Rational numbers room the collection of integers, entirety numbers, and also natural numbers. Rational numbers space numbers that have the right to be represented infractionform. Castle all have the right to be represented as rational numbers of the kind p/q or asterminating decimal numbers, or as non-terminating but repeating decimal numbers. Properties of rational numbers space the general propertiessuch as associative, commutative, distributive, and closure properties. Let us read about all the nature of reasonable numbers.

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1.What are The properties of rational Numbers?
2.Closure building of reasonable Numbers
3.Commutative property of rational Numbers
4.Associative residential or commercial property of reasonable Numbers
5.Distributive residential property of reasonable Numbers
6.Additive property of reasonable Numbers
7.Multiplicative home of rational Numbers
8.Properties that Rational numbers Examples
9.Practice Questions
10.FAQ's on nature of rational Numbers

WhatAre The properties of rational Numbers?


When numbers expressed in the type of p/q, then they room consideredrational numbers, hereboth p and q are integers and also q ≠ 0. Over there are 6 properties of rational numbers, i beg your pardon are noted below:

Closure PropertyCommutative PropertyAssociative PropertyDistributive PropertyMultiplicative PropertyAdditive Property

Let us discover these properties on the four arithmeticoperations (Addition, subtraction, multiplication, and division) in mathematics.


Closure Propertyof rational Numbers


The closure building of reasonable numbers states that when any two reasonable numbers are added, subtracted, or multipliedtheresult ofall three instances willalso it is in a rational number. Let us read about how the closure residential property of rational numbers functions on all the simple arithmeticoperations. Us will know this residential or commercial property on each operation using variousillustrations.

Let us take 2 rational numbers 1/3 and also 1/4, and perform an easy arithmetic operations on them.

For Addition: 1/3+ 1/4= (4+ 3)/12= 7/12. Here, the result is 7/12, which is a rational number. We say that rational numbers are closed under addition. The is, for any type of two rational number a and b, (a + b) is likewise a rational number.

For Subtraction: 1/3 -1/4 = (4-3)/12= 1/12.Here, the result is 1/12, i m sorry is a rational number. We say the rational numbers space closed under subtraction. The is, for any two rational number a and also b, (a -b) is additionally a reasonable number.

For Multiplication:1/3× 1/4= 1/12.Here, the result is 1/12, i beg your pardon is a rational number. Us say that rational numbers are closed under multiplication. That is, for any type of two rational numbers a and b, (a × b) is also a rational number.

For Division: 1/3 ÷ 1/4 = 4/3. Here, the an outcome is 4/3, i beg your pardon is a reasonable number. But we find that for any kind of rational number a, a ÷ 0 is not defined. So rational numbers are not closed under division. However, if us exclude zero climate the arsenal of, all various other rational numbers space closed under division.


CommutativePropertyof reasonable Numbers


The commutativeproperty of reasonable numbers claims that when any type of two reasonable numbers space addedor multiplied in any order it does not adjust theresult. However in the instance of subtraction and department if the bespeak of the numbers is adjusted then the an outcome will additionally change.We will know this home on each procedure using variousillustrations.

Let united state again take 2 rational number 1/3 and 1/4, and also perform an easy arithmetic to work on them.

For Addition: 1/3 + 1/4 = 1/4 + 1/3= 7/12.We speak that enhancement is commutative because that rational numbers. That is, for any kind of two rational number a and also b, a + b = b + a.

For Subtraction: 1/3 - 1/4 ≠1/4 - 1/3= 1/12≠ -1/12.You will discover that subtraction is no commutative because that rational numbers. That is, for any kind of two rational number a and b, a - b ≠ b - a.

For Multiplication:1/3 × 1/4 = 1/4× 1/3 = 1/12. Friend will uncover that multiplication is commutative because that rational numbers. In general, a × b = b × a for any type of two rational number a and also b.

For Division: 1/3 ÷ 1/4 ≠ 1/4 ÷ 1/3 = 4/3≠ 3/4. Friend will uncover that expressions on both sides room not equal. In general, a ÷ b ≠ b ÷ a for any kind of two rational number a and b. So department is not commutative because that rational numbers.


AssociativePropertyof reasonable Numbers


The associativeproperty of rational numbers says that when any threerational numbers are added or multiply the resultremains the exact same irrespective that the method numbers are grouped. Yet in the case of individually and division if the stimulate of the numbers is readjusted then the result will likewise change.We will understand this home on each procedure using assorted illustrations.

For Addition: For any kind of threerational number associative building for addition is given as A, B,and C, (A + B) + C = A + (B + C). Because that example, (1/3 + 1/4) + 1/2 = 1/4 + (1/3 + 1/2)= 13/12.We to speak that addition is associativefor rational numbers.

For Subtraction: For any three reasonable numbersassociative residential property for subtractionis given as A, B, and C, (A - B) - C ≠ A - (B - C). For example,(1/3 - 1/4) - 1/2 ≠1/3- (1/4- 1/2) = 1/24≠ 1/12.You will find that subtraction is not associative for rational numbers.

For Multiplication: For any three rational numbers associative home for multiplicationis provided as A, B,and C, (A × B) × C = A × (B × C). Because that example,(1/3 × 1/4)× 1/2 = 1/4× (1/3× 1/2) = 1/24 = 1/24. Girlfriend will find that multiplication is associative because that rational numbers.

