Which Two Whole Numbers Is The Following Irrational Number Between?$\[ \square \ \textless \ \sqrt{21} \ \textless \ \square \\]A. Between 1 And 2 B. Between 2 And 3 C. Between 3 And 4 D. Between 4 And 5
Introduction
Irrational numbers are real numbers that cannot be expressed as a finite decimal or fraction. They have an infinite number of digits after the decimal point, and these digits never repeat in a predictable pattern. One of the most well-known irrational numbers is the square root of a number that is not a perfect square. In this article, we will explore the concept of irrational numbers and square roots, and we will determine which two whole numbers the square root of 21 is between.
What are Irrational Numbers?
Irrational numbers are real numbers that cannot be expressed as a finite decimal or fraction. They have an infinite number of digits after the decimal point, and these digits never repeat in a predictable pattern. Examples of irrational numbers include the square root of 2, the square root of 3, and pi. These numbers are called irrational because they cannot be expressed as a simple fraction, and their decimal expansions go on forever without repeating.
What is a Square Root?
A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16. The square root of a number can be a rational or irrational number, depending on whether the number is a perfect square or not. Perfect squares are numbers that can be expressed as the product of an integer with itself, such as 1, 4, 9, 16, and so on.
The Square Root of 21
The square root of 21 is an irrational number because 21 is not a perfect square. To determine which two whole numbers the square root of 21 is between, we need to find the square roots of the two perfect squares that are closest to 21. The two perfect squares that are closest to 21 are 16 and 25, because 16 is less than 21 and 25 is greater than 21.
Finding the Square Roots of 16 and 25
The square root of 16 is 4, because 4 multiplied by 4 equals 16. The square root of 25 is 5, because 5 multiplied by 5 equals 25.
Determining Which Two Whole Numbers the Square Root of 21 is Between
Since the square root of 21 is between 4 and 5, we can conclude that the correct answer is between 4 and 5.
Conclusion
In conclusion, the square root of 21 is an irrational number because 21 is not a perfect square. To determine which two whole numbers the square root of 21 is between, we need to find the square roots of the two perfect squares that are closest to 21. The two perfect squares that are closest to 21 are 16 and 25, and the square roots of these numbers are 4 and 5, respectively. Therefore, the correct answer is between 4 and 5.
Final Answer
Introduction
In our previous article, we explored the concept of irrational numbers and square roots, and we determined which two whole numbers the square root of 21 is between. In this article, we will answer some frequently asked questions about irrational numbers and square roots.
Q: What is the difference between a rational and an irrational number?
A: A rational number is a number that can be expressed as a finite decimal or fraction, such as 3/4 or 0.5. An irrational number is a number that cannot be expressed as a finite decimal or fraction, such as the square root of 2 or pi.
Q: What is a perfect square?
A: A perfect square is a number that can be expressed as the product of an integer with itself, such as 1, 4, 9, 16, and so on.
Q: How do I determine which two whole numbers a square root is between?
A: To determine which two whole numbers a square root is between, you need to find the square roots of the two perfect squares that are closest to the number. For example, to determine which two whole numbers the square root of 21 is between, you would find the square roots of 16 and 25, which are 4 and 5, respectively.
Q: Can all irrational numbers be expressed as a decimal?
A: Yes, all irrational numbers can be expressed as a decimal, but the decimal expansion will go on forever without repeating.
Q: What is the significance of irrational numbers in mathematics?
A: Irrational numbers are significant in mathematics because they are used to describe many real-world phenomena, such as the ratio of a circle's circumference to its diameter (pi) and the square root of 2, which is used in geometry and trigonometry.
Q: Can irrational numbers be used in real-world applications?
A: Yes, irrational numbers are used in many real-world applications, such as engineering, physics, and computer science. For example, the use of pi in calculating the area and circumference of circles is essential in architecture and engineering.
Q: How do I calculate the square root of a number?
A: To calculate the square root of a number, you can use a calculator or a mathematical formula. The formula for calculating the square root of a number is: √x = x^(1/2).
Q: What is the difference between a square root and a cube root?
A: A square root is a value that, when multiplied by itself, gives the original number. A cube root is a value that, when multiplied by itself three times, gives the original number.
Conclusion
In conclusion, irrational numbers and square roots are fundamental concepts in mathematics that have many real-world applications. By understanding these concepts, you can solve problems and make calculations with ease.
Final Tips
- Always use a calculator or a mathematical formula to calculate square roots and other mathematical operations.
- Practice, practice, practice! The more you practice, the more comfortable you will become with irrational numbers and square roots.
- Don't be afraid to ask for help if you are struggling with a problem or concept.
Common Mistakes to Avoid
- Don't confuse rational and irrational numbers.
- Don't forget to use a calculator or a mathematical formula to calculate square roots and other mathematical operations.
- Don't be afraid to ask for help if you are struggling with a problem or concept.
Glossary of Terms
- Rational number: A number that can be expressed as a finite decimal or fraction.
- Irrational number: A number that cannot be expressed as a finite decimal or fraction.
- Perfect square: A number that can be expressed as the product of an integer with itself.
- Square root: A value that, when multiplied by itself, gives the original number.
- Cube root: A value that, when multiplied by itself three times, gives the original number.