A (rational) estimate for r also provides an estimate for, since if 0 < a It is a quirk of decimal notation that sometimes two expansions describe exactly the same number. Consider the number x = 4.999.... This is a repeating decimal, and every digit after the decimal point is a 9. (a) What is the decimal expansion of 10x? (b) Compute 10x - x. (In this case, subtraction works exactly the way you'd like it to, but you should stop and think about what it actually means to subtract infinite decimal expansions. How does this relate to our earlier discussion on lower and upper bounds?) (c) Solve the equation 9x - 45 = 0 for x, and conclude that 4.999... = 5.000....