write properties of integer over subtraction with example
Answers
Answer:
(- )+ (-) = +
(- )+ (+) = -
(+) + (-) = -
(+) + (±) = +
Step-by-step explanation:
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Answer:
CLOSURE PROPERTY OF INTEGERS - DEFINITION
Closure property says that if for any two integers a and b, a∗b is also an integer then the set of integers is closed under ∗
where ∗ represents +,−,× or ÷
For example:- Take 4 and 8
Now, 4+8=12 is an integer
4×8=32 is an integer
4−8=−4 is an integer
But 4÷8=
8
4
=
2
1
is not an integer
Hence, set of integer is closed under +,−× but not closed under ÷.
COMMUTATIVE PROPERTY OF INTEGERS - DEFINITION
Take any two numbers a and b in your mind. Now add a and b, which comes as a+b.
Add b and a, which comes to be b+a.
Aren't they same ?
Yes, they are equal.
This is because of commutative property.
So, let's have a look at commutative property of numbers which says that we can swap the numbers and still we get the same answer.
It is a property that associates with binary operations or functions like addition, multiplication.
COMMUTATIVE PROPERTY OF INTEGERS - DEFINITION
What about subtraction of numbers ?
Take a and b as two integers and subtract them i.e. a−b.
Now, subtract a from b i.e. b−a.
Are they same ?
No, they are not equal.
So, commutative property does not hold for subtraction.
Similarly, it does not hold for division.
ASSOCIATIVE PROPERTY OF INTEGERS - DEFINITION
Associative property states that, for any three elements(numbers) a,b and c we have
a∗(b∗c)=(a∗b)∗c, where ∗ represents a binary operation.
Let's take ∗ as addition(+)
Then, we have a+(b+c)=(a+b)+c
For eg:- For 2,5 and 11
2+(5+11)=2+16=18 and (2+5)+11=7+11=18
For multiplication
2×(5×11)=2×55=110 and (2×5)×11=10×11=110
Hence, a∗(b∗c)=(a∗b)∗c is true for addition and multiplication.
ASSOCIATIVE PROPERTY OF INTEGERS - EXAMPLE
What about subtraction and division ?
Associative property does not hold for subtraction and division
Let's take an example :
For 4,6 and 12
4÷(6÷12)=4÷
12
6
=4÷
2
1
=
2
1
4
=4×2=8 and
(4÷6)÷12=
6
4
÷12=
3
2
÷12=
12
3
2
=
3×12
2
=
3×6
1
=
18
1
=8
Therefore, a∗(b∗c)=(a∗b)∗c is not true for division.
Also, 4−(6−12)=4−(−6)=4+6=10 and (4−6)−12=−2−12=−14
=10
Hence, a∗(b∗c)=(a∗b)∗c is not true for subtraction as well.
CLOSURE PROPERTY IN REFERENCE TO INTEGERS - DEFINITION
System of Integers under Addition:
Addition of two Integers always results in an Integer.
Eg:
7+4=11, Result is an Integer.
Therefore, system is closed under addition.
System of Integers under Subtraction:
Subtraction of two Integers always results in an Integer.
Eg:
7−4=3, Result is an Integer, and
2−4=−2, Result is also an integer.
Therefore, system is closed under subtraction.
System of integers under Multiplication:
Multiplication of two integers always results in an integers.
Eg:
7×4=28, Result is an Integer
Therefore, system is closed under Multiplication.
System of Integers under Division:
Division of two integers does not always results in an integer.
Eg:
8÷4=2, Result is an integer, but
7÷4=
4
7
, Result is not an integer.
Therefore, system is not closed under division.
PROBLEMS ON NEGATIVE NUMBERS - EXAMPLE
Example:
Sunny walk 2 metre from his house towards the garden and then comes back 1.5 m. Then find the distance between the house and current position of Sunny.
Solution:-
Distance covered by Sunny towards garden =2 m
Negative distance from garden till Sunny =−1.5 m
So, the total distance covered =2+(−1.5)=0.5 m