/ Preparation Maths Number SystemWhat is Number System?
The number system is divided into two parts:
- Imaginary numbers
- Real numbers
Real numbers (R) are of two types:
- Rational numbers
- Irrational numbers
Rational numbers (Q)
A number that can be expressed as a/b is known as a rational number where a and b both are integers and b is not zero. Example, 5/7, -5/7, etc.
Properties of rational numbers
- The sum of rational numbers is always a rational number.
- The difference of rational numbers is always a rational number.
- The product of rational numbers is always a rational number.
- When you divide a rational number by a non-zero rational number, it gives you a rational number.
Irrational numbers (Q)
A number that cannot be expressed as a/b is known as an irrational number where a and b both are integers and b is not zero. For example, 'a' is irrational if its exact square root does not exist.
The decimal representation of rational numbers
(i) When you divide a rational number, and there is no remainder, the quotients of such divisions are called terminating decimals.
(ii) When dividing a rational number, if the division does not end, the quotients of such divisions are called non-terminating.
(iii) When a digit or a set of digits repeats continually in a non-terminating decimal, it is known as a recurring decimal.
Surds
If “y” is a positive rational integer and “a” is a positive integer, such that y1/a is irrational, y1/a is called a surd or a radical.
Rationalization
When a surd is rationalized by multiplying it with its rationalizing factor, it is known as rationalization.
- Weightage of Number System
- Illustrated examples on Number System
- FAQs on Number System
Weightage of Number System
The Number System is a basic chapter in mathematics. It is taught in Class 9 and carries eight marks.
Illustrated examples on Number System
1. Are the following statements true or false? Give reasons for your answers.
Solution.
(i). Every whole number is a natural number.
(ii) Every integer is a rational number.
(iii). Every rational number is an integer.
(i) False, because zero is a whole number but not a natural number.
(ii) True, because every integer m can be expressed in the form m/1, so it is a
rational number.
(iii) False, because ⅗ is not an integer.
2. Show that 0.3333... = 0.3 can be expressed in the form p/q, where p and
q are integers and q 0.
Solution.
Let x= 0.3333…
Now, 10x = 10 * (0.33…) = 3.333…
Now, 3.333.. = 3 + x, ( since x = 0.333…)
Thus, 10x = 3 + x
On solving, you get,
X = ⅓
3. Find an irrational number between 1/7 and 2/7.
Solution.
We know that 1/7 = 0.142857.
So we know that 2/7 = 0.285714.
A number that is non-terminating non-recurring that lies between these numbers is the required irrational number between 1/7 and 2/7.
There can be many such numbers that lie between these numbers. An example is 0.150150015000150000...
FAQs on Number System
Q: What makes real numbers?
Q: Is the negative of an irrational number also irrational?
Q: Is every irrational number a surd?
Q: Is the product of a non-zero rational number and an irrational number rational or irrational?
Q: How important is the chapter?
Maths Number System Exam
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