Question

Suppose 0 < = x < = 1 and that x = (O.a1a2a3. . .)2. Give a geometric meaning to the coefficients a1, a2, a3.... in terms of where x is located in the interval [0, 1].

Answer 1

The coefficients a1, a2, a3,... in the binary representation of x represent the digits after the decimal point. Each coefficient corresponds to a power of 2 and determines the value of x in the interval [0, 1].

Step-by-step explanation:

The coefficients a1, a2, a3,... represent the digits after the decimal point in the binary representation of the number x. Each coefficient can take on the value of either 0 or 1. For example, if a1 = 1, it means that the first digit after the decimal point is 1. Similarly, if a2 = 0, it means that the second digit after the decimal point is 0.

This binary representation of x determines its location in the interval [0, 1]. Each coefficient corresponds to a power of 2. For example, the first coefficient a1 corresponds to 2⁻¹, the second coefficient a2 corresponds to 2⁻², and so on. By summing up these powers of 2 for the non-zero coefficients, we can determine the value of x within the interval [0, 1].

For example, if x = 0.101_2, the coefficients are a1 = 1, a2 = 0, and a3 = 1.

This corresponds to the value of x = 2⁻¹ + 2⁻³ = 0.5 + 0.125 = 0.625, which lies between 0 and 1.