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10 Examples of Surds in Mathematics

October 17, 2023
written by Rida Mirza

Surds, also known as irrational numbers, are a essential component of mathematics. These numbers cannot be expressed as a simple fraction or decimal, and they often involve square roots or cube roots.

In this article, we will discuss ten examples of surds in mathematics.

image showing example of surds

Examples of Surds

These are 10 examples of surds.

1: Square Roots

Most common surd is the square root of a non-perfect square.

For example, √2, √3, and √5 are all surds since they cannot be expressed as simple fractions.

2: Cubic Roots

Cubic roots, such as ∛7 and ∛11, are also surds. These are numbers that cannot be exactly expressed as a fraction or decimal.

3: Addition of Surds

When you add or subtract surds with different values under the root, they remain surds.

For example, √2 + √3 is a surd, as it cannot be simplified further.

4: Multiplication of Surds

When you multiply surds with different values under the root, the result is also a surd.

For instance, √2 × √3 is a surd.

5: Division of Surds

Division of surds, like √6 ÷ √2, also results in a surd.

6: Rationalizing Denominators

Rationalizing the denominator of a fraction with surds, as in 1/√2, is a common mathematical technique.

7: Pythagorean Theorem

In geometry, the Pythagorean theorem often involves surds.

For example, in a right triangle, the length of the hypotenuse is given by √(a^2 + b^2), where ‘a’ and ‘b’ are the lengths of the two legs.

8: Trigonometric Ratios

Trigonometric functions like sine, cosine, and tangent often result in surd values.

For example, sin(45°) = √2/2, which is a surd.

9: Geometry and Area

In geometric calculations, surds are prevalent when determining the area of complex shapes, especially when non-perfect squares are involved.

10: Engineering and Design

In engineering and design, surds are used to calculate precise measurements and dimensions when working with non-standard shapes and angles.

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