h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

see, i never understood, nor i fear will i ever understand binary!
the enemy asked me to explain it yesterday.... major fail.

There'll be a few more of these "binary days" until the end of November 2011, then no more until 2100, so we might as well enjoy them.

I don't understand binary either - except from a telephonic point of view - how does 111010 equate to 58?

Reading from the right, the 1st digit tells you how many 1's there are. The next digit tells you how many 2's, the next, how many 4's and so on. So 111010 is one 32, one 16, one 8, no 4, one 2 and no 1. All this adds up to 58.

so, let me see if i have this straight.
it doubles each time?
1, 2, 4, 8, 16, 32, 64, 128...etc? or is there a limit to the number of columns?
yes, i AM 40, and i am a doofus!
there, said it.

I'm 63 next; and am about to give up on this until tomorrow.

Binary isn't all that exciting, but it's important for computer programmers.

A more generalized treatment on based notation, conventionally, the form is:

...[b^3], [b^2], [b^1], [b^0].[b-^1], [b-^2], [b-^3]...

You simply increase the exponent by 1 with every step to the left of the separatrix (it's technically a decimal point only in base 10) and decrease the exponent by one every step to the right of the separatrix, indicating fractions, whatever you want to make the base, 'b', usually any whole positive number greater than '1', |2, 3, 4,... .

So in base 10, which is what we use in everyday life, a number like 123.0 is:

1*10^2 + 2*10^1 + 3*10^0,

or, one times ten squared = one hundred, plus, two times ten to the first = twenty, plus, 3 times ten to the zeroth = three; one hundred twenty three.

Binary numbers, base 2, and hexadecimals, base 16, are bases commonly used in computer programing. In base 16, one substitutes the letters from 'A' to 'F' to provide numerals for 10 - 15, and '16' (base ten) is written '10' in base sixteen, while fifteen is 'F', fourteen is 'E',... ...ten is 'A'. Fractions to the right of the 'decimal point', (separatrix), in base sixteen are amounts like:

0.[sixteenths][two hundred fifty sixths][four thousand ninety sixths]...

Base two and base sixteen are popular in computer programming because base two numbers, binary numbers, are convenient for recording 'bits', single units of information, while base sixteen has a factorization compatible with base two (16 base ten = 2^4, or 10000 base two), and is convenient for recording larger units of information like 'nibbles' and 'bytes'.




Before the introduction of base ten Hindu-Arabic numbering, a common device for handling larger power of numbers was the 'exchequer', basically a checkerboard, where each column increased in multiples of five or ten, recorded in Roman numerals, with the count made by means of placing markers on the squares.




The notes like, ^, for "to the power of" or "exponent", and *, for "times", or "multiplied by" are standard, and if you use them that way in the Google search box and press 'search', the computer will do the arithmetic for you.




If you're interested in getting into fractional bases and exponents, you'll be working with logarithms and fractals.

ouch, ouch, ouch!

...forgot to throw in a note on traditional duodecimals, base twelve, for computing in dozens and gross. In traditional duodecimals, ten is represented by an 'X' and eleven by an 'E', while '10' becomes twelve.

Nowadays, people tend to simply take the alphabetical letters in sequence instead, ten and eleven are simply 'A' and 'B', just as in hexadecimals, but '10' in base twelve duodecimals still represents twelve, rather than ten as in base ten decimals, or sixteen as in base sixteen hexadecimals.

ITIWBS - for someone struggling with my "the number of 1's, the number of 2's", your explanation isn't going to mean much. The secret in writing a good explanation is to write it in a way that the reader doesn't have to understand the topic in order to understand the explanation.

Headache not intended.

When I was getting started on this, I really found the exchequer approach useful, simply using a checkerboard and checkers or poker chips for counters, two rows for base 2, when both rows have a counter, take them off and place a counter in the next column to the left.

One checker board is good up to base 8, add another one for base 16 so you've got eight columns of sixteen spaces.

Whenever a column fills up, whatever your base, simply clear it and place a counter in the next column to the left.

"Binary isn't all that exciting, but it's important for computer programmers."

Unless you want to always win at NIM. A20725760


*I was very pleased to see an h2g2 entry show up as the second return on a search for "binary coefficients, nim".

I know that it's been said too often lately, but there are only 10 sorts of people that understand binary ... Like lots of stuff, you either have a feel for number systems (or languages, or pastry baking, or riding a horse) or ya don't.

So to those who do, cheers to 58 Day.

What a coincidence:

F18390129?thread=7808964#p101721392

That was the only time I actually understood binary. I even managed to play that game - and win. :D

In an attempt to translate ITIWBS:

There are lots of ways to write numbers. The main method we use is based notation, which means we reuse the same small set of symbols, indicating different values by place: a symbol on the left of a number indicates a higher value than the same symbol on the left of a number.

For example, in the number 500050, the same symbol, 5, appears twice. The one on the left indicates a much higher value than does the one near the right.

The extent of the difference in value depends on the base. In the decimal system (base 10), the most common, the first column (ordering from right to left) indicates ones, the second indicates tens, the third indicates hundreds, etc.

For example, 5872 means two ones plus seven tens plus eight hundreds plus five thousands: 5872 = (2 * 1) + (7 * 10) + (8 * 100) + (5 * 1000). Or, 5872 = 2*10^0 + 7*10^1 + 8*10^2 + 5*10^3 (where ^ means "to the power of": x^n is x multiplied by itself n times). (Note that anything to the power of zero is one.)

So in binary, base 2, we have something like this:

101011 means 1*2^0 + 1*2^1 + 0*2^2 + 1*2^3 + 0*2^4 + 1*2^5 which is 1*1 + 1*2 + 0*4 + 1*8 + 0*16 + 1*32 which is 1 + 2 + 0 + 8 + 0 + 32 which is 43.

Generically, where b is the base, each column represents a value of b^(column number, counting from right to left, starting with zero).

Computers use base 2.

Programmers commonly use hexadecimal (base 16) and occasionally octal (base 8). These are both handy because it's very easy to convert from either of these to binary. (Converting between binary and decimal involved quite a bit of arithmetic, but converting from either octal or hexadecimal to binary is simple: each digit can be converted independently.)



There's a lie above. A685055. I said that the column on the right is has a value of b^0. This isn't true. There's the decimal point to consider. The number 123.4 has a value of 1*10^2 + 2*10^1 + 3*10^0 + 4*10^-1. So it's the column to the left of the decimal point which has the value 10^0, which is not necessarily the rightmost column, as there may be more numbers after the decimal point.

Of course, the name "decimal point" is appropriate only for decimal numbers. The more general term is "separatrix". (I didn't know this. I learned it from ITIWBS. It's my new knowledge of the day.)



On reflection, I think my explanation was just as obscure as ITIWBS'. But having the same thing explained in different ways can still be helpful. And hey, I enjoyed writing it.

TRiG.

I had some limited material on base 2 in elementary and high school, mandatory for the California state school system.

I didn't get on to hexadecimals until I was 39 and taking a course in Basic computer language.

Philosophy on math, complicated as it seems, it actually represents an effort to simplify a complex phenomenon so the mind can grasp it. Beyond that, it's just vocabulary.

That last line on logarithms and fractals, I just threw that in because it's an answer to an obvious question: What happens I don't restrict the base and exponents to whole counting numbers but instead substitute fractions?

Actually computing them is something more tedious than I would care to attempt, requiring phenomenal mathematical talent or a powerful computer program.

I'm going to give the thread a little work and move it over to Peer review, which is probably where I should have put it in the first place, Title, "Place Value Notation"