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Rational Numbers
Integers are closed under addition, multiplication and subtraction because sum, product and difference of two integers are again integers. But when we come across the division, the quotient of two integers is not always an integer. Therefore, there must be another set of numbers that includessuch quotients which are not integers which are called rational numbers. Thus, any numbers which can be expressed in the form of \(\frac{x}{y}\) , where x and y are integers and y ≠0 are called rational numbers. The set of rational number is denoted by Q. Natural numbers, whole numbers and integers are the subsets of the set of rational numbers.
Q= {...-5, \(\frac{-3}{2}\), -1, 0, 1,\(\frac{-3}{2}\), 4.....}
Irrational Numbers
Numbers which are not rational are called irrational.In other words, any number which cannot be expressed as the ratio of two integers is known as an irrational √2,√3, 3√5, π, e,...... are the examples of such numbers, an irrational number can be expressed as a non-terminating, non-recurring decimal. What about √4, √9 etc? These are not an irrational number because the square root of 4 is 2 and 9 is 3 which are rational numbers.
Integers
Natural numbers have closed the operations of addition and multiplication. But the set of natural numbers in not sufficient when we come across subtraction. That is, the difference between two natural numbers may not be a natural number, For example, 7-9=-2 which is not a natural number. So, in order to make the operation of subtraction meaningful, negative numbers and zero are introduced.
Integers are the set of natural numbers together with their negatives and including the zero number. The set of integers is denoted \(\mathbb{Z}\). So,
\(\mathbb{Z}\) = {.....-3,-2.-1,0,1,2,3....}
Real Numbers
The set of all rational and irrational numbers taken together form a new system of numbers known as a real number system. The set of all real numbers is denoted by \(\Re\).
Solution:
Here,
= 2\(\sqrt[3]{4}\)
= \(\sqrt[3]{2^3 × 4}\)
= \(\sqrt[3]{8 × 4}\)
= \(\sqrt[3]{32}\)
Solution:
Here,
= \(\sqrt[3]{16}\)
= \(\sqrt[3]{8 × 2}\)
= \(\sqrt[3]{2^3 × 4}\)
= 2\(\sqrt[3]{2}\)
Numbers which are not rational are called _______.
Integers are closed under addition, multiplication and subtraction in ______ number.
Natural numbers have closed the operations of addition and multiplication in ______.
The set of real number is denoted by _____.
The set of integers is denoted by ______.
The set of all rational and irrational numbers taken together form a new system of numbers known as ______.
An irrational number can be expressed as ______.
ASK ANY QUESTION ON Number System
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