Question

What values have binary representations in which only one of the bits is 1 ? List the binary representations for the smallest six values with this property.

Short answer

1, 10, 100, 1000, 10000, 100000.

Step-by-step solution

Understanding the Problem

First, we need to identify numbers whose binary representations contain exactly one '1'. In binary, numbers are composed of bits (0s and 1s). If there is only one '1' in the binary, then the number is a power of 2, like 1 (which is 2^0), 2 (which is 2^1), etc.

Listing Binary Representations

The smallest values, starting from 1, that have binary representations with only one '1' are as follows: - 1 has a binary representation of 1 (or 0001 in a 4-bit format). - 2 has a binary representation of 10 (or 0010 in a 4-bit format). - 4 has a binary representation of 100 (or 0100 in a 4-bit format). - 8 has a binary representation of 1000. - 16 has a binary representation of 10000. - 32 has a binary representation of 100000.

Key concepts

Powers of Two

The concept of "Powers of Two" is key to understanding binary numbers. A power of two is any integer of the form \(2^n\), where \(n\) is a non-negative integer. This means that when you square the base (which is 2 in this case), the outcome is one of the "special" numbers that play a pivotal role in computing.Some examples are:

• \(2^0 = 1\)
• \(2^1 = 2\)
• \(2^2 = 4\)
• \(2^3 = 8\)
• \(2^4 = 16\)
• \(2^5 = 32\)

Each of these results corresponds to a single '1' in a binary sequence, flanked by '0's. Powers of two are foundational because they help us understand how binary numbers are structured.

Binary Representation

Binary representation is how we express numbers using only two symbols: 0 and 1. This base-2 numeral system is essential in computing, because it is the foundation of all digital electronics.To translate a decimal number (our usual counting method with ten symbols, 0 through 9) into binary involves determining which powers of 2 sum up to the number. For example, the decimal number 10 can be represented in binary as 1010.This breaks down as:

• The '1' in the 2^3 place, meaning \(8\)
• The '0' in the 2^2 place, meaning \(0\)
• The '1' in the 2^1 place, meaning \(2\)
• The '0' in the 2^0 place, meaning \(0\)

Adding these gives \(8 + 0 + 2 + 0 = 10\), thus confirming that 1010 is the binary equivalent of the decimal 10.

Bits in Binary Numbers

Each digit in a binary number is called a "bit", short for "binary digit". In binary representation, each bit either represents a 0 or a 1. Understanding bits is crucial since they serve as the basic unit of storage and information in computing. The place of a bit in binary representation indicates its power of two, thereby contributing a specific value to the number it constructs:

• A '1' in the farthest right bit (2^0) accounts for 1, if present.
• A '1' in the next bit to the left (2^1) accounts for 2, if present.
• This pattern continues, with each shift left representing an increase in power (2^2 for 4, 2^3 for 8, etc.).

Thus, an understanding of individual bits allows one to reconstruct entire numbers. For example, the number '4' in binary, represented as '100', comprises one bit set at 2^2, with no contributions from lower powers.