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BinaryNumbersSystem

Introduction

When you hear that the binary system is used in computer science, andthat it involves long strings of ones and zeroes, it can seem very scary.The term "binary" alone simply refers to anything that is limited to one oftwo states. The term binary system refers to a system of counting byusing a series of ones and zeroes.

In our standard system, based on powers of 10, we use the numbers 0through 9 in each space. When I say a number like "1,302", I'm actuallyspeaking of 1 one-thousand, plus 3 hundreds, plus 0 tens, plus 2 ones.

Think of 1,302 our regular base 10 number system like this:

1000s 100s 10s 1s

1 3 0 2

In the binary system, only the numbers 0 and 1 are used in each space.The places themselves, instead of being powers of 10, as above, arepowers of 2. Just like our base 10 number system above, we start with a1s place at the rightmost place:

32s 16s 8s 4s 2s 1s

Just like our own 10s system, the places can go as high as is needed. Ifwe're given the binary number 11001010, we break it down like this:

128s 64s 32s 16s 8s 4s 2s 1s

1 1 0 0 1 0 1 0

So, in this number there are one 128, one 64, no 32s, no 16s, one 8, no4s, one 2 and no 1s. To find the decimal equivalent of 11001010, wesimply add up the spaces where we find ones. That gives us 128 + 64 + 8+ 2, or 202 as the equivalent in our base 10 number system.

Each place (1s, 2s, 4s, and so on) is referred to as a "bit" (short for"binary digit"). If you were to talk about 3 place, you would use the term
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"3-bit", and so on. Eight bits, as a group, is called a "byte". Although it'srarely-used, the term for a 4-bit number is a "nybble".

Binary numbers with their long strings of 1s and 0s can seem like a

more difficult challenge, but there are ways to tackle the task. Several

methods will be described below.

Lewis Jones' 3-Bit Method

Lewis Jonesoriginally developed this system for use with playing cards(seePlayingCardSystems,section 1.1), but works well with any type ofbinary information (including, obviously, binary numbers). In this systemthe binary numbers are broken into groups of three digits. With binary

numbers, there's only eight possible arrangements of a three-digit group:

000

001010011100101110111

Each group is then given a name that describes the locations of the 1s inthe number:

000 - None001 - Top010 - Middle011 - Upper

100 - Bottom101 - Outer110 - Lower111 - All

One of the advantages of the binary system is that we can focus on the

1s like this. After all, if it isn't a 1, it must be a 0.

You should also note that each group's label begins with a different

letter: N, T, M, U, B, O, L, A. This letter alone can be used to instantlyidentify any three-digit group of binary numbers. If you want to

remember several three-digit sequences of binary numbers, you can putthe letters together to form a memorable image.
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Let's say you want to remember the binary sequence 010101110001.First, you would break the sequence up into groups of three digits: 010

101 110 001. Next, you would convert each group to the appropriateletter:

010 - Middle101 - Outer110 - Lower001 - Top

To remember the sequence 010101110001, you simply remember thephrase "MOLT"!

Unfortunately, you may not always get a nice, neat word like "MOLT" inthis system. If this happens, you're free to insert extra i's and e's into the"words", as they have no meaning in this system. NMAU could become

NIMAU (which you can thing of as the name of an imaginary country),and TTLN becomes TITELINE.

Nybble (4-Bit) Method

To remember more bits at a single glance, the above method can beadapted to use 4-bit words instead of 3. With 4 bits, there are now 16possibilities to cover, so they will be described in small groups. Onceagain, the descriptions will focus on where the 1s in the number are.

The first two are the easiest:0000 - None1111 - Every

The next four all involve a single 1 in their number, and are also easy toremember:0001 - First0010 - Second0100 - Third1000 - Bottom(In this method, the leftmost bit is invariably considered to be "lower"

than the rightmost bit)

This group involves two 1s next to each other:0011 - Highest (The two highest numbers are both ones)0110 - Inside (The two ones are "inside" the zeroes)1100 - Minor (The two ones are in the most minor position)


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There are several 4-bit numbers in which have two 1s which aren't nextto each other:

1001 - Outer (The outer two digits are 1s)0101 - Rotating

1010 - Alternating

The "Alternating" and "Rotating" patterns are easily confused with eachother, so there's a built-in mnemonic in the words themselves. The firstvowel in the word "Alternating" is an "A", the 1st letter of the alphabet, sothe leftmost bit is a 1. The first vowel in the word "Rotating" is an "O",which looks like the number 0, therefore the leftmost bit is a 0.

