Binary notation

When you were first taught how to write num­bers and add them to­gether, you were prob­a­bly told some­thing about the “ones place”, the “tens place”, the “hun­dreds place”, and so on. Each digit fur­ther to the left rep­re­sents a larger mul­ti­ple of ten, and the mul­ti­ples are added to­gether to get the num­ber you want—so \(8207\) is counted as \((7 \times 10^0) + (0 \times 10^1) + (2 \times 10^2) + (8 \times 10^3)\), or “seven ones, zero tens, two hun­dreds, and eight thou­sands”. But why use pow­ers of ten? What’s so spe­cial about that num­ber?

Well, noth­ing, ac­tu­ally. We prob­a­bly only use pow­ers of ten be­cause we hap­pen to have evolved with ten fingers—any num­ber big­ger than one will work just as well, al­though it may look strange at first. Since two is the small­est in­te­ger larger than one, a bi­nary num­ber no­ta­tion—a no­ta­tion that uses pow­ers of two—is one of the sim­plest pos­si­ble. noteThe word “bi­nary” has the same root as the word “bi­cy­cle”—bi mean­ing “two”.

Bi­nary no­ta­tion uses only two digits, \(0\) and \(1\), and each “place” to the left goes up by a power of two in­stead of ten (in other words, it dou­bles). For ex­am­ple, in bi­nary no­ta­tion, the num­ber \(11010\) is counted as \((0 \times 2^0) + (1 \times 2^1) + (0 \times 2^2) + (1 \times 2^3) + (1 \times 2^4)\), or “zero ones, one two, zero fours, one eight, and one six­teen”. Trans­lat­ing back to the fa­mil­iar base ten, we would write it as \(26\).

You may no­tice that bi­nary no­ta­tion tends to be a bit longer than dec­i­mal (for ex­am­ple, “11010” takes more char­ac­ters to write than “26″). It’s also more difficult to read, un­less you have a lot of prac­tice with it. So why would any­one use it? Well, for one thing, bi­nary no­ta­tion is of­ten very con­ve­nient for talk­ing about pow­ers of two, for in­stance when work­ing with base-two log­a­r­ithms or log odds, or when work­ing with some quan­tity mea­sured in bits. It’s also es­sen­tial for work­ing with com­put­ers, as all mod­ern com­put­ers store and ma­nipu­late data, on the low­est level, ex­clu­sively us­ing bi­nary no­ta­tion.


  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.