{"id":4597,"date":"2021-11-26T18:48:50","date_gmt":"2021-11-26T18:48:50","guid":{"rendered":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/?page_id=4597"},"modified":"2021-12-13T14:55:45","modified_gmt":"2021-12-13T14:55:45","slug":"properties-of-rational-numbers","status":"publish","type":"page","link":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/math-concepts\/properties-of-rational-numbers\/","title":{"rendered":"properties of rational numbers"},"content":{"rendered":"

[et_pb_section fb_built=”1″ admin_label=”Section” module_class=”mainsec” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#e0f2fd” z_index=”1″ custom_padding=”5px||5px||true|false” locked=”off” collapsed=”off” global_colors_info=”{}”][et_pb_row column_structure=”3_5,2_5″ custom_padding_last_edited=”on|phone” _builder_version=”4.10.8″ _module_preset=”default” background_color=”#FFFFFF” width=”100%” max_width=”1310px” custom_padding=”|51px|40px|51px|false|true” custom_padding_tablet=”” custom_padding_phone=”|40px|30px|40px|false|true” border_radii=”on|10px|10px|10px|10px” global_colors_info=”{}”][et_pb_column type=”3_5″ admin_label=”Column L” _builder_version=”4.9.10″ _module_preset=”default” global_colors_info=”{}”][et_pb_text admin_label=”Acute Angles
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Properties of Rational Numbers with FAQs<\/b><\/h1>\n

[\/et_pb_text][et_pb_text admin_label=”Text” _builder_version=”4.13.1″ _module_preset=”default” text_font_size=”16px” header_2_font=”|600|||||||” header_2_text_color=”#a01414″ header_3_font=”|600|||||||” custom_padding=”15px|15px|54px|4px|false|false” hover_enabled=”0″ global_colors_info=”{}” sticky_enabled=”0″]<\/p>\n

The major properties of rational numbers are:(1) Closure, (2) Commutativity, (3) Associativity, (4) Distributive, (5) Identity Property and(6) Inverse Property.<\/p>\n

 <\/p>\n

Properties of Rational Numbers<\/b><\/h2>\n

Rational numbers are numbers that can be expressed in the fractional form i.e., p\/q, where both p and q are integers and
q<\/span> \u2260<\/span> 0. The rational numbers include integers, whole numbers and natural numbers. Rational numbers can be also described as terminating decimal numbers, or as non-terminating but repeating decimal numbers.\u00a0\u00a0<\/span><\/p>\n

Properties of rational numbers are:\u00a0<\/span><\/p>\n

(1) Closure property<\/span><\/p>\n

(2) Associative property<\/span><\/p>\n

(3) Commutative property<\/span><\/p>\n

(4) Distributive property<\/span><\/p>\n

(5) Identity property<\/span><\/p>\n

(6) Inverse Property<\/span><\/p>\n

<\/span><\/p>\n

1. Closure Property of Rational Numbers<\/b><\/h2>\n

This property states that when mathematical operations like addition, subtraction, and multiplication are applied on any two rational numbers then the result is also a rational number. Hence, rational numbers are closed under addition, subtraction and multiplication.
<\/b><\/p>\n

For Two Rational numbers\\frac{5}{2} \\text { and } \\frac{3}{2}<\/span><\/p>\n

 <\/p>\n

Closure under Addition<\/h2>\n

\\frac{5}{2}+\\frac{3}{2}=\\frac{8}{2}=4<\/span>, 4 is a rational number<\/p>\n

Hence we say that rational numbers are closed in addition<\/p>\n

Therefore, If a and b are rational numbers then (a+b) is also a rational number<\/p>\n

 <\/p>\n

closure under subraction<\/h2>\n

\\frac{5}{2}-\\frac{3}{2}=\\frac{2}{2}=1<\/span>, 1 is\u00a0 a rational number<\/p>\n

Hence we say that rational numbers are closed in subraction<\/p>\n

Therefore, If a and b are rational numbers then (a-b) is also a rational number<\/p>\n

<\/h2>\n

Closure under Multiplication<\/h2>\n

\\frac{5}{2} \\times \\frac{3}{2}=\\frac{15}{4}, \\frac{15}{4}<\/span>is a rational number<\/p>\n

