# 10 Examples of Surds in Mathematics

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.

**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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