√5

Definition

√5, or the square root of 5, is an irrational number that cannot be expressed as a fraction of two integers. It is approximately equal to 2.236 and falls under the category of non-repeating, non-terminating decimals. As an irrational number, √5 plays a significant role in various mathematical concepts, particularly in geometry, where it can appear in calculations involving right triangles or the Pythagorean theorem.

5 Must Know Facts For Your Next Test

1. √5 is approximately 2.236067977, making it an example of a non-repeating and non-terminating decimal.
2. The number √5 cannot be accurately represented as a fraction, which is why it is classified as an irrational number.
3. In geometry, √5 often arises in problems involving right triangles where one side length is 2 and the other side length is 1.
4. The irrationality of √5 can be proven using a method called proof by contradiction, demonstrating that no two integers can satisfy the equation x² = 5.
5. √5 plays a role in various mathematical formulas and can appear in calculations involving quadratic equations or roots.

Review Questions

• How can you demonstrate that √5 is an irrational number?
  • To show that √5 is irrational, you can use proof by contradiction. Assume that √5 can be expressed as a fraction in simplest form, say p/q. Squaring both sides gives 5 = p²/q², leading to p² = 5q². This indicates that p² is divisible by 5, implying p must also be divisible by 5. If p = 5k for some integer k, substituting back leads to a contradiction since it suggests q must also be even, meaning both p and q have a common factor of 5. This contradicts our original assumption that p/q was in simplest form.
• In what scenarios might you encounter √5 in practical applications?
  • You might come across √5 in various fields such as engineering or architecture when working with right triangles. For example, if you need to calculate the diagonal length of a rectangle with sides of lengths 2 and 1, applying the Pythagorean theorem would require determining the hypotenuse as √(2² + 1²) = √(4 + 1) = √5. This illustrates how understanding irrational numbers like √5 is essential for solving real-world problems involving geometry.
• Evaluate how understanding √5 contributes to broader mathematical concepts like algebra and geometry.
  • Understanding √5 helps in grasping broader mathematical principles such as algebraic manipulation and geometric relationships. In algebra, it aids in solving quadratic equations where irrational solutions are possible. In geometry, recognizing the significance of √5 allows for better comprehension of the properties of shapes and dimensions. This knowledge facilitates deeper insights into more complex mathematical theories and applications, highlighting how foundational concepts connect across different areas of mathematics.

Related terms

Irrational Numbers: Numbers that cannot be expressed as a simple fraction and have non-repeating, non-terminating decimal representations.

Square Root: A value that, when multiplied by itself, gives the original number; for example, √5 is the number that satisfies the equation x² = 5.

Pythagorean Theorem: A mathematical principle that relates the lengths of the sides of a right triangle, stating that the square of the hypotenuse equals the sum of the squares of the other two sides.