Question

the decimal expansion of the irrational number \\(\pi\\) begins 3.141592653589793.... use your calculator to determine which of the following rational numbers is the best approximation for the irrational number \\(\pi\\).\\(\frac{22}{7}, \frac{355}{113}, \frac{103,993}{33,102}, \frac{2,508,429,787}{798,458,000}\\)your calculator may tell you that some of these numbers are equal to \\(\pi\\), but that just indicates that the number agrees with \\(\pi\\) for as many decimal places as your calculator can handle (usually 10 - 14). no rational number is exactly equal to \\(\pi\\).choose the correct answer below.\\(\bigcirc\\) a. \\(\frac{2,508,429,787}{798,458,000}\\)\\(\bigcirc\\) b. \\(\frac{355}{113}\\)\\(\bigcirc\\) c. \\(\frac{103,993}{33,102}\\)\\(\bigcirc\\) d. \\(\frac{22}{7}\\)

Explanation:

Step1: Calculate each fraction's decimal

• For $\frac{22}{7}$: $22\div7\approx3.142857$
• For $\frac{355}{113}$: $355\div113\approx3.14159292$
• For $\frac{103993}{33102}$: $103993\div33102\approx3.141592653$
• For $\frac{2508429787}{798458000}$: $2508429787\div798458000\approx3.1415926535$

Step2: Compare with $\pi\approx3.141592653589793$

• $\frac{22}{7}\approx3.142857$ (differs at 3rd decimal)
• $\frac{355}{113}\approx3.14159292$ (differs at 7th decimal)
• $\frac{103993}{33102}\approx3.141592653$ (differs at 10th decimal)
• $\frac{2508429787}{798458000}\approx3.1415926535$ (matches $\pi$ up to 11th decimal)

Answer:

A. $\frac{2,508,429,787}{798,458,000}$