Converting Decimal to Binary

Since the binary system is the internal language of electronic computers, serious computer programmers should understand how to convert from decimal to binary.

we will start by discussing what are decimal and binary numbers.

      The decimal (base ten) numeral system  has ten possible values (0,1,2,3,4,5,6,7,8, or 9) for each place-value. In contrast, the binary  (base two) numeral system has two possible values, often represented as 0 or 1, for each place-value.

    To avoid confusion while using different numeral systems, the base  of each individual number may be specified by writing it as a subscript of the number. 

example: 

         *the decimal number 156 may be written as 15610 and 

          read as "one hundred fifty-six, base ten". 

        *The binary number 10011100 may be specified as               

    base two" by writing it as 100111002.

In a decimal system each digit has a value ten times greater than its previous number and this decimal numbering system uses a set of symbols, b, together with a base, q, to determine the weight of each digit within a number. 

For example, the six in sixty has a lower weighting than the six in six hundred and in a binary numbering system we need some way of converting Decimal to Binary. Any numbering system can be summarized bythe following relationship: 

N=b_i qⁱ

Where:

N is positive real number

B is symbol

Q is base value

N = b₂ q² + b₁ q¹  + b₀ q⁰ + b_-1 q^-1 ... etc

 In the decimal or denary system, the columns have values of units, tens, hundreds etc as we move from right to left and mathematically these values are written as 100, 101, 102, 103 etc. The decimal numbering system has a base of 10 or modulo-10 (sometimes called MOD-10) with the position of each digit in the decimal system indicating the magnitude orweight of the number. For example, 20 (twenty) is the same as saying 2 x 101 and 400 (four hundred) is the same as saying 4 x 102. Likewise, for fractional numbers the weight of the number is negative, 10-1, 10-2, 10-3 etc.

By adding together all the decimal number values from right to left at the positions that are represented by a "1" gives us:  (256) + (64) + (32) + (4) + (1) = 35710 or three hundred and fifty seven in decimal. Then, the binary array of digits 1011001012 is equivalent to 35710 in decimal or denary. As the decimal number is a weighted number, converting from decimal to binary will also produce a weighted binary number with the right-hand most bit being the Least Significant Bit or LSB, and the left-hand most bit being the Most Significant Bit or MSB. and we can represent these as

MSB	Binary Digit							LSB
28	27	26	25	24	23	22	21	20
256	128	64	32	16	8	4	2	1

Repeated Division-by-2 Method 

Another method of converting Decimal to Binary number equivalents is to write down the decimal number and to continually divide by 2 (two) to give a result and a remainder of either a "1" or a "0" until the final result equals zero.

Number	294				Dividing each number by "2" gives a result plus a remainder. The binary result is obtained by placing the remainders in order with the least significant bit (LSB) being at the top and the most significant bit (MSB) being at the bottom.
divide by 2
result	147	remainder	0        \(LSB)
divide by 2
result	73	remainder	1
divide by 2
result	36	remainder	1
divide by 2
result	18	remainder	0
divide by 2
result	9	remainder	0
divide by 2
result	4	remainder	1
divide by 2
result	2	remainder	0
divide by 2
result	1	remainder	0
divide by 2
result	0	remainder	1  (MSB)

the decimal to binary conversion gives the decimal number 

29410 = 1001001102 in binary, reading from right to left.

Then the main characteristics of a Binary Numbering System is that each "digit" or "bit" has a value of either "1" or "0" with each digit having a weight or value double that of its previous bit starting from the lowest or least significant bit (LSB) and this is called the "sum-of-weights" method. So we can convert a decimal number to binary either by using the sum-of-weights method or by using the repeated division-by-2 method.

Binary Number Names & Prefixes

Binary numbers can be added together and subtracted just like decimal numbers with the result being combined into one of several size ranges depending upon the number of bits being used. The classification of individual bits into larger groups are generally referred to by the following more common names of:

Number of Binary Digits (bits)	Common Name
1	Bit
4	Nibble
8	Byte
16	Word
32	Double Word
64	Quad Word

Also, when converting from Decimal to Binary or even from Binary to Decimal, we need to be careful that we do not mix up the two sets of numbers. 

For example, if we write the digits 10 on the page it could mean the number "ten" if we assume it to be a decimal number, or it could equally be a "1" and a "0" together in binary, which is equal to the number two in decimal from above.

 One way to overcome this problem and to identify whether the digits or numbers being used are decimal or binary is to write a small number called a subscript after the last digit to show the base of the number system being used. 

If we were using a binary number we would add the subscript 2 so the number would be written as 102, and if it were a decimal number we would add the subscript 10 so the number would be written as 1010.

Today, as micro-controller or microprocessor systems become increasingly larger, the individual binary digits (bits) are now grouped together into 8´s to form a single BYTE with most computer hardware such as hard drives and memory modules commonly indicate their size in Megabytes or even Gigabytes.

Number of Bytes	Common Name
1,024 (210)	kilobyte (kb)
1,048,576 (220)	Megabyte (Mb)
1,073,741,824 (230)	Gigabyte (Gb)
a very long number! (240)	Terabyte (Tb)

DECIMAL TO BINARY SUMMARY:

·        A "BIT" is the abbreviated term derived from BInary digiT

·         A Binary system has only two states, Logic "0" and Logic "1"

·        A Decimal system uses 10 different digits, 0 to 9

·       A Binary number is a weighted number who's weighted value increases from right to left

·       The weight of a binary digit doubles from right to left

·       A decimal number can be converted to a binary number by using the sum-of-weights method or the repeated division-by-2 method

·       When we convert numbers from decimal to binary, subscripts are used to avoid errors

Converting Decimal to Binary numbers can be done in a number of different ways as shown above. When converting decimal numbers to binary numbers it is important to remember which is the least significant bit (LSB), and which is the most significant bit (MSB). In the next tutorial about Binary Logicwe will look at converting binary numbers into Hexadecimal Numbers and vice versa and show that binary numbers can be represented by letters as well as numbers.