Hence we say that rational numbers are closed in subraction<\/p>\n

Therefore, If a and b are rational numbers then \\Rightarrow(\\mathbf{a} \\times \\mathbf{b})<\/span>is also a rational number<\/p>\n

 <\/p>\n

The division is not under closure property because division by zero is undefined.<\/span><\/p>\n

Hence, we can also say that except \u20180\u2019 all numbers are closed under division.<\/span><\/p>\n

<\/span><\/p>\n

Closure under division (only for non zero denominator)<\/span><\/p>\n

\\left(\\frac{5}{2} \\div \\frac{3}{2}\\right)=\\frac{5}{2} \\times \\frac{2}{3}=\\frac{5}{3}, \\frac{5}{3}<\/span><\/span><\/p>\n

Hence, we say that ratonal numbers are closed under division.<\/span><\/p>\n

Therefore, If a and b are rational numbers then \\Rightarrow(a \\div b)<\/span> for \\mathbf{b} \\neq 0<\/span><\/p>\n

 <\/p>\n

2. Commutative Property\u00a0<\/b><\/h2>\n

This property states that when two rational numbers are added or multiplied in any order the outcome of the operation is equal. But in the case of subtraction and division, the outcome values will not remain equal if the order of the numbers is changed.<\/span><\/p>\n

For Two Rational numbers\\frac{5}{2} \\text { and } \\frac{3}{2}<\/span><\/span><\/p>\n

<\/span><\/p>\n

Commutative under addition<\/span><\/h2>\n

\\frac{5}{2}+\\frac{3}{2}=\\frac{8}{2}=4 <\/span><\/span><\/p>\n\\frac{3}{2}+\\frac{5}{2}=\\frac{8}{2}=4<\/span>\n

 <\/p>\n

Hence we say that rational numbers are commutative under the addition<\/span><\/p>\n

Therefore, If a and b are rational numbers \\Rightarrow(\\mathbf{a}+\\mathbf{b})=(\\mathbf{b}+\\mathbf{a})<\/span><\/span><\/p>\n

<\/span><\/p>\n

Noncommutative under subtraction<\/span><\/h2>\n

\\frac{5}{2}-\\frac{3}{2}=\\frac{2}{2}=1 <\/span>
<\/span><\/p>\n

\\frac{3}{2}-\\frac{5}{2}=\\frac{-2}{2}=-1<\/span><\/span><\/p>\n

 <\/p>\n

Hence we say that rational numbers are not commutative under the subraction<\/span><\/p>\n

Therefore, If a and b are rational numbers\\Rightarrow \\mathbf{a}-\\mathbf{b} \\neq \\mathbf{b}-\\mathbf{a}<\/span><\/span><\/p>\n

<\/span><\/h2>\n

commutative under multiplication<\/span><\/h2>\n

\n\\frac{5}{2} \\times \\frac{3}{2}=\\frac{15}{4} <\/span><\/span><\/p>\n


\\frac{3}{2} \\times \\frac{5}{2}=\\frac{15}{4}<\/span><\/span><\/p>\n

Hence we say that the rational numbers are commutative under multiplication<\/p>\n

Therefore a + b are rational numbers \\Rightarrow(\\mathbf{a} \\times \\mathbf{b})=(\\mathbf{b} \\times \\mathbf{a})<\/span><\/p>\n

 <\/p>\n

Noncommutative under Division<\/span><\/h2>\n

\\frac{5}{2} \\div \\frac{3}{2}=\\frac{5}{3} <\/span><\/span><\/p>\n


\\frac{3}{2} \\div \\frac{5}{2}=\\frac{3}{5}<\/span><\/span><\/p>\n

 <\/p>\n

Hence we say that rational numbers are not commutative under the division<\/span><\/p>\n

Therefore, If a and b are rational numbers\\Rightarrow \\mathbf{a} \\div \\mathbf{b} \\neq \\mathbf{b} \\div \\mathbf{a}<\/span><\/span><\/p>\n

<\/span><\/p>\n

3. Associative Property<\/b><\/h2>\n

This property states that when any three rational numbers are added or multiplied by grouping in any manner the outcome is equal. But in the case of subtraction and division, the outcome value will not be equal when the order of the numbers are reversed or grouped differently.<\/span><\/p>\n

For three rational numbers \u00a0\\frac{7}{2}, \\frac{5}{2} \\text { and } \\frac{3}{2}<\/span><\/p>\n

