Mastering the Binary System: Exercises I The binary system is the foundational language of modern computing. Every digital photo, video, and text file breaks down into a series of 1s and 0s. Understanding how to read and convert binary numbers is an essential skill for computer science, networking, and digital electronics.
This guide provides practical, step-by-step exercises designed to build your fluency in binary-to-decimal and decimal-to-binary conversions. Understanding the Base-2 Grid
Unlike the decimal system (Base-10), which uses ten digits (0–9) and values that increase by powers of 10, the binary system (Base-2) uses only two digits (0 and 1) and values that increase by powers of 2.
To convert numbers quickly, always write out your binary positional value grid from right to left, starting at 1: Exercise Set A: Binary to Decimal Conversion
To convert a binary number to a decimal number, place the binary digits into your grid. Add together the positional values where a “1” appears. Ignore the positions with a “0”. Example: Convert 10101000 to decimal. There is a 1 under 128, 32, and 8. Calculation: 128 + 32 + 8 = 168. Answer: 168 Practice Problems: Convert 00001101 to decimal. Convert 00111010 to decimal. Convert 11000111 to decimal. Exercise Set B: Decimal to Binary Conversion
To convert a decimal number to binary, use the subtraction method. Start from the largest positional value on your grid (128) and move left to right.
If the decimal number is equal to or greater than the grid value, put a “1” in that column and subtract the grid value from your number.
If the decimal number is smaller, put a “0” and move to the next column. Repeat until you reach 0. Example: Convert 75 to binary. Can we subtract 128? No → 0 Can we subtract 64? Yes → 1 (Remainder: 75 – 64 = 11) Can we subtract 32? No → 0 Can we subtract 16? No → 0 Can we subtract 8? Yes → 1 (Remainder: 11 – 8 = 3) Can we subtract 4? No → 0 Can we subtract 2? Yes → 1 (Remainder: 3 – 2 = 1) Can we subtract 1? Yes → 1 (Remainder: 1 – 1 = 0) Answer: 01001011 Practice Problems: Convert 19 to an 8-bit binary number. Convert 84 to an 8-bit binary number. Convert 242 to an 8-bit binary number. Answer Key and Explanations Set A Answers 13 (8 + 4 + 1) 58 (32 + 16 + 8 + 2) 199 (128 + 64 + 4 + 2 + 1) Set B Answers 00010011 (16 + 2 + 1) 01010100 (64 + 16 + 4) 11110010 (128 + 64 + 32 + 16 + 2) Next Steps to Mastery
Mastering binary requires muscle memory. Practice creating your own 8-bit numbers, converting them back and forth, and verifying your results using a standard calculator set to “Programmer” mode. In our next installment, Exercises II, we will explore binary addition and how computers represent negative numbers using Two’s Complement.
If you want to keep practicing, let me know if you would like to try larger numbers, learn how to convert fractional binary numbers, or see how hexadecimal shortcuts work.
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