Properties of Surds

We will discuss about the different properties of surds.

If a and b are both rationals and √x and √y are both surds and a + √x = b + √y then a = b and x = y

If a not equal to b, let us assume, b = a + m, where m (m ≠ 0) is a rational.

Now, by question, a + √x = b + √y

⇒ a + √x = a + m + √y  

⇒ √x = m + √y, which is impossible (since a simple quadratic surd cannot be equal to the sum of a rational quantity and a simple quadratic surd).

Therefore, we must have, a = b.

When a = b then a + √x = b + √y ⇒ √x = √y ⇒ x = y.

Notes:

1. If a - √x = b - √y where a, b are both rationals and √x, √y are both surds, then proceeding as above we can show a = b and x = y.

2. If √x and √y are actually rationals (in the form of surds), then the relation a + √x = b + √y does not imply a = b and x = y.

 For example, we have,

10 = 6 + 4 = 6 + √16 and 10 = 4 + 6 = 4 + √36

⇒ 6 + √16 = 4 + √36

Evidently we cannot have, 6 = 4 or 16 = 36.

This is due to the fact that √16 and √36 are not surds, they represent rational numbers.


3. If a + √x = b + √y where a, b are both rationals and √x, √y are both surds then, a = b i.e. rational parts of two sides are equal and x = y i.e., irrational parts of two sides are equal.

4. If a - √x = b - √y where a, b are both rationals and √x, √y are both surds then, a = b i.e. rational parts of two sides are equal and x = y i.e., irrational parts of two sides are equal.

5. If a + √x = 0, then a = 0 and x = 0.

6. If a - √x = 0, then a = 0 and x = 0.

7. If a + √x = b + √y then, a - √x = b - √y

8. If √(a + √x) = √b + √y then √(a - √x) = √b - √y

9. Identically, if √(a - √x) = √b - √y then √(a - √x) = √b - √y.





11 and 12 Grade Math

From Properties of Surds to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. What is Area in Maths? | Units to find Area | Conversion Table of Area

    Jul 17, 25 01:06 AM

    Concept of Area
    The amount of surface that a plane figure covers is called its area. It’s unit is square centimeters or square meters etc. A rectangle, a square, a triangle and a circle are all examples of closed pla…

    Read More

  2. Worksheet on Perimeter | Perimeter of Squares and Rectangle | Answers

    Jul 17, 25 12:40 AM

    Most and Least Perimeter
    Practice the questions given in the worksheet on perimeter. The questions are based on finding the perimeter of the triangle, perimeter of the square, perimeter of rectangle and word problems. I. Find…

    Read More

  3. Formation of Square and Rectangle | Construction of Square & Rectangle

    Jul 16, 25 11:46 PM

    Construction of a Square
    In formation of square and rectangle we will learn how to construct square and rectangle. Construction of a Square: We follow the method given below. Step I: We draw a line segment AB of the required…

    Read More

  4. Perimeter of a Figure | Perimeter of a Simple Closed Figure | Examples

    Jul 16, 25 02:33 AM

    Perimeter of a Figure
    Perimeter of a figure is explained here. Perimeter is the total length of the boundary of a closed figure. The perimeter of a simple closed figure is the sum of the measures of line-segments which hav…

    Read More

  5. Formation of Numbers | Smallest and Greatest Number| Number Formation

    Jul 15, 25 11:46 AM

    In formation of numbers we will learn the numbers having different numbers of digits. We know that: (i) Greatest number of one digit = 9,

    Read More