Associative under addition<\/p>\n\\left(\\frac{7}{2}+\\frac{5}{2}\\right)+\\frac{3}{2}=\\frac{12}{2}+\\frac{3}{2}=\\frac{15}{2} <\/span>\n\\frac{7}{2}+\\left(\\frac{5}{2}+\\frac{3}{2}\\right)=\\frac{7}{2}+\\frac{8}{2}=\\frac{15}{2}<\/span>\n

Hence we say that rational numbers are associative under addition<\/p>\n

Therefore a b and c are rational numbers\\Rightarrow(\\mathbf{a}+\\mathbf{b})+\\mathbf{c}=\\mathbf{a}+(\\mathbf{b}+\\mathbf{c})<\/span><\/p>\n

 <\/p>\n

nonassociative under subractionl<\/p>\n\\left(\\frac{7}{2}-\\frac{5}{2}\\right)-\\frac{3}{2}=\\frac{2}{2}-\\frac{3}{2}=\\frac{-1}{2} <\/span>\n

 <\/p>\n\\frac{7}{2}-\\left(\\frac{5}{2}-\\frac{3}{2}\\right)=\\frac{7}{2}-\\frac{2}{2}=\\frac{5}{2}<\/span>\n

Hence we say that rational numbers are not associative under subraction<\/p>\n

Therefore if a b and c are rational numbers\\Rightarrow(\\mathbf{a}-\\mathbf{b})-\\mathbf{c} \\neq \\mathbf{a}-(\\mathbf{b}-\\mathbf{c})<\/span><\/p>\n

 <\/p>\n

Associative under multiplication<\/p>\n\\left(\\frac{7}{2} \\times \\frac{5}{2}\\right) \\times \\frac{3}{2}=\\frac{35}{4} \\times \\frac{3}{2}=\\frac{105}{8} <\/span>\n

 <\/p>\n\\frac{7}{2} \\times\\left(\\frac{5}{2} \\times \\frac{3}{2}\\right)=\\frac{7}{2} \\times \\frac{15}{4}=\\frac{105}{8}<\/span>\n

 <\/p>\n

Hence\u00a0 Hence we say that rational numbers are associative under multiplication<\/p>\n

therfore a b and c are rational numbers \\Rightarrow(\\mathbf{a} \\times \\mathbf{b}) \\times \\mathbf{c}=\\mathbf{a} \\times(\\mathbf{b} \\times \\mathbf{c})<\/span><\/p>\n

 <\/p>\n

Nonassociative under Division<\/span><\/h2>\n

\\left(\\frac{7}{2} \\div \\frac{5}{2}\\right) \\div \\frac{3}{2}=\\frac{7}{5} \\div \\frac{3}{2}=\\frac{14}{15} <\/span><\/span><\/p>\n\\frac{7}{2} \\div\\left(\\frac{5}{2} \\div \\frac{3}{2}\\right)=\\frac{7}{2} \\div \\frac{5}{3}=\\frac{21}{10}<\/span>\n

 <\/p>\n

Hence we say that rational numbers are not associative under division<\/span><\/p>\n

therfore a b and c are rational numbers\\Rightarrow(\\mathbf{a} \\div \\mathbf{b}) \\div \\mathbf{c} \\neq \\mathbf{a} \\div(\\mathbf{b} \\div \\mathbf{c})<\/span><\/span><\/span><\/p>\n

 <\/p>\n

 <\/p>\n

4. Distributive Property<\/b><\/h2>\n

This property states that<\/span><\/p>\n

If a b and c are rational numbers then<\/span><\/p>\n

a \\times(b+c)=(a \\times b)+(a \\times c) <\/span><\/span><\/p>\n


a \\times(b-c)=(a \\times b)-(a \\times c)<\/span><\/span><\/p>\n

 <\/p>\n

For three rational numbers \u00a0\\frac{7}{2}, \\frac{5}{2} \\text { and } \\frac{3}{2}<\/span><\/span><\/p>\n

\\frac{7}{2} \\times\\left(\\frac{5}{2}+\\frac{3}{2}\\right)=\\frac{7}{2} \\times \\frac{8}{2}=\\frac{7}{2} \\times 4=14 <\/span><\/span><\/p>\n


