Question

Translate \(45 \mathrm{~B}_{16}\) to a base 10 number.

Short answer

69

Step-by-step solution

Identify the base-16 number components

The number given is 45 in base-16. This means it consists of two digits: 4 and 5.

Write down the value of each digit

In base-16, the rightmost digit represents 16^0, the next digit to the left represents 16^1, and so on. Thus, the digits 4 and 5 represent the following: 4 represents 4 * 16^1 and 5 represents 5 * 16^0.

Calculate the decimal (base-10) value of each digit

Calculate the individual contributions of each digit in decimal: - For the digit 4: 4 * 16^1 = 4 * 16 = 64 - For the digit 5: 5 * 16^0 = 5 * 1 = 5

Summing the values

Add the calculated values from the previous step: 64 + 5

Finalize the base-10 value

The sum of the values is 64 + 5 = 69. Thus, 45 in base-16 is 69 in base-10.

Key concepts

base-16 to base-10 conversion

When converting a number from base-16 (hexadecimal) to base-10 (decimal), it's important to understand the value each position holds. In base-16, each digit represents a power of 16, starting from 16^0 at the rightmost digit.
For instance, in the number 45B₁₆:

• The digit B (which is 11 in decimal) is in the 16^0 place.
• The digit 5 is in the 16^1 place.
• The digit 4 is in the 16^2 place.

To convert this to base-10, you calculate the contribution of each digit multiplied by its positional value and then sum these contributions.
For 45B₁₆:

• 4 * 16^1 = 64
• 5 * 16^0 = 5

Adding these gives you: 64 + 5 = 69. Therefore, 45 in base-16 is 69 in base-10.

hexadecimal system

The hexadecimal system, or base-16, is a numeral system used extensively in computing and digital electronics. It uses sixteen symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F.
Each symbol represents a value from 0 to 15.

• A = 10
• B = 11
• C = 12
• D = 13
• E = 14
• F = 15

The main advantage of the hexadecimal system is that it can represent large numbers compactly. For example, the decimal number 255 is FF in hexadecimal.
This system is particularly useful in computing because each hexadecimal digit translates directly into a 4-bit binary sequence.

• For example, the hexadecimal number A3 translates to binary as 1010 0011.

Understanding and using the hexadecimal system is fundamental for professionals dealing with low-level programming and hardware design.

number systems

A number system defines how we write and understand numbers. The most commonly known number system is the decimal (base-10) system, which uses 10 symbols (0-9). However, other number systems like binary (base-2), octal (base-8), and hexadecimal (base-16) are also widely used in various fields, especially in computing.
Each number system has a different base, which determines how place values work.

• In decimal, each place represents a power of 10.
• In binary, each place represents a power of 2.
• In octal, each place represents a power of 8.
• In hexadecimal, each place represents a power of 16.

For example, the binary number 1011 converts to decimal as:

• (1 * 2^3) + (0 * 2^2) + (1 * 2^1) + (1 * 2^0) = 8 + 0 + 2 + 1 = 11.

Understanding different number systems is crucial because it helps in areas like computer science, electrical engineering, and related fields where different systems are frequently used.