In math a rational number is a number that can be articulated as a ratio of 2 integers, which is in the form of 'p/q' where p and q are integers and q should not be equal to zero. Likewise, any repeating or ending decimal represents a rational number. In this article we shall argue is a repeating decimal a rational number.
Examples of rational numbers:
The following are the examples of rational numbers:
'1/4' is a rational number (1 divided by 4, or the ratio of 1 to 4)
The decimal 0.5 is a rational number ('1/2' )
1 is a rational number ('1/1' )
4 is a rational number ('4/1' )
The decimal 3.14 is a rational number ('314/10' )
The decimal 1.13 is a rational number ('113/100' )
The negative decimal -3.3 is a rational number ('-33/10' )
The repeating decimal 1.1212121212... is a rational number.
The repeating decimal 1.3333333333... is a rational number.
Is a repeating decimal a rational number?
Yes a repeating decimal is a rational number. Because the decimal extension of a rational number always either terminates after finitely many digits or begins to replicate the same series of digits more and more. Therefore any repeating decimal is a rational number.
Let us discuss the properties of rational numbers
Commutative property a+b = b+a
Associative property (a+b)+c = a+(b+c)
Additive identity a+0 = a
Multiplicative identity a (1) = a
Additive inverse a+ (-a) = 0
Multiplicative inverse (1/a) =1
Multiplication property of zero a(0)=0
Distributive property a(b+c) = ab + ac
Example problems for rational number:
Identify the rational numbers from the following.
5, 78, 5.6, 784, 0.55, 6.57575757..., 3.645548349... , 'sqrt2' , 'sqrt3' , 'pi'
Solution:
5 can be written as a ratio of 5 and 1, which is '5/1' where 1 and 5 are integers.
78 can be written as a ratio of 78 and 1, which is '78/1' where 1 and 78 are integers.
5.6 can be written as a ratio of 56 and 10, which is '56/10' where 10 and 56 are integers.
0.55 can be written as a ratio of 55 and 100, which is '55/100' where 100 and 55 are integers.
6.57575757... has a repeating decimal 5 and 7 therefore it is a rational number.
3.645548349... has a repeating decimal 3 therefore it is a rational number.
sqrt2 is equal to 1.4142135.. is not a rational number because the decimal is not repeating.
sqrt3 is equal to 1.732050.. is not a rational number because the decimal is not repeating.
'pi ' is equal to 3.14159265.... is not a rational number because the decimal is not repeating.
Here in this page we are going to discuss about operations with rational numbers. In abstract algebra, the concept of a polynomial is extended to include formal expressions in which the coefficients of the polynomial can be taken from any field. In this setting given a field F and some indeterminate X, a rational expression is any element of the field of fractions of the polynomial ring F [X].
Source: Wikipedia
Operations on rational numbers
Rational number is the quotient of two integers. Therefore, a rational number is a number that can be write in the form 'w/x', where w and x are integers, and x is not zero. A rational number written in this way is commonly called a fraction.
'w/x'
Where
w 'rArr' an integer
x 'rArr' a nonzero integer
'17/15', '(14)/(9x)' 'rArr' Rational numbers
An integer can be marking it as the quotient of the integer and 1, every integer is a rational number.
Note : A rational number written as a fraction can be written in decimal notation.
Examples
Below are the examples on operations with rational numbers -
Example 1:
Write '48/4' as a decimal.
Solution:
12 'rArr' This is called a terminating decimal.
4 | 48
4
08
8
0'rArr' The remainder is Zero.
'48/4' = 12
Adding operations with same denominators:
Example 2:
'8/2' + '4/2' = ?
Solution:
'8/2' + '4/2' = '8/2' + '4/2'
= '(8+4)/2'
= '(12/2)'
= '6'
Adding operations with different denominators:
Just as we add fractions, rational numbers with different denominators can also be extra. By finding out the LCM, we can take the denominators to the same number.
Example 3:
'4/3' + '3/6'
Solution:
= '4/3' + '3/6'
6 is the LCM of 3 and 6.
= '8/6' + '3/6'
= '(8+3)/6'
= '11/6'
= '11/6'
Subtraction operations with same denominators:
Just as we subtract fractions, we can subtract rational numbers with same denominator.
Example 4:
'5/6' - '2/6' = ?
Solution:
= '5/6' - '2/6'
= '(5-2)/6'
= '4/6'
= '2/3'
Subtraction operations with different denominators:
Just as we subtract fractions, rational numbers also can be taken off with different denominators. The common denominator is achieved by finding out the LCM.
Example 5:
'-5/12' + '2/6'
Solution:
= '-5/12' + '2/6'
= '-5/12' + '4/12'
= '(-5+4)/12'
= '-1/12'
Multiplication operations with same denominators:
Just alike the multiplication of whole numbers and integers, multiplication of rational number are also repeated addition.
Example 6:
'5/4*10/5'
Solution:
= '5/4*10/5'
= '(5*10)/(4*5)'
= '50/20'
= '5/2'
Multiplication operations with different denominators:
Example 7:
'5/2*10/3'
Solution:
= '5/2*10/3'
= '(5*10)/(2*3)'
= '50/6'
Practices problems
Problem 1:
'2/6+1/3' = ?
Answer: '2/3'
Problem 2:
'2/5' + '1/5'
Answer: '3/5'
Problem 3:
'7/5' - '5/5'
Answer: '2/5'
Problem 4:
'30/3' -('5/5')
Answer:'135/15' or '9'
Problem 5:
'5/3*2/3'
Answer:'10/9'
Problem 6:
'2/5*4/2'
Answer:'4/5'
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