MATHEMATICS
, Key – core concepts
REAL NUMBERS NATURAL NUMBERS
NUMBERS “FAMILY”/SYSTEM
The number line consists of all the Rational
(Q) and Irrational (Q') numbers which
Even numbers : 2;4;6;8….
together form the set of Real (ℝ) numbers.
Odd numbers : 1;3;5;7…
UNREAL NUMBERS
Prime numbers : 2;3;5;7;11;13… (1 is not a prime
√−25 is an example of a non-real number. number)
There is no number that, when squared, will
Composite numbers : 4;6;8;9;10;12 (numbers with
equal -25.
more than 2 factors)
√−25 does not exist on the number line. Square numbers : 1;4;9;16;25…
The real and non-real numbers together
form the complex numbers. Cubic numbers : 1;8;27;64;125…
COUNTING NUMBERS
INTEGERS
RASIONAL NUMBER
any number that can be written as a fraction, i.e. a
𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝐴
or [where A & B ∈ Z ; B ≠ 0]
𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝐵
W All integers.
W All fractions.
W All terminating decimal fractions.
W All repeating decimal fractions.
IRRASIONAL NUMBERS
W All non-terminating, non-repeating decimal
Can only be written in a number form with infinite, numbers
non-repeating digits after the decimal point, the W Pi (π)
numbers can NOT be written as a FRACTION W √Positive non − square
3
W √Positive non − cubic
, Key – core concepts
BODMAS CALCULATIONS WITH BRACKETS ()
NB ! At minus - and brackets ()
Inside the bracket (-3)² = (-3)(-3) =+9
No bracket -3² =-(3)(3) = -9
Outside the bracket -(3)² =-(3)(3) = -9
CALCULATIONS WITH ROOTS √
+ or – under √ [first SIMPLIFY under the √]
W √16 + 9=√25
W √16 + 9≠√16 + √9
x or ÷ under the √ [divide √ and work out each
part's √ separately]
W √16 × 9=√144
SIGN RULES [X ÷] W √16 × 9=√16 × √9
, Key – core concepts
ADDITION + * SUBTRACTION - SCM * LCD
W Same signs (add together and KEEP the sign) SCM (Smallest Common Multiple )
W Different signs (subtract and keep the largest
(ALSO SCD = smallest common denominator)
number's sign)
MULTIPLY THAT WHICH APPEARS WITH THE BIGGEST ONE
Characters inside and outside brackets (multiply together to
AND THAT WHICH IS SHORT [FRACTIONAL] (largest
get a new character. Two signs next to each other must
number's factors and what remains with the other)
become one sign)
Determine the SCM of 9 and 12:
INVERSES
Multiples of 9 are: 9; 18; 27; 36; 45; 54; 63; …
addition inverse of -4 = 4
Multiples of 12 are: 12; 24; 36; 48; 60; 72; …
multiplicative inverse of -2 = -½ The SCM of 9 and 12 is: 36
MULTIPLES * FACTORS GCF / HCF (Greatest [Highest] common denominator /
factor)
MULTIPLES MULTIPLY ONLY WHAT IS COMMON TO ALL
[count in the number] (numbers common to all)
E.G : 2, 4, 6, 8, 10 [MULTIPLES OF 2] Determine the SCM of 8 and 10.
FACTORS Step 1. What is the LCF of the 2 numbers: It is 2
Factors of 8: 1; 2; 4; 8 Factors of 10: 1; 2; 5; 10
[numbers that can divide exactly into the number]
Step 2. Determine the product of 8 and 10: 8 x 10 = 80
E.G : 1, 2, 3, 6, 9, 18 [FACTORS OF 18]
Step 3. Divide the product of the numbers by the LCF: 80 ÷ 2 = 40
PRIME NUMBERS
Step 4. The SCM of 8 and 10 is 40
[number with only 2 factors namely 1 and the number itself]
[2, 3, 5, 7, 11, 13, 17, 19]
, Key – core concepts
REAL NUMBERS NATURAL NUMBERS
NUMBERS “FAMILY”/SYSTEM
The number line consists of all the Rational
(Q) and Irrational (Q') numbers which
Even numbers : 2;4;6;8….
together form the set of Real (ℝ) numbers.
