Multiplying Integers
 

Integers for KS3



To put it simple, integers are just extended natural numbers, the set of natural numbers, add 0 and negative natural numbers. The set of integers is denoted by:




but




which is exactly , while




and we see that it is and we say that the set of natural numbers consists of the set of positive integers (which is the set of natural integers), the set of negative integers, and the number zero.


Presenting integers on a number line



Since integers are just extended natural numbers, they can be represented on number line. We extend it by mirroring all natural numbers over the starting point - making all integers to the left of the starting point negative.


Extend the line with natural numbers to the left:




A unit to the left, we get




One unit to the left of , we get




We repeat the process by applying the unit evenly to the left to obtain the remaining negative numbers.




The number is called the additive inverse of . Natural numbers are positive integers, and mirrored natural numbers are additive inverse.





Arithmetic operations of integers



Addition and multiplication of integers are defined similarly to addition and multiplication of natural numbers. In addition to these two basic arithmetic operations, we also define subtraction in the set of integers.


Subtraction of Integers



The subtraction is defined as the addition of the additive inverse: , where is called the minuend, is the subtrahend and the result of is the difference.


Properties of arithmetic operations



For integers, in addition to properties of arithmetic operations natural numbers, some rules also apply.


Existence of an identity element



When adding to any integer, the result does not change.


Identity element:




for each of



The sum of any integer and its additive inverse



The sum of any integer and its additive inverse equal .




The negation of an additive inverse is its positive value



The additive inverse of is .




The additive inverse of a sum



The negation of a sum is equal to the sum of its additive inverses.




Identity element for multiplication



The number is the multiplication identity element.




Multiplication by (-1)



Multiplying by gives the additive inverse of the number .




Multiplication by 0



When multiplying any number by , the result is always equal to .




The product of positive and negative integers



The product of two negative integers is positive.




The product of two positive integers is positive.




The product of a negative and a positive integer



The product of a negative and a positive is negative.




Removing brackets



When calculating, we follow the order of arithmetic operations (multiplication takes precedence over addition and subtraction), and in the case of brackets, we always eliminate these first. In doing so, we must pay attention to the signs:


If the brackets are preceded by a plus sign (+), the numbers in the brackets retain their sign:




and





If the brackets are preceded by a minus sign (-), the numbers in brackets change their sign:




and




Order of integers



In a set of integers, we can compare any two numbers with each other and determine whether they are the same (=) or which of them is greater (>) or less (<). The inequalities less than or equal to (<=) or greater than or equal to (> =) can also apply.


Greater than " > "



For any integers and , is greater than , which is written with symbols:




if and only if




or if lies to the right of on the number line.



Example

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Less than " < "



For any integers and , is less than , which is written with symbols:




if and only if




or if lies on the left side of on the number line.



Example

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Equals to " = "



For any integers and , is equal to , which is written with symbols:




if and only if




or if coincides with on the number line.



Example

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Less than or equal to " <= "



For any integers and , is less than or equal to , which is written with symbols:




if and only if




or if lies on the left side of or coincides with it on the number line.



Example

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Greater than or equal to " >= "



For any integers and , is greater than or equal to , which is written with symbols:




if and only if




or if lies on the right side of or coincides with it on the number line.



Example

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Properties of "Less than"



  • If we add the same number on both sides of the inequality, the inequality is maintained:


    if




    then it is true for




  • "Less than" is transitive:


    if and , then



  • When multiplying inequalities by a positive number, the inequality sign is retained:


    if and then



  • When multiplying inequality by a negative number, the inequality sign is reversed:


    if and then




Properties of "less than or equal to"



  • "Less than or equal to" is reflexive:


    if



  • "Less than or equal to" is antisymmetric:


    if and , then



  • "Less than or equal to" is transitive:


    if and , then




Properties of "greater than or equal to"



  • "Greater than or equal to" is reflexive:


    if



  • "Greater than or equal to" is antisymmetric:


    if and , then



  • "Greater than or equal to" is transitive:


    if and , then




material editor: ISMAIL SHOBOLA