\\left(\\frac{7}{2} \\times \\frac{5}{2}\\right)+\\left(\\frac{7}{2} \\times \\frac{3}{2}\\right)=\\frac{35}{4}+\\frac{21}{4}=\\frac{56}{4}=14 <\/span><\/span><\/p>\n


\\therefore \\frac{7}{2} \\times\\left(\\frac{5}{2}+\\frac{3}{2}\\right)=\\left(\\frac{7}{2} \\times \\frac{5}{2}\\right)+\\left(\\frac{7}{2} \\times \\frac{3}{2}\\right)<\/span><\/span><\/p>\n

 <\/p>\n

similiarly\u00a0<\/span><\/p>\n

\\frac{7}{2} \\times\\left(\\frac{5}{2}-\\frac{3}{2}\\right)=\\frac{7}{2} \\times \\frac{2}{2}=\\frac{7}{2} <\/span><\/span><\/p>\n


\\left(\\frac{7}{2} \\times \\frac{5}{2}\\right)-\\left(\\frac{7}{2} \\times \\frac{3}{2}\\right)=\\frac{35}{4}-\\frac{21}{4}=\\frac{14}{4}=\\frac{7}{2} <\/span><\/span><\/p>\n\\therefore \\frac{7}{2} \\times\\left(\\frac{5}{2}-\\frac{3}{2}\\right)=\\left(\\frac{7}{2} \\times \\frac{5}{2}\\right)-\\left(\\frac{7}{2} \\times \\frac{3}{2}\\right)<\/span>\n

 <\/p>\n

<\/span><\/p>\n

5. Identity Property<\/b><\/h2>\n

For any rational number, the additive identity is \u20180\u2019 and the multiplicative identity is \u20181\u2019.<\/span><\/p>\n

For the rational number\u00a0 <\/span>\\frac{3}{4}<\/span><\/p>\n

\\frac{3}{4}+0=\\frac{3}{4} \\Rightarrow 0<\/span>is the additive identity<\/span><\/p>\n

\\frac{3}{4} \\times 1=\\frac{3}{4} \\Rightarrow 1<\/span>is the multiplicative identity<\/span><\/p>\n

<\/span><\/p>\n

6. Inverse Property<\/b><\/h2>\n

For any rational number, the additive inverse is the negative of that number and the multiplicative inverse is its reciprocal value.<\/span><\/p>\n

For the rational number \\frac{3}{4}<\/span><\/span><\/p>\n

\\frac{3}{4}+\\left(-\\frac{3}{4}\\right)=0 \\Rightarrow-\\frac{3}{4}<\/span> is the additive inverse of \\frac{3}{4}<\/span><\/span><\/p>\n

\\frac{3}{4} \\times \\frac{4}{3}=1 \\Rightarrow \\frac{4}{3}<\/span>is the multiplicative inverse of \\frac{3}{4}<\/span><\/span><\/p>\n

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Frequently Asked Questions\u00a0<\/span><\/span><\/h1>\n
    <\/ol>\n

    1. What are the Properties of Rational Numbers?
    <\/strong><\/h3>\n

    Ans:<\/strong>The major properties of rational numbers are:
    (1) Closure Property, (2) Commutative Property, (3) Associative Property, (4) Distributive Property, (5) Identity Property and(6) Inverse Property.
    <\/span><\/p>\n

    <\/span><\/p>\n

    2. When two rational numbers are added then which property is used?<\/strong><\/h3>\n

    Ans: <\/strong>By closure property of addition when two rational numbers are the result is also a rational number.
    For Example \\frac{3}{5}+\\frac{4}{5}=\\frac{7}{5}, \\frac{7}{5}<\/span> is a rational number<\/span><\/p>\n

    <\/span><\/p>\n

    3. What is the distributive property of rational numbers?<\/strong><\/h3>\n

    Ans: <\/strong>The distributive property states,\u00a0<\/span><\/p>\n

    If a b and c are rational numbers then,<\/span><\/p>\n

    a \\times(b+c)=(a \\times b)+(a \\times c) <\/span><\/span><\/p>\n


    a \\times(b-c)=(a \\times b)-(a \\times c)<\/span><\/p>\n

    <\/span><\/p>\n

    [\/et_pb_text][\/et_pb_column][\/et_pb_row][\/et_pb_section]<\/p>\n","protected":false},"excerpt":{"rendered":"

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