### Introduction to the Binary and Other Number Systems

[Note to persons reading this in Braille. This section contains some mathematical notation which may not display correctly when transcribed to Braille 2. We would appreciate feedback on any problems.]

Place-value and the decimal system. Our decimal or base ten number system is a place-value system. This means that the place or location where you put a numeral determines its corresponding numerical value. (This is analogous to braille codes in that the place you put a cell sometimes determines its meaning.) A two in the one's place means two times one or two. A two in the one-thousand's place means two times one thousand or two thousand.

The place values increase from right to left. The first place just before the decimal point—or the right-most place if there isn't any decimal point—is the one's place, the second place or next place to the left is the ten's place, the third place is the hundred's place, and so on.

Figure 1 shows the place values for the first six whole number places in the decimal system.

```      Whole number places: 6th    5th   4th  3rd 2nd 1st
Place values:        100000 10000 1000 100 10  1```

Fig. 1. Place values (in decimal) for the decimal number system.

The place-value of the place immediately to the left of the "decimal" point is one in all place-value number systems. (The point itself should have different names in different number systems but this distinction isn't usually made.)

The place-value of any place to the left of the one's place is a whole number computed from a product (multiplication) in which the base of the number system is repeated as a factor one less number of times than the position of the place.

For example, the fourth place in our decimal or base ten system will have the value one thousand since ten is repeated in the product four minus one or three times:

`        10 x 10 x 10 = 1000.`

The place-value of any place to the right of the decimal point is a fraction computed from a product in which the reciprocal of the base—or a fraction with one in the numerator and the base in the denominator—is repeated as a factor exactly as many times as the place is to the right of the decmal point.

For example, the place where I've put an "X" in 0.00X00, which is the third place to the right of the decimal point, has the place value of one-thousandth since one-tenth is repeated in the product three times:

`        (1/10) x (1/10) x (1/10) = (1/1000).`

Other systems. The number system with base 16 is called the hexadecimal system, the one with base 8 is called the octal system, and the one with base 2 is called the binary system.

If you write a number, 101, say, and want to make it clear that it is a binary number, you can either write

101 (base 2) or 1012 .

Analogous rules to the decimal system work for these and any other place-value number system. The place-values for all the places except the one's place depend on the base of the number system; the highest value you can express using only the one's place is always one less than the base. This means that you need as many different numerals, counting zero, as the base. For example, the binary or base 2 system only uses two numerals: one and zero.

Figure 2 gives the place values, expressed in decimal, for the first six whole number places in the octal system. The fourth place, for example, has the value 512, which is 8 x 8 x 8.

```      Whole number places: 6th   5th   4th 3rd 2nd 1st
Place values:        32768 4096  512 64  8   1```

Fig. 2. Place values (in decimal) for the octal number system.

Figure 3 gives the places values, again expressed in decimal, for the first six whole number places in the binary system. The same place-value rules apply to the binary system: the right-most place just before what is called the binary point in a binary number is the one's place, the next place to the left is the two's place, the next place to the left of that is the two times two or four's place, and so on.

```      Whole number places: 6th 5th 4th 3rd 2nd 1st
Place values:        32  16  8   4   2   1```

Fig. 3. Place values (in decimal) for the binary number system.

The largest number that you can write in binary using only six places or six digits is 111111 (base 2) which, since there is a numeral one in each place, has the decimal value 32+16+8+4+2+1 = 63. This is, of course, consistent with their being 63 different possible braille cells.

Binary-coded octal numbers. So far we've used decimal numbers to give the place values for the various number systems but it is convenient to use octal numbers to give the place values for the binary number system when interpreting the braille cells according to this system.

```      Whole number places: 6th 5th 4th 3rd 2nd 1st
Place values:        408 208 108  48  28  18```

Fig. 4. Place values (in octal) for the binary number system. (Compare to Fig. 3.) Fig. 5. NUMBRL place values.
Although it is common to write numbers horizontally, as in Figure 4, this isn't necessary. They can even be written in a grid like the braille cell as shown in Figure 5.

There are a number of different ways of assigning place values to positions. The most useful correspondence is to let the upper position of the right-hand column be the 1st or one's place, the middle position be the 2nd or two's place, and the bottom position be the 3rd or four's place as is done in NUMBRL.

