Question

List the hexadecimal representations of the integers from 20 to 22.

Short answer

20: 14, 21: 15, 22: 16.

Step-by-step solution

Understand Problem

We need to convert the integers 20, 21, and 22 from decimal to hexadecimal representation.

Convert 20 to Hexadecimal

Divide 20 by 16. The quotient is 1 and the remainder is 4. The hexadecimal representation of 20 is thus 14, where '1' is from the quotient and '4' is the remainder.

Convert 21 to Hexadecimal

Divide 21 by 16. The quotient is 1 and the remainder is 5. The hexadecimal representation of 21 is 15, where '1' is the quotient and '5' is the remainder.

Convert 22 to Hexadecimal

Divide 22 by 16. The quotient is 1 and the remainder is 6. The hexadecimal representation of 22 is 16, where '1' is the quotient and '6' is the remainder.

Key concepts

Binary and Hexadecimal Systems

In computing, numbers are often represented in different systems, particularly binary and hexadecimal. The binary system, or base-2, uses only two digits: 0 and 1. Computers use this system natively because circuits can easily represent these two states with their on (1) and off (0) positions.

The hexadecimal system, or base-16, is used mainly because it provides a more human-friendly way of representing binary-coded values. It uses sixteen symbols: 0 through 9 to represent values zero to nine and the letters A through F to represent ten to fifteen. This system is beneficial in computer science for simplifying lengthy binary representations. For instance, a single hexadecimal digit can represent four binary digits (bits), making it efficient for conveying large amounts of binary information succinctly.

Number Systems Conversion

Converting between different number systems is a crucial skill in computer science. A common conversion task is translating a decimal number, which is base-10, into a hexadecimal number. The steps are straightforward:

• Divide the decimal number by 16.
• Write down the quotient and the remainder separately.
• The remainder represents the last digit of the hexadecimal number.
• Continue dividing the quotient by 16 until you reach zero, noting the remainders.
• Read the remainders from bottom to top to get the final hexadecimal representation.

This method is not only limited to decimal to hexadecimal conversions, as it can be adapted for other bases by altering the divisor accordingly.

Mathematics in Computer Science

In computer science, mathematics plays a critical role in designing and understanding various programming and computational systems. When dealing with number system conversions, particularly from decimal to hexadecimal or binary, mathematical concepts offer a foundation for efficient data representation.

Mathematics helps optimize algorithms that handle this data, ensuring calculations are performed quickly and accurately. Computer scientists often use these conversions when programming, as computers operate in binary, but developers often code using hexadecimal for simplicity.

Understanding mathematical concepts like number bases enables problem-solving and builds the necessary skills for fields such as cryptography, algorithm design, and data compression, all of which rely heavily on efficient data handling and manipulation at the binary and hexadecimal level.