with the help of numbers show that a - b is greater than a + (-b). Give two examples. Can we use integers to show a - b is greater than b - a. Explain in your answer
Answers
hi friend
Step-by-step explanation:
nteger Numbers and Mathematical Operations
Mathematical operations include addition, subtraction, multiplication, and division of any number. When we perform these operations with integer numbers we always keep in mind the sign before every number.
As we already know that an integer includes a number with a positive or negative sign, therefore, these have to be dealt with different perceptions. Before delving into further operations, we first need to know the properties related to these mathematical operations.
Addition and Subtraction of Integer numbers
integer numbers
Source: emaze
Property of Closure in Addition and Subtraction
When we add two whole numbers, the answer is a whole number. This is called the closure property of additions. Now, do integers also follow this property? Let’s see:
10 + (-12) = -2
Is -2 an integer number. Since it has a negative sign with it, therefore we call it an integer number. From the above example, we can say that the sum of two integers is also an integer. So when an integer a is added with another integer b the answer is always an integer.
Let’s take some more examples to support our observation:
19+10= +29
15 + (-20)= -5
In both the examples, the solution is an integer. Therefore every integer when summed up with an integer gives an integer. What happens when an integer is subtracted from an integer. Let’s see the following examples:
10- 3 = +7
15 – (-7) = -22
In both the examples, the difference between two integers gives an integer in solution. This means that integers are also closed under subtraction. Hence, in the case of subtraction of two integers a and b the answer to a-b is also an integer.
Commutative Property
Commutative property states the following:
a+b =b+a
For whole numbers, we already know that the sum of two whole numbers is always the same. But is the case true for integer numbers as well? The sum of two integer numbers is also, always the same. This means that integer numbers also follow the commutative property like whole numbers. Let’s see the following examples:
15 + 20 =35; 20 +15=35
-10 + (-5) = -15; -5 + (-10) = -15
The above examples prove that the addition of integers is commutative. Is the case true with subtractions? Are subtractions also commutative? The following examples will let us know this:
5-(-3) = +8
-3-5 = -8
Are the integer numbers same? The answer is a no! This brings us to a conclusion that subtractions of integers are not commutative. Therefore, a-b ≠ b-a
Associative Property
According to associative property of whole numbers a,b,c,
[a+b]+c =a+[b+c]
Does this property also apply to integer numbers? Let’s see:
[(-4)+(-6)]+(-2) = -12
(-4) + [(-6)+(-2)] = -12
The answer in both the cases is the same. So the sum of integer numbers abides the associative property of addition.
Additive Identity
The additive identity of any number is checked with the help of zero. For all the whole numbers, zero proves to be their additive identity. This means that when a whole number is added with a zero, the answer is the whole number itself. Is the case same with integer numbers as wel?. The answer here is a yes. The following examples prove our observation:
+10 +0 = +10
-9+0=-9
Therefore for any integer, x,
x+0 = x = 0+x
Multiplication of Integers
After addition and subtraction, we need to understand the multiplication of two integer numbers. Let’s look at some of its properties.
Closure Property
Is the product of two integer numbers also an integer number? Well, yes the product of every integer is an integer.
Commutative Property
The product of two same integers is always the same. This means that
a × b = b× a
Multiplication with Zero
Every integer, when multiplied with a zero, gives zero as the answer.
a×0 = 0
Multiplicative Identity
Every integer number, when multiplied with 1, gives an integer number in an answer.
a×1 = a
(-a) ×1 = -a
Associative Property
Integer numbers also abide by the associative property of multiplication. This implies that like whole numbers the grouping of integers does not affect the product of integers. Thus,
(a×b)×c = (a) ×(b×c)
Distributive Property
Here we shall check the distributivity of integers over addition is true or not.
(-4) × (2 + 6) = -32
(-4) × (2) + (-4) ×(+6) =(-8) +(-24) = -32
The example above shows that multiplication of integers also shows distributivity of multiplication over addition. Thus,
a × (b+c) = (a×b) + (a×c)
plzzz mark theanswer as braniliest
Step-by-step explanation:
Operations on Integer Numbers
Integer numbers are whole numbers. These can be negative, positive or a zero. When we perform mathematical operations of integers, we follow a different set of rules that are specific to the character of the number. Not every rule of mathematical operation applies to the integers. Let’s see how handling integers are different from the normal mathematical operations.
Introduction to Integers
Properties of Integers
Addition and Subtraction of Integers
Integer Numbers and Mathematical Operations
Mathematical operations include addition, subtraction, multiplication, and division of any number. When we perform these operations with integer numbers we always keep in mind the sign before every number.
As we already know that an integer includes a number with a positive or negative sign, therefore, these have to be dealt with different perceptions. Before delving into further operations, we first need to know the properties related to these mathematical operations.
Addition and Subtraction of Integer numbers
integer numbers
Source: emaze
Property of Closure in Addition and Subtraction
When we add two whole numbers, the answer is a whole number. This is called the closure property of additions. Now, do integers also follow this property? Let’s see:
10 + (-12) = -2
Is -2 an integer number. Since it has a negative sign with it, therefore we call it an integer number. From the above example, we can say that the sum of two integers is also an integer. So when an integer a is added with another integer b the answer is always an integer.
Let’s take some more examples to support our observation:
19+10= +29
15 + (-20)= -5
In both the examples, the solution is an integer. Therefore every integer when summed up with an integer gives an integer. What happens when an integer is subtracted from an integer. Let’s see the following examples:
10- 3 = +7
15 – (-7) = -22
In both the examples, the difference between two integers gives an integer in solution. This means that integers are also closed under subtraction. Hence, in the case of subtraction of two integers a and b the answer to a-b is also an integer.
Commutative Property
Commutative property states the following:
a+b =b+a
For whole numbers, we already know that the sum of two whole numbers is always the same. But is the case true for integer numbers as well? The sum of two integer numbers is also, always the same. This means that integer numbers also follow the commutative property like whole numbers. Let’s see the following examples:
15 + 20 =35; 20 +15=35
-10 + (-5) = -15; -5 + (-10) = -15
The above examples prove that the addition of integers is commutative. Is the case true with subtractions? Are subtractions also commutative? The following examples will let us know this:
5-(-3) = +8
-3-5 = -8
Are the integer numbers same? The answer is a no! This brings us to a conclusion that subtractions of integers are not commutative. Therefore, a-b ≠ b-a
Associative Property
According to associative property of whole numbers a,b,c,
[a+b]+c =a+[b+c]
Does this property also apply to integer numbers? Let’s see:
[(-4)+(-6)]+(-2) = -12
(-4) + [(-6)+(-2)] = -12
The answer in both the cases is the same. So the sum of integer numbers abides the associative property of addition.
Additive Identity
The additive identity of any number is checked with the help of zero. For all the whole numbers, zero proves to be their additive identity. This means that when a whole number is added with a zero, the answer is the whole number itself. Is the case same with integer numbers as wel?. The answer here is a yes. The following examples prove our observation:
+10 +0 = +10
-9+0=-9
Therefore for any integer, x,
x+0 = x = 0+x
Multiplication of Integers
After addition and subtraction, we need to understand the multiplication of two integer numbers. Let’s look at some of its properties.
Closure Property
Is the product of two integer numbers also an integer number? Well, yes the product of every integer is an integer.
Commutative Property
The product of two same integers is always the same. This means that
a × b = b× a
every incorrect answer.