The following table shows the decimal values from zero to seven written as both binary and octal numbers. These are all the numbers that can be written in binary using only the first three places.
 Binary Sum of place values (in octal) Octal 000 0+0+0 00 001 0+0+1 01 010 0+2+0 02 011 0+2+1 03 100 4+0+0 04 101 4+0+1 05 110 4+2+0 06 111 4+2+1 07
The binary numbers have all been written as three-digit numerals and the octal numbers have been written as two-digit numerals by using leading zeroes which, of course, are a typesetting convenience that don't affect the values.

If we think of the filled dot positions in a braille cell as representing one's and the empty positions as representing zeroes, then these eight binary or octal numbers represent all eight possible dot patterns (including no dots) for one column of a braille cell.

If we assign the place values to dot positions as shown in Figure 5, the relationship between the dot patterns the octal numbers are as follows.

```          dot  1       10   dot  4        1
dot  2       20   dot  5        2
dots 1-2     30   dots 4-5      3
dot  3       40   dot  6        4
dots 1-3     50   dots 4-6      5
dots 2-3     60   dots 5-6      6
dots 1-2-3   70   dots 4-5-6    7
```

The next table shows the decimal values from eight to fifty-six, counting by eights, again written as both binary and octal numbers. These are all the numbers that can be written in binary using only the second three places.(The binary digits are written in groups of three to make them more readable.)

 Binary Sum of place values (in octal) Octal 001 000 0+0+10+0+0+0 10 010 000 0+20+0+0+0+0 20 011 000 0+20+10+0+0+0 30 100 000 40+0+0+0+0+0 40 101 000 40+0+10+0+0+0 50 110 000 40+20+0+0+0+0 60 111 000 40+20+10+0+0+0 70

Comparing this table with the previous one, we can see the symmetry. The same numeral or digit corresponds to the same pattern of one's and zeroes in each case; the only difference is that the pattern has been shifted to the left.

[This is the end of the information on number systems that is related to six-dot braille cells. Note that 77 (base 8) = (7x8) + (7x1) = 63.]

You may have thought previously about the additional characters needed for the one-digit numerals for the decimal values 10-15 using the hexadecimal number system. The standard choice is to use the letters A-F for these values! Also, it is more common to indicate a hexadecimal number by prefacing it with a pound sign, "#", rather than writing "(base 16)" or using a subscript of "16" with the number as mentioned previously for other bases. Thus #A=10, #B=11, #C=12, #D=13, #E=14, and #F=15. Once again, we see an analogy with braille.

#### What are the different ways of assigning place values in binary representations?

In Jonathan Swift's famous book, Gulliver's Travels, there is a story about two camps of Lilliputians which differed only in the way they ate soft-boiled eggs: one cracked open the Big End and the other the Little End.

Computer scientists, who occasionally have a sense of humor, have adopted these names to indicate how numbers are stored in a computer. A number stored in the ordinary way with its lower-valued or "little" end toward the lower-order part of a computer word is said to be in Little Endian form whereas a number stored in the reverse order is said to be in Big Endian form. This means that the NUMBRL mapping of place values to the dot positions of braille cells, shown above in Figure 5, is either "Little Top-ian" or "Big Bottom-ian" unless we open out the braille cell so all six dot positions are linear.

 Big-Endian Binary Sum of place values (in octal) Octal 000 0+0+0 00 100 1+0+0 01 010 0+2+0 02 110 1+2+0 03 001 0+0+4 04 101 1+0+4 05 011 0+2+4 06 111 1+2+4 07

Although I made up the last two names, the term Middle-Endian is an actual technical term that refers to any mixed order, such as the American way of writing dates in the form mm/dd/yy—rather than the European dd/mm/yy or the Japanese use of yy/mm/dd (for Western-style dates).

This term is also correctly applied to the assignment of dot positions 3-2-1-4-5-6 to the keys of a brailler so that, for example, both index fingers are used for the upper dots of the columns. The picture of the brailler shown at the last link has the corresponding NUMBRL values, 40-20-10-1-2-4, on labels above its keys. These labels show that when the brailler method for making a braille cell linear is applied to the NUMBRL position values, NUMBRL is an example of a "Middle-Endian" form of associating place values to the dot positions of the braille cells since the first three places are reversed as shown in Table 3.