Representation of Irrational Numbers on Number Line
Representation of irrational numbers on a number line can be done by first identifying the origin point and then moving towards the positive or the negative numbers. A number line is a straight line with both positive and negative numbers visually represented. Positive, negative, zero, and decimal numbers can be represented on a number line. In this article, we are going to learn the steps of representation irrational numbers on a number line and solve a few examples to understand the concept better.
| 1. | What are Irrational Numbers? |
| 2. | Steps on Representation of Irrational Numbers on Number Line |
| 3. | FAQs on Representation of Irrational Numbers on Number Line |
What are Irrational Numbers?
Irrational numbers are real numbers that cannot be represented as a simple fraction. These cannot be expressed in the form of ratio, such as p/q, where p and q are integers, q≠0. It is a contradiction of rational numbers. Some of the commonly known examples are π, √2, √5, etc. Irrational numbers consist of non-terminating and non-recurring decimals. Addition, subtraction, multiplication, and division of two irrational numbers may or may not be a rational number.
Definition of a Number Line
A visual representation of numbers on a straight line drawn either horizontally or vertically is known as a number line. Writing down numbers on a number line makes it easy for us to compare them and to perform basic arithmetic operations on them. There are three parts of a number line which are - negative side, zero, and positive side. Numbers to the left of 0 are negative numbers and numbers to the right of 0 are all positive numbers. So, we can say that on a number line, the numbers present on the right are larger than the numbers on their left. For example, 3 comes to the right of 1, so 3 > 1. Look at the image of a number line given below.
Steps on Representation of Irrational Numbers on Number Line
Real numbers can be considered as rational numbers or irrational numbers. Hence, a unique point is considered to represent them on the number line. Some irrational numbers in the form of √n, where n is a positive integer can be represented on a number line by using the following steps.
- Step 1: Split the number inside the square root such that the sum adds up to the number.
- Step 2: The distance between these two natural numbers should be equal on the number line starting from the origin. One line should be perpendicular to the other.
- Step 3: Use Pythagoras Theorem
- Step 4: Represent the area as the desired measurement.
Let us look at an example to understand this better. Represent √2 on a number line.
Step 1: Draw a number line with the center as zero, left of zero as -1, and right of zero as 1.
Step 2: Keeping the same length as between 0 and 1, draw a line perpendicular to point (1), such that the new line has a length of 1 unit.
Step 3: Draw a line from 0 to the end of the perpendicular line constructing a right-angled triangle ABC. With AB as height, BC as the base, and AC as the hypotenuse. See image below.
Step 4: Find the length of the hypotenuse i.e. AC by applying the Pythagoras theorem.
(AC)2 = (AB)2 + (BC)2
(AC)2 = (1)2 + (1)2
(AC)2 = 2
AC = √2
Step 5: Keeping AC as radius with C as the center, cut an arc on the same number line naming it as D. CD will also become the radius of the arc with length √2.
Step 6: Therefore, point D represents √2 on the number line as shown in the image below.
Important Notes
- To represent an irrational number on a number line, we break the number into two parts inside the square root.
- Through the Pythagoras theorem, the two parts make the sides of the triangles and the hypotenuse is the number inside the square root.
- While plotting the triangle on the number line, remember to have one vertex as zero.
- The hypotenuse needs to be on the number line made by a compass determining the value on the number line.
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Examples on Representation of Irrational Numbers on Number Line
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Example 1: Represent √5 on a number line.
Solution:
Use the same steps as mentioned above in this article.
Split the number into two inside the square root. 5 = 2 + 1. Where 2 is the length of the perpendicular line.
Using the Pythagoras theorem, find the length of the hypotenuse.
(hypotenuse)2 = (2)2 + (1)2
(hypotenuse)2 = 4 + 1
hypotenuse = √5
Keeping the center the endpoint of the hypotenuse as the center, an arc is drawn creating a point on the number line attaching the end of the perpendicular. Making that point the same length as √5.
Therefore, √5 can be represented on a number line.
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Example 2: Represent y ≥ 2 on a number line.
Solution:
y ≥ 2 means that the value of a variable y is 2 or greater than 2. This can be easily represented on the number line as:
The black region on the line represents y ≥ 2.
FAQs on Representation of Irrational Numbers on Number Line
Can Irrational Numbers be Represented on Number Line?
Irrational numbers are considered as the subset of real numbers and they can be represented on a number line. However, while representing an irrational number, the location can be estimated and not have an accurate location.
How to Represent Irrational Numbers on a Number Line?
Real numbers can be considered rational numbers or irrational numbers. Hence, a unique point is considered to represent them on the number line. Some irrational numbers in the form of √n, where n is a positive integer can be represented on a number line by using the following steps.
- Step 1: Split the number inside the square root such that the sum adds up to the number.
- Step 2: The distance between these two natural numbers should be equal on the number line starting from the origin. One line should be perpendicular to the other.
- Step 3: Use Pythagoras Theorem
- Step 4: Represent the area as the desired measurement.
Can we Represent all Real Numbers on a Number Line?
Yes, we can represent all real numbers on a number line. But, sometimes, mostly in the case of irrational numbers, we represent the approximate value of the number on the number line.
What is the Difference Between Rational and Irrational Numbers?
Rational numbers are those that are terminating or non-terminating repeating numbers, while irrational numbers are those that neither terminate nor repeat after a specific number of decimal places. In other words, numbers that can be expressed in a/b or fraction form is a rational number, a number that cannot be expressed in a ratio of two numbers is irrational numbers.
Why Pi is an Irrational Number?
Pi is defined as the ratio of a circle's circumference to its diameter. The value of Pi is always constant. Pi (π) approximately equals 3.14159265359... and is a non-terminating non-repeating decimal number. Hence 'pi' is an irrational number.
Why Rational Numbers and Irrational Numbers Are in the Set of Real Numbers?
The numbers which can be expressed in the form of decimals are considered real numbers. If we talk about rational and irrational numbers both the forms of numbers can be represented in terms of decimals, hence both rational numbers and irrational numbers are in the set of real numbers.