10, 20, 40: cd(40, 160) = 40
Least common multiple (c): the smallest positive integer divisible by both numbers (The product of 2
integers is always a common multiple, but not necessarily the smallest).
Multiples of 4: 4, 8, 12, 16...
Multiples of 6: 6, 12, 18, 24...
Common multiples: 12, 24, 36...: c(4, 6) = 12
Note: c is always defined as greater than 0!
,2-Negative numbers
Adding & subtracting negative numbers
Adding a negative number = subtracting a positive number -> result is smaller
6 + (−2) = 6 − 2 = 4
Subtracting a negative number = adding a positive number -> result is bigger
6 − (−2) = 6 + 2 = 8
Multiplying negative numbers
Rules:
Positive x Positive = Positive
Positive x Negative = Negative
Negative x Positive = Negative
Negative x Negative = Positive
Dividing negative numbers
Division follows the same sign rules as multiplication (since division = multiplication by a fraction):
Positive ÷ Positive = Positive
Positive ÷ Negative = Negative
Negative ÷ Positive = Negative
Negative ÷ Negative = Positive
Opposite numbers
2 numbers are opposites if their sum equals 0 (e.g., -2 & 2, -5 & 5) - equally far from 0 on the number
line. The opposite of a number is found by placing a minus sign in front of it. Note: −(−8) = +8, since
−− = +.
Absolute value
The absolute value of a number (written x) is always non-negative:
If the number is negative -> its opposite: − 5 = 5
If the number is non-negative -> itself: 5 = 5
e.g., − 65 = 65, 89 = 89, 12 − 19 = − 7 = 7
, 3-Fractions
Fractions
A fraction consists of a numerator (top) & denominator (bottom), separated by a line (e.g., 4/5 means 4
out of 5 equal parts).
Note: the denominator can never equal 0 (division by 0 is undefined).
Equivalent fractions
A fraction doesn't change if you multiply or divide both numerator & denominator by the same number.
e.g., 2/6 = 1/3, 3/6 = 1/2, 4/6 = 2/3
Negative fractions
Same sign rules as division:
Positive/Positive = Positive
Positive/Negative = Negative
Negative/Positive = Negative
Negative/Negative = Positive
Simplifying fractions
Simplification: reducing numerator & denominator to smaller numbers without changing the fraction's
value. A fraction is fully simplified when it contains no minus sign & can't be reduced further.
Minus signs: even number of minus signs -> cancel out; odd number -> one minus sign placed in front of
the fraction
Method: divide numerator & denominator by common factors until no further reduction is possible.
15/35 = 3/7
48/60 = 24/30 = 12/15 = 4/5
4/ − 6 = 2/ − 3 = −2/3
−3/ − 7 = 3/7
Adding & subtracting fractions with like denominators
When fractions share the same denominator, only the numerators change - the denominator stays the
same.
Addition: 1/5 + 3/5 = 4/5
Subtraction: 5/7 − 3/7 = 2/7
Writing fractions with like denominators
To add/subtract fractions with different denominators, first convert them to a common denominator by
multiplying numerator & denominator by the same number.
Easy method: multiply the 2 denominators together
,1-Integers
Integers
Integers: whole numbers placed on a number line.
Positive: right of 0
Negative: left of 0
0: non-positive & non-negative
Conventions:
Non-positive: negative or equal to 0
Non-negative: positive or equal to 0
Ordering of integers
Moving left on the number line: numbers get smaller (e.g., -3<2)
Moving right on the number line: numbers get bigger (e.g., 3>1)
Comparing larger positive numbers: align last digits - at the first differing position, the highest digit =
biggest number.
23,150 vs. 23,148: 5>4: 23,150>23,148
Comparing larger negative numbers: align last digits - at the first differing position, the highest digit =
smallest number.
-23,150 vs. -23,148: 5>4: -23,150<-23,148
Sum, terms, product & factors
Addition & subtraction:
Addition = sum; the numbers added are terms (e.g., 2+3=5; terms are 2 & 3)
Subtraction = difference; the numbers subtracted are terms (e.g., 5-3=2; terms are 5 & 3).
Multiplication & division:
Multiplication = product; the numbers multiplied are factors (e.g., 3x8=24; factors are 3 & 8)
Division = quotient; the numbers divided are factors (e.g., 4÷2=4x1/2=2; factors: 4 & 1/2)
Divisors
Integer a is a divisor of integer b if b a yields an integer q (i.e., a × q = b). The result q is the quotient.
