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    Name - Samarjeet Baliyan

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    Samarjeet Baliyan
    Rational numbers are closed under addition, multiplication, and subtraction, but not division. For addition and multiplication, rational numbers are commutative and associative, but subtraction and division are not commutative. Multiplication is distributive over addition and subtraction for rational numbers.

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    Name - Samarjeet Baliyan

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    Samarjeet Baliyan
    39 views7 pages
    Rational numbers are closed under addition, multiplication, and subtraction, but not division. For addition and multiplication, rational numbers are commutative and associative, but subtraction and division are not commutative. Multiplication is distributive over addition and subtraction for rational numbers.

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    Rational numbers are closed under addition, multiplication, and subtraction, but not division. For addition and multiplication, rational numbers are commutative and associative, but subtraction and division are not commutative. Multiplication is distributive over addition and subtraction for rational numbers.

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    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    39 views7 pages

    Name - Samarjeet Baliyan

    Uploaded by

    Samarjeet Baliyan
    Rational numbers are closed under addition, multiplication, and subtraction, but not division. For addition and multiplication, rational numbers are commutative and associative, but subtraction and division are not commutative. Multiplication is distributive over addition and subtraction for rational numbers.

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    © All Rights Reserved
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    Download as pptx, pdf, or txt
    of 7

    (a) Addition – As we know how to add two rational numbers. Let us add a pair.

    (3/8)+((-5)/7)= (21+(-40))/56= -19/56


    We say that rational numbers are closed under addition. That is for any two rational numbers
    a & b, a+b is also a rational number.

    (b) Subtraction- As we know how to add two rational numbers. Let us subtract a pair.
    (-5/7)-(2/3)= (-5x3-2x7)/21 = -29/21
    We say that rational numbers are closed under subtraction. That is for any two rational
    numbers, a & b, a-b is also a rational number.

    (c) Multiplication – As we know how to multiply rational numbers. Let us multiply a pair.
    (-2/3)x(4/5)= -8/15
    We say that rational numbers are closed under multiplication. That is for any two rational
    numbers, a & b, axb is also a rational number.

    (d) Division – We note that


    (-5/3) ÷(2/5) = -25/6 ( a rational number)
    We find that for any rational number a,a÷0 is not defined.
    So rational numbers are not closed under division. However if we exclude zero then the
    collection of all other rational numbers is closed under division
    (a) Addition- As we know how to add two rational numbers. Let us add a pair.
    (-2/3)+(5/7)=1/21 and (5/7)+(-2/3)=1/21
    So, (-2/3)+(5/7)=(5/7)+(-2/3)
    We say that addition is commutative for rational numbers. That is , for any two rational
    numbers a and b, a+b = b+a.

    (b) Subtraction-
    (2/3)-(5/4)= (5/4)-(2/3) False
    We say that Subtraction is not commutative for rational numbers.

    (c) Multiplication-
    (-7/3)x(6/5)= (-42/15) = (6/5)x(-7/3)
    We say that Multiplication is commutative of rational numbers. In general, axb = bxa for
    any two rational numbers a and b.

    (d) Division-
    (-5/3) ÷ (3/7) = (3/7)÷(-5/4)
    Expressions on both side are not equal. So division is not commutative for rational
    numbers.
    (a) Addition-
    (-2/3)+[(3/5)+(-5/6)] = (-2/3)+(-7/30) = (-27/30) = (-9/10)
    [(-2/3)+(3/5)]+(-5/6) = (-1/15)+(-5/6) = (-27/30) = (-9/10)
    So, (-2/3)+[3/5+(-5/6)] = [-2/3+3/5]+(-5/6)
    We say that Addition is associative for rational numbers. That is for any three rational numbers
    a, b and c, a+(b+c) = (a+b)+c

    (b) Subtraction-
    -2/3-[-4/5-1/2] = [2/3-(-4/5)]-1/2 False
    We say that Subtraction is not associative for rational numbers.

    (c) Multiplication -
    -7/3x(5/4x2/9) = -7/3x10/36 = -70/108= -35/54
    (-7/3x5/4)x2/9 = -35/12x2/9 = -70/108= -35/54
    We say that Multiplication is associative for rational numbers. That is for any three rational
    numbers a, b and c, ax(bxc) = (axb)xc

    (d) Division -
    1/2÷[-1/3÷2/5] = [1/2÷(-1/3)] ÷2/5 False
    LHS ≠ RHS
    We say that division is not associative for rational numbers.
    (a) Multiplication over addition and subtraction –

    For all rational numbers a, b & c,

    a(b+c) = ab+ac
    a(b-c) = ab-ac

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