Binary Numbers (College Board AP® Computer Science Principles): Revision Note

Number systems & base conversion

What are number systems?

• A number base defines how many unique digits are used to represent values in a system

• Decimal (base 10) uses ten digits: 0 to 9

• Binary (base 2) uses two digits: 0 and 1

• Any value that can be represented in decimal can also be represented in binary, and vice versa

• Conversion between number bases is the process of expressing the same value in a different representation

Converting decimal to binary

• Divide the decimal number by 2 repeatedly

• Record the remainder (0 or 1) at each step

• Read the remainders from bottom to top to get the binary value

Example: convert 45 to binary

• Reading the remainders from bottom to top: 101101

• 45 in decimal = 101101 in binary

Converting binary to decimal

• Write out the place value for each bit position (powers of 2, right to left)

• Multiply each binary digit by its place value

• Add the results together

Example: convert 11010110 to decimal

• 128 + 64 + 16 + 4 + 2 = 214

• 11010110 in binary = 214 in decimal

Place value & calculations

How does place value work in binary?

• Each position in a binary number has a place value equal to a power of 2

• The rightmost position is 2^0 (= 1), the next is 2^1 (= 2), then 2^2 (= 4), and so on

• The base of the number system determines the exponent pattern: in binary, each position is worth twice as much as the position to its right

Example: convert 10010011 to decimal

• 128 + 16 + 2 + 1 = 147

• 10010011 in binary = 147 in decimal