Divisors of 12: 1, 2, 3, 4, 6, 12
Division with remainder:
When a is not a divisor of b: b = a × q + r, where r is the remainder - a non-negative integer smaller
than a. To find q & r:
Find the largest q such that a × q ≤ b
Then r = b − a × q
,Order of operations
When multiple operations appear in a calculation, apply these rules in order:
1. Brackets first - always calculate what's inside brackets before anything else
2. Multiplication & division before addition & subtraction
3. Addition & subtraction worked left to right
Factorization
Writing a positive integer as a product of smaller positive integers is a factorization of that integer (Note:
12 × 1 = 12 is not a factorization since 12 is not smaller than 12; not every number can be factorized, e.g.,
7)
Factorizations of 12: 2 × 6, 3 × 4, 4 × 3, 6 × 2
Relation to divisors: the factors of an integer are the same as its divisors (e.g., 3 & 4 are factors of 12
because 3 × 4 = 12).
Prime numbers
A prime number is a positive integer with exactly 2 divisors: 1 & itself (Note: 1 is not prime - it has only
one divisor). Prime numbers can't be factorized. First few primes:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97...
Prime factorization
A prime factorization expresses an integer as a product of prime factors only (e.g., 60 = 2 × 2 × 3 × 5 ).
Every positive integer has a unique prime factorization.
Every positive integer that is not itself prime can be written as a prime factorization. For prime numbers,
we conventionally say their prime factorization is the number itself.
e.g., 4 = 2 × 2, 6 = 2 × 3, 8 = 2 × 2 × 2, = 3 × 3, 10 = 2 × 5, 12 = 3 × 4, 15 = 3 × 5, 16 = 2 × 2 × 2 × 2
Greatest common divisor & least common multiple
Greatest common divisor (cd): the largest integer that divides both numbers.
Common divisors of 40 & 160: 1, 2, 4, 5, 8, 10, 20, 40: cd(40, 160) = 40
Least common multiple (c): the smallest positive integer divisible by both numbers (The product of 2
integers is always a common multiple, but not necessarily the smallest).
Multiples of 4: 4, 8, 12, 16...
Multiples of 6: 6, 12, 18, 24...
Common multiples: 12, 24, 36...: c(4, 6) = 12
Note: c is always defined as greater than 0!
,2-Negative numbers
Adding & subtracting negative numbers
Adding a negative number = subtracting a positive number -> result is smaller
6 + (−2) = 6 − 2 = 4
Subtracting a negative number = adding a positive number -> result is bigger
6 − (−2) = 6 + 2 = 8
Multiplying negative numbers
Rules:
Positive x Positive = Positive
Positive x Negative = Negative
Negative x Positive = Negative
Negative x Negative = Positive
Dividing negative numbers
Division follows the same sign rules as multiplication (since division = multiplication by a fraction):
Positive ÷ Positive = Positive
Positive ÷ Negative = Negative
Negative ÷ Positive = Negative
Negative ÷ Negative = Positive
Opposite numbers
2 numbers are opposites if their sum equals 0 (e.g., -2 & 2, -5 & 5) - equally far from 0 on the number
line. The opposite of a number is found by placing a minus sign in front of it. Note: −(−8) = +8, since
−− = +.
Absolute value
The absolute value of a number (written x) is always non-negative:
If the number is negative -> its opposite: − 5 = 5
If the number is non-negative -> itself: 5 = 5
e.g., − 65 = 65, 89 = 89, 12 − 19 = − 7 = 7
, 3-Fractions
Fractions
A fraction consists of a numerator (top) & denominator (bottom), separated by a line (e.g., 4/5 means 4
out of 5 equal parts).
Note: the denominator can never equal 0 (division by 0 is undefined).
Equivalent fractions
A fraction doesn't change if you multiply or divide both numerator & denominator by the same number.
e.g., 2/6 = 1/3, 3/6 = 1/2, 4/6 = 2/3
Negative fractions
Same sign rules as division:
Positive/Positive = Positive
Positive/Negative = Negative
Negative/Positive = Negative
Negative/Negative = Positive
Simplifying fractions
Simplification: reducing numerator & denominator to smaller numbers without changing the fraction's
value. A fraction is fully simplified when it contains no minus sign & can't be reduced further.
Minus signs: even number of minus signs -> cancel out; odd number -> one minus sign placed in front of
the fraction
Method: divide numerator & denominator by common factors until no further reduction is possible.
15/35 = 3/7
48/60 = 24/30 = 12/15 = 4/5
4/ − 6 = 2/ − 3 = −2/3
−3/ − 7 = 3/7
Adding & subtracting fractions with like denominators
When fractions share the same denominator, only the numerators change - the denominator stays the
same.
Addition: 1/5 + 3/5 = 4/5
Subtraction: 5/7 − 3/7 = 2/7
Writing fractions with like denominators
To add/subtract fractions with different denominators, first convert them to a common denominator by
multiplying numerator & denominator by the same number.
Easy method: multiply the 2 denominators together
