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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.

## 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, âˆš2, âˆš3, and âˆš5 are all surds since they cannot be expressed as simple fractions.

### 2: Cubic Roots

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

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

For example, âˆš2 + âˆš3 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, âˆš2 Ã— âˆš3 is a surd.

### 5: Division of Surds

Division of surds, like âˆš6 Ã· âˆš2, also results in a surd.

### 6: Rationalizing Denominators

Rationalizing the denominator of a fraction with surds, as in 1/âˆš2, 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 âˆš(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(45Â°) = âˆš2/2, 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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