Please help me with the below questions.
Write a rational approximation, to the nearest thousandth, for each number:
1) Square root 14
2) Square root 1.4
3) Pie
4) Pie -3
1: square root 14 = 3.74165..... so to the thousandth would be 3.74
2 square root 1.4 = 1.18
3 3.14159265357..... would be 3.14
4 .14
To find rational approximations for the given numbers, we need to find fractions that are close to the decimal values. Here's how you can approximate each number:
1) Square root of 14:
- Start by estimating the square root of 14. As 14 is between the perfect squares of 9 (3^2) and 16 (4^2), you can expect the square root of 14 to be between 3 and 4.
- Now, use a calculator or a table to find a more accurate decimal value. In this case, the square root of 14 ≈ 3.741
- To find the rational approximation, convert the decimal value to a fraction. In this case, it becomes 3.741 ≈ 3731/1000.
2) Square root of 1.4:
- Since 1.4 is not a perfect square, we can use estimation to find the rational approximation.
- Recognize that 1.4 is between the perfect squares of 1 (1^2) and 4 (2^2). So, the square root of 1.4 will be between 1 and 2.
- Using a calculator or a table, the square root of 1.4 ≈ 1.183.
- Convert this decimal to a fraction: 1.183 ≈ 1183/1000.
3) Pi (π):
- Pi is an irrational number, which means it cannot be expressed as a finite decimal or fraction. It is approximately equal to 3.14159.
- Since π is already a decimal representation, there is no need to find a rational approximation.
4) Pi - 3:
- Subtracting 3 from an irrational number like π does not change its irrationality, so the result is also an irrational number. Therefore, there is no rational approximation for π - 3.
Remember, rational approximations provide an estimate in fraction form that is close to the decimal value.