The natural numbers are the ordinary counting numbers.
The set of natural numbers is an infinite set, meaning that it is never-ending.
The letter N is used to label the set of natural numbers.
The set of natural numbers is an infinite set, meaning that it is never-ending.
The letter N is used to label the set of natural numbers.
2.1 Addition and Subtraction
Addition and Subtraction are known as operations. Adding natural numbers is finding their sum or total. Subtracting natural numbers is finding the difference between the two numbers or taking one from the other. Subtraction is the inverse operation to addition. Addition is the inverse operation to subtraction. An inverse operation reverses the effect of another operation. | Key Words
|
2.2 Factors and Multiples
Factors : A factor of a natural number is any natural number that divides evenly into the given number. e.g : The factors of 12 are {1, 2, 3, 4, 6, 12} Every factor is part of a pair. The number 12 has three factor pairs : 1 x 12 = 12 => Factor pairs are : 1 and 12 2 x 6 = 12 2 and 6 3 x 4 = 12 3 and 4 Indices (Powers) : If five 2's are multiplied together, i.e. 2 x 2 x 2 x 2 x 2, it can be written as 25 (pronounced '2 to the power of 5'). In this example, the 5 and the 6 are called indices or powers. 2 is a factor of 25. Prime Numbers :
Highest Common Factor (HCF) : The highest common factor of two numbers is the highest number that can divide into both of these numbers. Multiples : The multiples of 6 are {6, 12, 18, 24, 30, 36...} As you can see, it is an infinite set, i.e. it goes on forever. Lowest Common Multiple (LCM) : The LCM of 3 and 4 is 12, as 12 is the lowest number in which 3 and 4 both divide evenly into. |
The singular of indices is index. Prime numbers are natural numbers that have two factors only. Whole numbers that are not prime are called composite numbers. A multiple of a natural number is a number into which the natural number divides, leaving no remainder. The lowest common multiple of two numbers is the smallest multiple that both numbers share. |
2.3 Multiplication and Division
If you multiply two natural numbers together, then you are carrying out the operation of multiplication on two natural numbers. If we wish to find the area of a rectangle, we multiply the length of the rectangle by the width of the rectangle. So we can look at the multiplication of two numbers as finding an area. Multiplication is the inverse operation to division. Division is the inverse operation to multiplication. For example : 8 divided by 2 is 4; 8/2 = 4 4 multiplied by 2 is 8; 4 x 2 = 8 | As with addition and subtraction, multiplication and division are also known as operations. Multiplication reverses the operation of division. |
2.4 Commutative, Associative and Distributive Properties
There are three basic properties (laws) that we use with the operations of addition, subtraction, multiplication and division. Commutative Property Addition and subtraction are commutative operations on the natural numbers. 3 + 4 = 4 + 3 3 x 4 = 4 x 3 However, subtraction and division are not commutative operations on the natural numbers. 8 - 2 is not = 2 - 8 8 / 2 is not = 2 / 8 Associative Property Addition and subtraction are associative operations on the natural numbers. (3 + 4) + 5 = 3 + (4 + 5) (3 x 4) x 5 = 3 x (4 x 5) Subtraction and division are not associative operations on natural numbers. (8 - 4) - 2 is not = 8 - (4 - 2) (8 / 4) / 2 is not = 8 / (4 / 2) Distributive Property This property means that : 5(4 + 3) = (5 x 4) + (5 x 3) as 5(4 + 3) = 5(7) = 35 And (5 x 4) + (5 x 3) = 20 + 15 = 35 Also, 5(4 - 3) = 5(4) - 5(3) 5 (1) = 20 - 15 5 = 5 | An operation is commutative if a change in the order of the numbers does not change the result. An operation is associative if a change in grouping does not change the result. The distributive properties says that multiplication distributes over addition and subtraction. |
2.5 Order of Operations
In maths, the order of operations is the order in which things are done. It is important to have order, otherwise answers will be different. For example : is 2 + 3 x 4 = 5 x 4 = 20? or = 2 + 12 = 14? BIMDAS We can use the guide shown to help us remember the order in which operations are carried out. These letters stand for : Brackets Index Multiplication Division Addition Subtraction Therefore, Brackets come first, then Indices (powers), Multiplication/Division and finally Addition/Subtraction. | For MD and AS read left to right. |