About Real Numbers
What are Real Numbers?
Real numbers are the combination of rational numbers and irrational numbers.
We can classify numbers further as:
1. Natural numbers:
Natural numbers are numbers such as 1, 2, 3, 4, 5, .. and so on. It is denoted by ‘N’.
2. Whole numbers:
The whole numbers start from 0, then include all positive numbers up to infinity. Whole numbers are 0, 1, 2, 3, 4, 5, … and so on. It is denoted by ‘W’.
3. Integers:
Integers are the set of whole numbers and negative numbers. It is denoted by ‘Z’. So, Z={…-5, -4, -3, -2, -1, 0, 1, 2, 3 , 4 , 5, …}
4. Rational numbers:
The numbers that are in the form \(\frac{p}{q}\) where, \(p\) and \(q\) are integers and \(q \neq 0\) are called rational numbers.
5. Irrational numbers:
If the quotient of two integers is non-terminating and non-repeating then it is an irrational number.
Real numbers are numbers that can be represented as points on an infinitely long line known as the number line.
Rapid Recall
Solved Examples on Real Numbers
Example 1: Plot only rational numbers on the number line among the given numbers:
\(-\frac{3}{2},\ \ 1,\ \ \sqrt9,-\sqrt4,\ \ \sqrt{35},\ \sqrt[3]{64},-\sqrt3,-\sqrt[3]{125}\)
Solution:
Among the given numbers,
Irrational numbers are: \(-\sqrt3, \sqrt{35}\)
Now, \(\sqrt9=3\)
\(-\sqrt4\ =-2\)
\(\sqrt[3]{64}=4\)
\(-\sqrt[3]{125}=-5\)
Hence, the rational numbers are: \(-\frac{3}{2},\ 1,\ \sqrt9,\ -\sqrt4,\ \sqrt[3]{64},\ -\sqrt[3]{125}\)
Hence, the rational numbers are plotted on the number line.
Example 2: John was walking on a straight line, from the starting point he took 3 steps to the left and then took 7 steps to the right and again walked \(\frac{5}{2}\) steps to the right. How many steps is he away from the start point?
Solution:
Let’s graph the steps covered by John.
Let the starting point at which John starts from be 0. Let each step be 1 unit.
He took 3 steps in the left direction and reached at – 3, then he walked in the right direction and took 7 steps, that is, \(-3+7=4\). Then again he took \(\frac{5}{2}\) steps in the right direction, that is,
\(4+\frac{5}{2}=4+2.5=6.5.\)
Therefore, John is 6.5 steps away from the starting point.
Example 3: Circle the irrational numbers in the given figure.
Solution:
Irrational numbers: Any number that is non-terminating and non-recurring is termed as an irrational number.
\(\sqrt5=2.236067977\ldots\sqrt2=1.41421356237\ldots\)
\(e=2.71828182845\ldots\)
These are irrational numbers.
Real numbers are numbers that can be represented on a number line. Real numbers consist of negative numbers, zero and positive numbers. Hence 0 is a real number.
- Draw a horizontal line with arrows on both the ends, this denotes that numbers extend infinitely on both sides. Mark ‘0’ in the middle. The number 0 is known as the origin.
- Mark numbers on the line such as -2, -1, 0, 1, 2 … at equidistant points.
- The positive numbers will be on the right hand side and negative numbers will be on the left hand side of the number 0.