Odd numbers : 1;3;5;7…
UNREAL NUMBERS
Prime numbers : 2;3;5;7;11;13… (1 is not a prime
√−25 is an example of a non-real number. number)
There is no number that, when squared, will
Composite numbers : 4;6;8;9;10;12 (numbers with
equal -25.
more than 2 factors)
√−25 does not exist on the number line. Square numbers : 1;4;9;16;25…
The real and non-real numbers together
form the complex numbers. Cubic numbers : 1;8;27;64;125…
COUNTING NUMBERS
INTEGERS
RASIONAL NUMBER
any number that can be written as a fraction, i.e. a
𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝐴
or [where A & B ∈ Z ; B ≠ 0]
𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝐵
W All integers.
W All fractions.
W All terminating decimal fractions.
W All repeating decimal fractions.
IRRASIONAL NUMBERS
W All non-terminating, non-repeating decimal
Can only be written in a number form with infinite, numbers
non-repeating digits after the decimal point, the W Pi (π)
numbers can NOT be written as a FRACTION W √Positive non − square
3
W √Positive non − cubic
, Key – core concepts
BODMAS CALCULATIONS WITH BRACKETS ()
NB ! At minus - and brackets ()
Inside the bracket (-3)² = (-3)(-3) =+9
No bracket -3² =-(3)(3) = -9
Outside the bracket -(3)² =-(3)(3) = -9
CALCULATIONS WITH ROOTS √
+ or – under √ [first SIMPLIFY under the √]
W √16 + 9=√25
W √16 + 9≠√16 + √9
x or ÷ under the √ [divide √ and work out each
part's √ separately]
W √16 × 9=√144
SIGN RULES [X ÷] W √16 × 9=√16 × √9
, Key – core concepts
ADDITION + * SUBTRACTION - SCM * LCD
W Same signs (add together and KEEP the sign) SCM (Smallest Common Multiple )
W Different signs (subtract and keep the largest
(ALSO SCD = smallest common denominator)
number's sign)
MULTIPLY THAT WHICH APPEARS WITH THE BIGGEST ONE
Characters inside and outside brackets (multiply together to
AND THAT WHICH IS SHORT [FRACTIONAL] (largest
get a new character. Two signs next to each other must
number's factors and what remains with the other)
become one sign)
Determine the SCM of 9 and 12:
INVERSES
Multiples of 9 are: 9; 18; 27; 36; 45; 54; 63; …
addition inverse of -4 = 4
Multiples of 12 are: 12; 24; 36; 48; 60; 72; …
multiplicative inverse of -2 = -½ The SCM of 9 and 12 is: 36
MULTIPLES * FACTORS GCF / HCF (Greatest [Highest] common denominator /
factor)
MULTIPLES MULTIPLY ONLY WHAT IS COMMON TO ALL
[count in the number] (numbers common to all)
E.G : 2, 4, 6, 8, 10 [MULTIPLES OF 2] Determine the SCM of 8 and 10.
FACTORS Step 1. What is the LCF of the 2 numbers: It is 2
Factors of 8: 1; 2; 4; 8 Factors of 10: 1; 2; 5; 10
[numbers that can divide exactly into the number]
Step 2. Determine the product of 8 and 10: 8 x 10 = 80
E.G : 1, 2, 3, 6, 9, 18 [FACTORS OF 18]
Step 3. Divide the product of the numbers by the LCF: 80 ÷ 2 = 40
PRIME NUMBERS
Step 4. The SCM of 8 and 10 is 40
[number with only 2 factors namely 1 and the number itself]
[2, 3, 5, 7, 11, 13, 17, 19]
