Question

Convert \((A B 6 C 7 D)_{16}\) to decimal notation.

Short answer

Answer: The decimal equivalent of the hexadecimal number (AB6C7D)₁₆ is 11,234,339.

Step-by-step solution

Identify each digit of the hexadecimal number

The given hexadecimal number is \((AB6C7D)_{16}\). We can see that it has 6 digits. From left to right, the digits are \(A\), \(B\), \(6\), \(C\), \(7\), and \(D\).

Convert the hexadecimal digits to decimal

Before we can multiply the hexadecimal digits by the powers of 16, we first need to convert them to decimal digits. Here are the conversions for the digits \(A\), \(B\), \(C\), and \(D\): - \(A_{16} = 10_{10}\) - \(B_{16} = 11_{10}\) - \(C_{16} = 12_{10}\) - \(D_{16} = 13_{10}\)

Multiply each digit by the corresponding power of 16

Now, we will multiply each decimal digit by the corresponding power of 16 based on its position in the number (starting from 0 for the rightmost digit). The powers are as follows: - \(10_{10} × 16^5\) - \(11_{10} × 16^4\) - \(6_{10} × 16^3\) - \(12_{10} × 16^2\) - \(7_{10} × 16^1\) - \(13_{10} × 16^0\) Calculating the products, we have: - \(10 × 1048576 = 10485760\) - \(11 × 65536 = 720896\) - \(6 × 4096 = 24576\) - \(12 × 256 = 3072\) - \(7 × 16 = 112\) - \(13 × 1 = 13\)

Add the products

Lastly, we add all the products together to find the decimal equivalent of the given hexadecimal number: \(10485760 + 720896 + 24576 + 3072 + 112 + 13 = 11234339\) Therefore, \((AB6C7D)_{16} = 11234339_{10}\).

Key concepts

Hexadecimal Number System

The hexadecimal number system is a base-16 numbering system, which means it uses 16 different symbols to represent values. These symbols are the digits 0 to 9 and the letters A to F. In this system, each digit represents a power of 16, with the rightmost digit indicating 16 to the power of zero, similar to how the decimal system works by powers of 10.

• The digits 0 through 9 represent the same available values in both hexadecimal and the common decimal system.
• The letters A through F represent the decimal values 10 to 15 respectively.

The hexadecimal system is often used in computing and digital electronics because it can represent large binary numbers in a more concise and readable form. Each hexadecimal digit represents four binary digits, also known as bits, which simplifies conversion and representation of binary-coded values.
When handling problems with hexadecimal numbers, understanding these conversions between different bases is crucial for seamless computations.

Number Base Conversion

Number base conversion involves changing a number from one numbering system to another. For hex to decimal conversion, each digit of the hexadecimal number is expanded to its decimal form first and then is multiplied by its respective power of base 16.

• Identify each digit and determine its decimal equivalent.
• Multiply each decimal equivalent by 16 raised to power of its position (index position starts from 0 at the rightmost digit).
• Sum all these results to get the number in the decimal system. This represents the total value of the number when expressed in base 10.

For example, in the hexadecimal number \((AB6C7D)_{16}\), the conversion requires multiplying each digit by its positional power of 16, and then adding these products together. This systematic approach helps ensure the conversion captures the cumulative value represented by each digit's specific place in the overall number.

Power of 16

The concept of 'Power of 16' is central to understanding hexadecimal numbers because it underpins how values are weighted in different positions of the hexadecimal system. Each digit in a hexadecimal number is multiplied by 16 raised to a power that corresponds to the digit's position in the number.

• The rightmost position is 16⁰, which is equal to 1.
• The second position from the right is 16¹, equal to 16.
• The third position is 16², and it increases exponentially with every step to the left.

For the number \((AB6C7D)_{16}\), the powers used are from 16⁰ up to 16⁵, as the number has six digits. Calculating these powers gives weight and significance to each digit based on its position. Understanding and using the power of 16 correctly is essential when converting hexadecimal to decimal, as it reflects how much each digit contributes to the whole number's value.