This next group contains three 1s next to each other:

0111 - Upper (the three uppermost numbers are all 1s)1110 - Lower (the three lowermost numbers are all 1s)

The final two remaining combinations contain three 1s each, with a zerosomewhere in the middle:1011 - Growing (If you break up this 4-bit combination, it looks like thenumbers are growing - "10...11...")

1101 - Countdown (Think of a rocket ship countdown from "11" to "01")

As with the previous 3-bit method, each pattern has a name beginningwith a different letter (N, E, F, S, T, B, H, I, M, O, R, A, U, L, G or C), soeach pattern can be recalled just by its first letter. When rememberingletter combinations together, however, you no longer have freedom toplace unused vowels among the letters, as all five of the regular vowels

(A, E, I, O, U) have a particular meaning in this system.

There are two ways to deal with this. First, you could get lucky and havethe letters you're recalling make a word (such as ACHE, ALIEN orORANGES). The second is to remember the numbers in pairs, with theimportant letters being the first and last letters of a word (If you have toremember F and S, you might think of the word "FrieS?", for example). Inthis way, you're free to add any letters you wish to make a word, becausethe only letters that matter will be the first and last ones. With thisapproach, you'll be able to remember long strings of binary digits as

simply as rememberinglinked lists.

"Conversion" Method

Like the Lewis Jones method above, this system works with groups ofthree digits. In this system, however, we start by converting each groupof three to its binary equivalent:
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000 - 0001 - 1

010 - 2011 - 3

100 - 4

101 - 5110 - 6111 - 7

These equivalents must be memorized before proceeding any further. Youcan use theMajorSystemor theDominicSystemto link each binarygroup to its binary equivalent.

To remember a sequence in this manner, you would once again breakdown the number into three digit groups, and then label each group withthe appropriate number.

For example, let's use the number 011100001000. Breaking this up intogroups of three, we get 011 100 001 000. These groups convert into3410.

It is important to realize, at this point, that 3410 is a result of the way we

broke the number up, and that 3410 is NOTthe binary equivalent of011100001000 (the actual base 10 equivalent of this binary number is1800).

With theMajorSystem,you would remember this number as "MARTS".

With theDominicSystem,you would remember the first two groups ofthree as the person you associate with CD (the equivalent of 34). Youwould then picture CD performing the action or using the prop of AO.

Presentation of Binary Memory Feats

The feat of being able to remember long strings of 1s and 0s is certainly

impressive, but 1s and 0s themselves have no real meaning. To make theconcept of memorizing more interesting, there are many interesting ways

to give binary information a real-world meaning:

Red or black cards Picture or number cards High or low cards Odd or even cards Face-up or face-down cards Good or bad gambling hands
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Heads or tails on coins Odd or even spots on dice

Here are some more concepts to spark ideas for various binary memory

demonstrations:

add/subtractalive/deadalone/in a crowdbig/smallday/nightearly/lateeasy/difficultenlarge/reduceequal/unequalgood/badhard/soft

hot/coldin/outin front/behindlong/shortloose/compactlove/hate

male/femalemotion/stillnessmultiply/dividenear/faron/off

open/closeover/underpass/failpast/futurerich/poorright/leftright/wrongsafe/dangerousstart/finishstop/gotall/short

true/falseup/downus/themwet/drywide/narrow

young/old


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A little imagination here can yield a wealth of meaningful and amazingbinary memory demonstrations.

There are two different approaches to binary memory demonstrations, as

well. In the first, you simply remember the binary combination, and later

recall it perfectly (such as memorizing the order of reds and blacks in adeck). In the second, you memorize the binary combination (such as theheads-or-tails status of several coins), have someone alter a few of thefactors (someone turns some coins from heads to tails and other coinsfrom tails to heads), and then you're able to identify which factors havechanged