## binary
Each place to the left in a binary number represents the next highest power of two. The binary number 10110 _{2}, for example, means 1 × 2^{4}
+ 0 × 2^{3} + 1 × 2^{2} + 1 × 2^{1} + 0 × 2^{0},
or 22_{10} in the familiar decimal
notation. Non-integers can be represented by using negative powers, which
are set off from the other digits by means of a radix point
(called a decimal point in base 10). The binary number
11.01_{2} thus means 1 × 2^{1} + 1 × 2^{0}
+ 0 × 2^{-1} + 1 × 2^{-2} which equals 3.25_{10}.
A number that terminates in decimal doesn't necessarily do so in binary
(for example, 0.310 = 0.0100110011001..._{2}), and vice versa. An
irrational number, however, is
non-periodic in both systems; for example, π
= 3.1415926..._{10} = 11.001001000011111..._{2}. As the scientist-hero of Fred Hoyle's science fiction novel A For Andromeda explains: It's arithmetic expressed entirely by the figures 0 and 1, instead of the figures 1 to 10, which we normally use and which we call denary. 0 and 1, you see, could be dot and dash... the binary system is basic; it's based on positive and negative, yes and no ... – it's universal.Because of its fundamental nature it has been suggested that binary may represent a lingua franca for communication between technological civilizations in the Galaxy. ## Working in binary TweetAt first, long numbers in the binary system seem a little difficult to understand. If we look at the diagram below it will make it clearer. Just as in the decimal system, the right-hand column is for units. The next column is for the number of 10s, the next for (10 × 10)s, and the left-hand column for (10 × 10 × 10)s. If we want to change a binary number into a decimal number we must remember that 10 in the binary system is the equivalent of 2 in the decimal system. Let's change the number 1111 in the binary system into its equivalent in the decimal system. The right-hand number is 1. The next number, the binary equivalent of 10, is 2. The next (10 × 10) equals 2 × 2, or 4. And the left-hand number corresponds to (10 × 10 × 10), so in the decimal system it is 2 × 2 × 2, or 8. Add these together, and we get 15. When adding in the binary system, it's important to remember that 1 × 1, which would equal 2 in the decimal system, equals 10 in the binary system. When multiplying in the binary system, we write it out as in long multiplication. ## Related categories• ARITHMETIC• TYPES OF NUMBER • COMPUTERS, ARTIFICIAL INTELLIGENCE, AND CYBERNETICS | |||||||||||

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