Internet, Networking, & Security Home Networking 38 38 people found this article helpful Working with Binary Numbers By Bradley Mitchell Writer An MIT graduate who brings years of technical experience to articles on SEO, computers, and wireless networking. our editorial process LinkedIn Bradley Mitchell Updated April 11, 2019 Paul Conrath/Getty Images Home Networking The Wireless Connection Routers & Firewalls Network Hubs ISP Broadband Ethernet Installing & Upgrading Wi-Fi & Wireless Tweet Share Email Binary and hexadecimal numbers are two alternatives to the traditional decimal numbers we use in daily life. Critical elements of computer networks like addresses, masks, and keys all involve binary or hexadecimal numbers. Understanding how such binary and hexadecimal numbers work is essential in building, troubleshooting, and programming any network. Bits and Bytes This article series assumes a basic understanding of computer bits and bytes. Binary and hexadecimal numbers are the natural mathematical way to work with the data stored in bits and bytes. Binary Numbers and Base Two Binary numbers all consist of combinations of the two digits '0' and '1'. These are some examples of binary numbers:11010101111101111000000 10101000 00001100 01011101 Engineers and mathematicians call the binary numbering system a base-two system because binary numbers only contain the two digits '0' and '1'. By comparison, our normal decimal number system is a base-ten system that uses the ten digits '0' through '9'. Hexadecimal numbers are a base-sixteen system. Converting From Binary to Decimal Numbers All binary numbers have equivalent decimal representations and vice versa. To convert binary and decimal numbers manually, you must apply the mathematical concept of positional values. The positional value concept is simple: With both binary and decimal numbers, the actual value of each digit depends on its position ("how far to the left") within the number. For example, in the decimal number 124, the digit '4' represents the value "four," but the digit '2' represents the value "twenty," not "two." The '2' represents a larger value than the '4' in this case because it is positioned further to the left in the number. Likewise in the binary number 1111011, the rightmost '1' represents the value "one," but the leftmost '1' represents a much higher value ("sixty-four" in this case). In mathematics, the base of the numbering system determines how much to value digits by position. For base-ten decimal numbers, multiply each digit on the left by a progressive factor of 10 to calculate its value. For base-two binary numbers, multiply each digit on the left by a progressive factor of 2. Calculations always work from right to left. In the above example, the decimal number 123 works out to: 3 + (10 * 2) + (10*10 * 1) = 123 and the binary number 1111011 converts to decimal as: 1 + (2 * 1) + (2*2 * 0) + (4*2 * 1) + (8*2 * 1)+ (16*2 * 1) + (32*2 * 1) = 123 Therefore, the binary number 1111011 is equal to the decimal number 123. Converting From Decimal to Binary Numbers To convert numbers in the opposite direction, from decimal to binary, requires successive division rather than progressive multiplication. To manually convert from a decimal to a binary number, start with the decimal number and begin dividing by the binary number base (base "two"). For each step the division results in a remainder of 1, use '1' in that position of the binary number. When the division results in a remainder of 0 instead, use '0' in that position. Stop when the division results in a value of 0. The resulting binary numbers are ordered from right to left. For example, the decimal number 109 converts to binary as follows: 109 / 2 = 54 remainder 154 / 2 = 27 remainder 027 / 2 = 13 remainder 113 / 2 = 6 remainder 16 / 2 = 3 remainder 03 / 2 = 1 remainder 11 / 2 = 0 remainder 1 The decimal number 109 equals the binary number 1101101.