Question

The decimal equivalent of \((11001)_{2}\) is (a) 25 (b) 50 (c) 35 (d) 15

Short answer

The decimal equivalent of \( (11001)_{2} \) is 25, hence option (a) is correct.

Step-by-step solution

Identify Powers of 2 for Each Digit

Starting from the right side (Least significant bit), assign the power of 2 to each digit. For the binary digit \(11001\), powers of 2 will be \(2^0, 2^1, 2^2, 2^3, 2^4\) moving from right to left.

Multiply Powers of 2 with Corresponding Digits

Multiply each digit by its corresponding power of 2. For '1's multiply with the associated power of 2, for '0's multiply with 0. This gives the sequence: \(1*2^4, 1*2^3, 0*2^2, 0*2^1, 1*2^0\)

Add up All Values

Add the values we get from step 2, which are: \(1*2^4 = 16\), \(1*2^3 = 8\), \(0*2^2 = 0\), \(0*2^1 = 0\), \(1*2^0 = 1\). Summing these up: 16+8+0+0+1 = 25

Key concepts

Powers of 2

When dealing with binary to decimal conversion, it is crucial to understand the concept of 'powers of 2.' In binary arithmetic, each digit represents an increasing power of 2, starting with the least significant bit (LSB) on the right.
Each position represents a successive power, much like in the decimal system where each position is a power of 10. For example, the binary number \(11001\) is read as:

• The rightmost digit, or LSB, is multiplied by \(2^0\).
• The next digit is multiplied by \(2^1\).
• This pattern continues moving left, increasing the power by one for each position.

So, in our example of \(11001\), these correspond to the powers \(2^4, 2^3, 2^2, 2^1,\) and \(2^0.\)
The "1" or "0" in each binary position determines whether we apply the associated power of 2 to the final sum, as we will add it when calculating the decimal equivalent.

Binary Arithmetic

Binary arithmetic involves calculating numerical values using the binary number system, which includes only two digits – 0 and 1. Converting binary numbers to decimal numbers requires specific steps that leverage these binary digits. Each binary digit represents a power of 2, and to convert, you multiply each binary digit by its corresponding power of 2.
For instance, with the binary number \(11001\):

• The digit '1' at the leftmost position is multiplied by \(2^4\).
• The next '1' is multiplied by \(2^3\).
• The middle '0' means we skip multiplying by \(2^2\) for this digit.
• Similarly, the following '0' results in no contribution from \(2^1\).
• The final '1' at the rightmost position is multiplied by \(2^0\).

Each non-zero binary digit effectively "activates" its associated power of 2 in the decimal calculation. By performing these multiplications and summing the results, you transition from the binary representation to its decimal equivalent.

Decimal System

The decimal system is a base-10 numerical system most commonly used in daily life. Unlike binary, which is base-2, decimal uses ten digits ranging from 0 to 9. Each position in a decimal number represents a power of 10, increasing from right to left.
In the context of converting binary to decimal, understanding the decimal system helps us interpret and use the result of the conversion. When we've calculated the number \(25\) from a binary value, it means that the computed sum of powers of 2 aligns with the decimal number 25.
This corresponds to:

• Two lots of 10 (as \(2 \times 10\)) and
• Another 5 as individual units, summing up to a total of 25,

illustrating how binary values transform neatly into familiar decimal numbers through arithmetic operations—specifically, using powers of 2 for binary versus powers of 10 for decimal.