![ Addition is associative for rational numbers.
That is for any three rational numbers a, b and c, :
a + (b + c) = (a + b) + c.
For Example
Since, -9/10 = -9/10
Hence, L.H.S. = R.H.S.
Therefore, the property
has been proved.
Subtraction is Not Associative for rational numbers
Multiplication is associative for rational numbers.
That is for any rational numbers a, b and c :
a* (b*c) = (a*b) * c
For Example –
Since, -5/21 = -5/21
Hence, L.H.S. = R.H.S
Division is Not Associative for Rational numbers.
ASSOCIATIVE PROPERTY
L.H.S. R.H.S.
-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
L.H.S. R.H.S.
-2/3* (5/4*2/7)
= -2/3 * 10/28
= -2/3 * 5/14
= -10/42
= -5/21
(-2/3*5/4) * 2/7
= -10/12 * 2/7
= -5/6 * 2/7
= -10/42
= -5/21](https://image.slidesharecdn.com/rationalnumbers-140509050200-phpapp01/75/Rational-Numbers-6-2048.jpg)
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Rational Numbers
Rational numbers are defined as integers in the form of p/q where q is not zero and can be added, subtracted, and multiplied with closure, but division is not closed. Addition and multiplication are commutative and associative for rational numbers, while subtraction and division are neither commutative nor associative. The document also discusses properties like the additive inverse and the reciprocal of rational numbers.
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Definition of rational numbers as p/q where q ≠ 0. Examples include 5/8, -3/14.
Rational numbers are closed under addition, subtraction, multiplication, but not division. Discusses commutative and associative properties for addition and multiplication, and their lack for subtraction and division.
Distributive law explained with an example, and definitions of additive inverse and reciprocal (multiplicative inverse) in rational numbers.
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Rational Numbers
- 1.
- 2. The integers whichare in the form of p/q where q is not equal to 0 are known as Rational Numbers. Examples : 5/8; -3/14; 7/-15; -6/-11
- 3. Rational numbersare closed under addition. That is, for any two rational numbers a and b, a+b s also a rational number. For Example - 8 + 3 = 11 ( a rational number. ) Rational numbers are closed under subtraction. That is, for any two rational numbers a and b, a – b is also a rational number. For Example - 25 – 11 = 14 ( a rational number. ) Rational numbers are closed under multiplication. That is, for any two rational numbers a and b, a * b is also a rational number. For Example - 4 * 2 = 8 (a rational number. ) Rational numbers are not closed under division. That is, for any rational number a, a/0 is not defined. For Example - 6/0 is not defined.
- 4. • Rational numberscan be added in any order. Therefore, addition is commutative for rational numbers. For Example :- • Subtraction is not commutative for rational numbers. For Example - Since, -7 is unequal to 7 Hence, L.H.S. Is unequal to R.H.S. Therefore, it is proved that Subtraction is not commutative for rational numbers. L.H.S. R.H.S. - 3/8 + 1/7 L.C.M. = 56 = -21+8 = -13 1 /7 +(-3/8) L.C.M. = 56 = 8+(-21) = -13 L.H.S. R.H.S. 2/3 – 5/4 L.C.M. = 12 = 8 – 15 = -7 5/4 – 2/3 L.C.M. = 12 = 15 – 8 = 7
- 5. • Rational numberscan be multiplied in any order. Therefore, it is said that multiplication is commutative for rational numbers. FOR EXAMPLE – Since, L.H.S = R.H.S. Therefore, it is proved that rational numbers can be multiplied in any order. • Rational numbers can not be divided in any order. Therefore, division is Not Commutative for rational numbers. FOR EXAMPLE – Since, L.H.S. is not equal to R.H.S. Therefore, it is proved that rational numbers can not be divided in any order. L.H.S. R.H.S. -7/3*6/5 = -42/15 6/5*(7/3) = -42/15 L.H.S. R.H.S. (-5/4) / 3/7 = -5/4*7/3 = -35/12 3/7 / (-5/4) = 3/7*4/-5 = -12/35
- 6. Addition isassociative for rational numbers. That is for any three rational numbers a, b and c, : a + (b + c) = (a + b) + c. For Example Since, -9/10 = -9/10 Hence, L.H.S. = R.H.S. Therefore, the property has been proved. Subtraction is Not Associative for rational numbers Multiplication is associative for rational numbers. That is for any rational numbers a, b and c : a* (b*c) = (a*b) * c For Example – Since, -5/21 = -5/21 Hence, L.H.S. = R.H.S Division is Not Associative for Rational numbers. ASSOCIATIVE PROPERTY L.H.S. R.H.S. -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 L.H.S. R.H.S. -2/3* (5/4*2/7) = -2/3 * 10/28 = -2/3 * 5/14 = -10/42 = -5/21 (-2/3*5/4) * 2/7 = -10/12 * 2/7 = -5/6 * 2/7 = -10/42 = -5/21
- 7. DISTRIBUTIVE LAW For allrational numbers a, b and c, a (b+c) = ab + ac a (b-c) = ab – ac For Example – Since, L.H.S. = R.H.S. Hence, Distributive Law Is Proved. L.H.S. R.H.S. 4 (2+6) = 4 (8) = 32 4*2 + 4*6 = 8 = 24 = 32
- 8. Additive Inverse Additiveinverse is also known as negative of a number. For any rational number a/b, a/b+(-a/b)= (-a/b)+a/b = 0 Therefore, -a/b is the additive inverse of a/b and a/b is the Additive Inverse of (-a/b). Reciprocal Rational number c/d is called the reciprocal or Multiplicative Inverse of another rational number a/b if a/b * c/d = 1