For Division:For any threerational number associative building for divisionis offered as A, B, and also C, (A ÷ B) ÷ C ≠ A ÷ (B ÷ C). Because that example, (1/3 ÷ 1/4)÷ 1/2≠ 1/4 ÷ (1/3 ÷ 1/2) = 8/3≠ 2/3. Friend will uncover that expressions on both sides room not equal. So division is not associative for rational numbers.


Distributive Propertyof rational Numbers


The distributiveproperty of reasonable numbers claims that any expression with three rational number A, B, and C,givenin formA(B+ C) climate it is solved as A× (B+ C) = ab + AC or A (B – C) = ab – AC. This method operand A is distributed among the other two operands i.e., B and also C. This building is also known together the distributivity the multiplication over addition orsubtraction. Let us learnhow the distributiveproperty of rational numbers works. We will recognize this residential or commercial property using theillustration offered below.

Solve: 1/2(1/6+ 1/5)

Solution:The provided expression is that the formA (B + C) = A × (B + C) = abdominal + AC1/2(1/6 + 1/5) = 1/2× 1/6 + 1/2× 1/5 = 11/60

Let us deal with the exact same expression through subtraction.

Solve: 1/2(1/6 -1/5)

Solution:The offered expression is of the formA (B -C) = A × (B -C) = abdominal - AC1/2(1/6 -1/5) = 1/2× 1/6 -1/2× 1/5 = -1/60


Additive residential property of rational Numbers


There are two simple additive properties of rational numbers, additive identity, and also additive inverse.

For any kind of rational number a/b, b≠ 0 the relationship between additive identity and also the additive inverse is provided as:

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This home is an extremely useful in solving facility calculations.Let us know the additive identity and additive inversewith the help of examples.

Additive Identity

The additive identification property of reasonable numbers claims that thesum of any type of rational number (a/b) and also zero is the reasonable number itself. Suppose a/b is any kind of rational number, climate a/b + 0 = 0 + a/b = a/b. Here, 0is the additive identity for rational numbers. Let us understand this through an example:3/7 + 0 = 0 + 3/7 = 3/7

Additive Inverse

The additive inverseproperty of reasonable numbers claims that if a/b is a rational number, climate there exists a rational number (-a/b) together that, a/b + (-a/b)= (-a/b) + a/b = 0.For example,the additive train station of 3/7is (-3/7).(3/7) + (-3/7) = (-3/7) + 3/7 = 0.


MultiplicativePropertyof rational Numbers


There are two simple multiplicative properties of rational numbers, multiplicative identity, and also multiplicative inverse. This residential property is also an extremely useful in solving complex calculations. Permit us recognize the two through examples.

Multiplicative Identity

The additive identification property of rational numbers claims that the product of any kind of rational number and also 1 is the reasonable number itself. Here, 1is the multiplicative identification for reasonable numbersthat expressed in a/b form. If a/b is any kind of rational number, then a/b × 1 = 1 × a/b= a/b. Because that example:5/3×1= 1 × 5/3= 5/3.

Multiplicative Inverse

The multiplicative inverseproperty of reasonable numbers claims that for every reasonable number a/b, b ≠ 0, over there exists a reasonable number b/asuch that a/b × b/a= 1. In this case, rational number b/ais the multiplicative train station of a rational number a/b. Because that example, the multiplicative train station of 7/3 is 3/7. (7/3× 3/7 = 1).

Note: Every rational number multiplied v 0 offers 0. If a/b is any kind of rational number, then a/b × 0 = 0 × a/b = 0. Because that example, 7/2× 0 = 0 ×7/2 = 0.

Related short articles onProperties of rational Numbers

Check the end these interesting articles to learn more about the nature of reasonable numbers.


Example 1: using the commutative property of rational numbers, justifywhether the given cases are commutative or not:a)You initially had actually 2/3 juicein your bottle, and you included 1/6 moreb)You initially had 1/6 juicein your bottle, and also you included 2/3 more

Solution:

By addition, we can calculate the complete quantity the juicein your bottle.Total amount of Juice= Initial quantity of Juice+ included quantity= 2/3 + 1/6 = 5/6.Let's do the very same calculation because that the 2nd case,Total quantity of Juice = Initial quantity of Juice + included quantity = 1/6 + 2/3 = 5/6.Here,in both cases, the full quantity that juice isthe same.We understand that the commutative home of rational numbers onaddition operationstates that, for any two rational number a and also b, a + b = b + a. Here additionally 2/3 + 1/6 = 1/6 + 2/3 = 5/6.


Example 2: aid Jack in solving 7/2(1/6+ 3/7) by using the distributive building of rational numbers.

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Solution:

Using the distributive building of rational numbers let us write the given expression in the formA (B + C) = A × (B + C) = ab + AC= 7/2(1/6+ 3/7)=7/2 × (1/6+ 3/7)= (7/2 × 1/6) + (7/2 × 3/7)= 25/12


Example 3:If 8/3 × (7/6 × 5/4) = 35/9, then discover (8/3 × 7/6) × 5/4.

Solution:

The associative building of rational numbers states that for any type of three rational numbers (A, B, and C) expression deserve to be express as(A × B) × C = A × (B × C)Given =8/3 × (7/6 × 5/4) = 35/9

Using the associative residential property of reasonable numbers, we can evaluate(8/3 × 7/6) × 5/4 is also equal to 35/9.To verify: (8/3 × 7/6) × 5/4. First, deal with the terms inside parentheses.= 56/18× 5/4= 35/9Hence,8/3 × (7/6 × 5/4) = (8/3 × 7/6) × 5/4 = 35/9.