Hello guys... My math exam is on 22 of September and I need a chapter full recap on rational numbers ;-; btw I'm ready to give away my 50 points to the people who answer sooooo... Be quick and plz answer it properly ;-;. I need to know about the all the properties plz.
:')
Answers
Step-by-step explanation:
Properties of Rational Numbers
The major properties of rational numbers are:
Closure Property
Commutativity Property
Associative Property
Distributive Property
Let us now study these properties in detail.
Closure Property
Properties of Rational Numbers
Source: Solving math problems
1) Addition of Rational Numbers
The closure property states that for any two rational numbers a and b, a + b is also a rational number.
12 + 34
= 4+68
= 108
Or, = 54
The result is a rational number. So we say that rational numbers are closed under addition.
2) Subtraction of Rational Numbers
The closure property states that for any two rational numbers a and b, a – b is also a rational number.
12 – 34
= 4–68
= −28
Or, = −14
The result is a rational number. So the rational numbers are closed under subtraction.
3) Multiplication of Rational Numbers
The closure property states that for any two rational numbers a and b, a × b is also a rational number.
12 × 34
= 68
The result is a rational number. So rational numbers are closed under multiplication.
4) Division of Rational Numbers
The closure property states that for any two rational numbers a and b, a ÷ b is also a rational number.
12 ÷ 34
= 1×42×3
= 23
The result is a rational number. But we know that any rational number a, a ÷ 0 is not defined. So rational numbers are not closed under division. But if we exclude 0, then all the rational numbers are closed under division.
Commutative Property
1. Addition
For any two rational numbers a and b, a + b = b+ a
−23+ 57 and 57+ −23 = 121
so, −23+ 57 = 57+ −23
We see that the two rational numbers can be added in any order. So addition is commutative for rational numbers.
2. Subtraction
For any two rational numbers a and b, a – b ≠ b – a. Given are the two rational numbers 53 and 14,
53 – 14 = 20−312
= 1712
But, 14 – 53 = 3−2012
= −1712
So subtraction is not commutative for ratioanl numbers.
3. Multiplication
For any two rational numbers a and b, a × b = b × a
−73+ 65 = 65+ −73
= −4215 = −4215
We see that the two ratrional numbers can be multiplied in any order. So multiplication is commutative for ratioanl numbers.
4. Division
For any two rational numbers a and b, a ÷ b ≠ b ÷ a. Given are the two rational numbers 53 and 14
53 ÷ 14 = 5×43×1
= 203
But, 14 ÷ 53 = 1×34×5
= 320
We see that the expressions on both the sides are not equal. So divsion is not commutative for ratioanal numbers.
Associative Property
Take any three rational numbers a, b and c. Firstly add a and b and then add c to the sum. (a + b) + c. Now again add b and c and then a to the sum, a + (b + c). Is (a + b) + c and a + (b + c) same? Yes and this is how associative property works. It states that you can add or multiply numbers regardless of how they are grouped.
For example, given numbers are 5, -6 and 23
( 5 – 6 ) + 23
= -1 + 23
= – 13
Now, 5 + ( -6 + 23 )
= – 13
In both the groups the sum is the same.
Addition and multiplication are associative for rational numbers.
Subtraction and division are not associative for rational numbers.
Distributive Property
Distributive property states that for any three numbers x, y and z we have
x × ( y + z ) = (x × y) +( x × z)
Solved Examples for You
Question: …………….. are not associative for rational numbers.
Addition and multiplication
Subtraction and multiplication
Subtraction and division
Addition and division
Solution: C. When all three rational numbers are subtracted or divided in an order, the result obtained will change if the order is changed. So, subtraction and division are not associative for rational numbers.
Step-by-step explanation:
please refer to the attachment
