# Binary & Decimal Conversion Part of the Electronics Merit Badge Digital Resource Guide: [Introduction & Overview](https://merit-badge.university/merit-badges/electronics/guide/index.md) *4. Circuits & Digital Logic* > **4b.** Show how to change three decimal numbers into binary numbers and three binary numbers into decimal numbers. In [Req 4a](https://merit-badge.university/merit-badges/electronics/guide/req4a/), you learned that digital electronics speak in binary — a number system with only two digits: 0 and 1. Now you need to become fluent in translating between the decimal system (the base-10 system you use every day) and binary (the base-2 system that every computer uses internally). ## Why Binary Matters Computers use binary because transistors — the tiny switches inside every chip — have two states: on (1) and off (0). A single binary digit is called a **bit**. Eight bits grouped together form a **byte**, which can represent any number from 0 to 255. Every number, letter, color, and sound in a computer is ultimately stored as a pattern of bits. ## Understanding Place Values The key to conversion is understanding **place values**. In the decimal system, each position is worth 10 times more than the one to its right: | Position | Thousands | Hundreds | Tens | Ones | |---|---|---|---|---| | Place value | 1000 | 100 | 10 | 1 | | Power of 10 | 10^3 | 10^2 | 10^1 | 10^0 | Binary works the same way, but each position is worth **2 times** more than the one to its right: | Position | 128s | 64s | 32s | 16s | 8s | 4s | 2s | 1s | |---|---|---|---|---|---|---|---|---| | Place value | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | | Power of 2 | 2^7 | 2^6 | 2^5 | 2^4 | 2^3 | 2^2 | 2^1 | 2^0 | ## Decimal to Binary — The Division Method To convert a decimal number to binary, repeatedly divide by 2 and record the remainders. Read the remainders from bottom to top. ### Example: Convert 42 to binary | Step | Division | Quotient | Remainder | |---|---|---|---| | 1 | 42 / 2 | 21 | 0 | | 2 | 21 / 2 | 10 | 1 | | 3 | 10 / 2 | 5 | 0 | | 4 | 5 / 2 | 2 | 1 | | 5 | 2 / 2 | 1 | 0 | | 6 | 1 / 2 | 0 | 1 | Reading the remainders from bottom to top: **42 in decimal = 101010 in binary.** ### Example: Convert 13 to binary | Step | Division | Quotient | Remainder | |---|---|---|---| | 1 | 13 / 2 | 6 | 1 | | 2 | 6 / 2 | 3 | 0 | | 3 | 3 / 2 | 1 | 1 | | 4 | 1 / 2 | 0 | 1 | Reading bottom to top: **13 = 1101** ### Example: Convert 200 to binary | Step | Division | Quotient | Remainder | |---|---|---|---| | 1 | 200 / 2 | 100 | 0 | | 2 | 100 / 2 | 50 | 0 | | 3 | 50 / 2 | 25 | 0 | | 4 | 25 / 2 | 12 | 1 | | 5 | 12 / 2 | 6 | 0 | | 6 | 6 / 2 | 3 | 0 | | 7 | 3 / 2 | 1 | 1 | | 8 | 1 / 2 | 0 | 1 | Reading bottom to top: **200 = 11001000** ## Binary to Decimal — The Addition Method To convert binary to decimal, write out the place values above each bit, then add up the place values wherever you see a 1. ### Example: Convert 110101 to decimal | Place value | 32 | 16 | 8 | 4 | 2 | 1 | |---|---|---|---|---|---|---| | Binary digit | 1 | 1 | 0 | 1 | 0 | 1 | Add the place values where the digit is 1: 32 + 16 + 4 + 1 = **53** ### Example: Convert 10010 to decimal | Place value | 16 | 8 | 4 | 2 | 1 | |---|---|---|---|---|---| | Binary digit | 1 | 0 | 0 | 1 | 0 | Add: 16 + 2 = **18** ### Example: Convert 11111111 to decimal | Place value | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | |---|---|---|---|---|---|---|---|---| | Binary digit | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | Add: 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = **255** > **Did You Know?** > > The number 255 (11111111 in binary) appears constantly in computing. It is the maximum value a single byte can hold. That is why color values on screens range from 0 to 255 — each color channel (red, green, blue) is stored in one byte. Pure white is (255, 255, 255) and pure black is (0, 0, 0). ## Quick Check — Place Value Method for Decimal to Binary There is a faster method that some people prefer. Start with the largest power of 2 that fits into your number, subtract it, and continue: **Convert 45 to binary:** - 45 >= 32? Yes. Write **1**. Remainder: 45 - 32 = 13 - 13 >= 16? No. Write **0**. - 13 >= 8? Yes. Write **1**. Remainder: 13 - 8 = 5 - 5 >= 4? Yes. Write **1**. Remainder: 5 - 4 = 1 - 1 >= 2? No. Write **0**. - 1 >= 1? Yes. Write **1**. Remainder: 0 Result: **45 = 101101** ![A visual reference card showing the powers of 2 from 1 to 128 with worked examples of decimal-to-binary and binary-to-decimal conversions](https://ik.imagekit.io/doulacoop/tr:w-800/mbu-assets/merit-badges/electronics/guide/binary-conversion-cheatsheet?v=4bd4e23a) > **Tip** > > Practice with numbers you know. Your age, your house number, the year you were born — convert them all to binary and back. The more you practice, the faster it gets. Many Scouts find that the place-value method feels more intuitive than the division method after a few tries. [RapidTables Binary-Decimal Converter](https://www.rapidtables.com/convert/number/binary-to-decimal.html): Online converter to check your work. Convert back and forth between binary and decimal to verify your manual calculations. --- - Previous: [Control, Logic & Analog](https://merit-badge.university/merit-badges/electronics/guide/req4a/index.md) - Next: [Build a Circuit Project](https://merit-badge.university/merit-badges/electronics/guide/req4c/index.md)