Question

1. Brief properties of Addition and subtraction of integers with examples
on A-4 sheets and attach with this worksheet.​

Answer 1

Answer:

Step-by-step explanation:

A minus in front of a number changes the sign of the number.

To get a grasp of this rule, we’ll call a couple of old friends to our aid – the number line and the multiplication of natural numbers. Remember how multiplying a number by the number 1 gives you that same number as a result? Well, putting a minus in front of a number is shorthand for multiplying that number by -1. The distance from the origin point on the number line stays the same, but the minus shifts it to the opposite side of the number line.

 

So, if we put a minus in front of a positive integer, we’ll get a negative version of that same integer. And if we put the minus in front of a negative integer, we’ll get its positive version as a result.

Using just mathematical language, that means that:

2⋅(−1)=−2

and

−2⋅(−1)=2.

 

2. If a negative integer is behind an operator, it has to be surrounded by parentheses.

This one is here to avoid confusion, because the minus sign is also the operator for subtraction. If we put two operators next to each other, it is unclear if:

one of them is a sign, and not an operator

one of them is a typo, or

a number or a variable is missing between them.

To make things easier, a rule has been created to put negative integers into brackets (parentheses). That way, everybody knows that the minus is there on purpose and that it is a sign.

For example: −3+(−5)=−8⇒–3–5=−8

Although mistakes can be avoided during addition and subtraction by using rule number one, this rule will be indispensable during multiplication.

 

3. Adding two negative integers together will always give you a negative integer as a result.

A negative integer represents the distance from a single point positioned left of the point of origin on the number line to the point of origin itself. When we add two negative integers together, we basically get the sum of their distances. But, since both of them are positioned left of the point of origin on the number line, we keep that direction. Like this:

hope this will help u

Answer 2

Mathematical equations have their own manipulative principles. These principles or properties help us to solve such equations. The properties of integers are the basic principle of the mathematical system and it will be used throughout the life. Hence, it’s very essential to understand how to apply each of them to solve math problems. Basically, there are three properties which outline the backbone of mathematics. They are:

*Associative property

*Commutative property

*Distributive property

All properties and identities for addition and multiplication of whole numbers are applicable to integers also. Integers include the set of positive numbers, zero and negative numbers which can be represented with the letter Z.

Z = {……….−5,−4,−3,−2,−1,0,1,2,3,4,5………}

Some of the Properties of Integers are given below:

Property 1: Closure property

Among the various properties of integers, closure property under addition and subtraction states that the sum or difference of any two integers will always be an integer i.e. if x and y are any two integers, x + y and x − y will also be an integer.

Ex: 3 – 4 = 3 + (−4) = −1;

(–5) + 8 = 3,

The results are integers.

Closure property under multiplication states that the product of any two integers will be an integer i.e. if x and y are any two integers, xy will also be an integer.

Ex: 6 × 9 = 54 ; (–5) × (3) = −15, which are integers.

Division of integers doesn’t follow the closure property, i.e. the quotient of any two integers x and y, may or may not be an integer.

Ex : (−3) ÷ (−6) = 12, is not an integer.

Property 2: Commutative property

Commutative property of addition and multiplication states that the order of terms doesn’t matter, result will be same. Whether it is addition or multiplication, swapping of terms will not change the sum or product. Suppose, x and y are any two integers, then

⇒ x + y = y + x

⇒ x × y = y × x

Ex: 4 + (−6) = −2 = (−6) + 4;

10 × (−3) = −30 = (−3) × 10

But, subtraction (x − y ≠ y − x) and division (x ÷ y ≠ y ÷ x) are not commutative for integers and whole numbers.

Ex: 4 − (−6) = 10 ; (−6) – 4 = −10

⇒ 4 − (−6) ≠ (−6) – 4

Ex: 10 ÷ 2 = 5 ; 2 ÷ 10 = 15

⇒ 10 ÷ 2 ≠ 2 ÷ 10

Property 3: Associative property

Associative property of addition and multiplication states that the way of grouping of numbers doesn’t matter; the result will be same. One can group numbers in any way but the answer will remain same. Parenthesis can be done irrespective of the order of terms. Let x, y and z be any three integers, then

⇒ x + (y + z) = (x + y) +z

⇒ x × (y × z) = (x × y) × z

Ex: 1 + (2(−3)) = 0 = (1 + 2) + (−3);

1 × (2 × (−3)) =−6 = (1 × 2) × (−3)

Subtraction of integers is not associative in nature i.e. x − (y − z) ≠ (x − y) − z.

Ex: 1 − (2 − (−3)) = −4; (1 – 2) – (−3) = −21 – (2 – (−3)) ≠ (1 − 2) − (−3)

Property 4: Distributive property

Distributive property explains the distributing ability of an operation over another mathematical operation within a bracket. It can be either distributive property of multiplication over addition ordistributive property of multiplication over subtraction. Here integers are added or subtracted first and then multiplied or multiply first with each number within the bracket and then added or subtracted.This can be represented for any integers x, y and z as:

⇒ x × (y + z) = x × y + x × z

⇒ x × (y − z) = x × y − x × z

Ex: −5 (2 + 1) = −15 = (−5 × 2) + (−5 × 1)

Property 5: Identity Property

Among the various properties of integers, additive identity property states that when any integer is added to zero it will give the same number. Zero is called additive identity. For any integer x,

x + 0 = x = 0 + x

Multiplicative identity property for integers says that whenever a number is multiplied by the number 1 it will give the integer itself as the product. Therefore, the integer 1 is called the multiplicative identity for a number. For any integer x,

x × 1 = x = 1 × x

If any integer multiplied by 0, product will be zero:

x × 0 = 0 =0 × x

If any integer multiplied by -1, product will be opposite of the number:

x × (−1) = −x = (−1) × x<

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