9. Binary Numbers

Unlike a linear, or analogue circuit which contains signals that are constantly changing from one value to another, such as amplitude or frequency, digital circuits process signals that contain just two voltage levels or states, labelled logic "0" and logic "1". Generally, a logic "1" represents a higher voltage, which is referred to as a HIGH and a logic "0" is referred to as a LOW. These discrete voltage levels are commonly known as BInary digiTS and are normally referred to as BITS. Because there are only two valid Boolean values for representing either a logic "1" or a logic "0", makes the system of using Binary Numbers ideal for use in digital or electronic circuits and systems. The Binary Numbers system is a Base-2 system which follows the same rules in mathematics as the common decimal system meaning instead of powers of ten, for example 1, 10, 100, 1000 etc, binary numbers use powers of two, doubling the value of each successive bit, 1, 2, 4, 8, 16, 32 etc.

The voltages used to represent a digital circuit are called "logic levels" and ideally one voltage level represents a HIGH and another represents a LOW. Digital waveforms or signals consist of discrete voltage levels that are changing back and forth between these HIGH and LOW levels or states. But what makes a signal or voltage "Digital" and how can we represent these voltage levels. Electronic circuits can be divided into two main categories.

• Analogue Circuits - Analogue or Linear circuits amplify or respond to continuously varying voltage levels over a period of time.
• Digital Circuits - Digital circuits produce or respond too two distinct voltage levels representing either a Logic level "1" or a Logic level "0".

a. Convert Decimal to Binary

The Decimal or "denary" counting system uses the Base of 10 numbering system where each digit in a number takes on one of ten possible values from 0 to 9, eg 213₁₀ (Two Hundred and Thirteen). 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.

Then, any numbering system can be summarised by the following relationship:

	N = bi qi
		where:	N is a real positive number b is the symbol q is the base value and integer (i) can be positive, negative or zero
		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. Then 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 or weight 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.

The value of a decimal number is equal to the sum of the digits multiplied by their respective weights. For example:  N = 6163₁₀  (Six Thousand One Hundred and Sixty Three)  in a decimal format is equal to:

(6×10³) + (1×10²) + (6×10¹) + (3×10⁰) = 6163

Unlike the decimal numbering system which uses the base of 10, digital logic uses just two values or states, a logic level "1" or a logic level "0", so each "0" and "1" is considered to be a single digit in a Base of 2 or Binary numbering system. In the binary numbering system, each digit has a value twice that of the previous digit but can only have a value of either "1" or "0" therefore, q = "2" and the position of either a "0" or a "1" indicates its weight.

For example:

Decimal Digit Value	256	128	64	32	16	8	4	2	1
Binary Digit Value	1	0	1	1	0	0	1	0	1

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. 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 into 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.

Example.  Convert the decimal number 294₁₀ into its binary number equivalent.

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)

Then, the Decimal number 294₁₀ is equivalent to 100100110₂ in the binary format reading it 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.

b. 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 classifications 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)

c. 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

Next topic will be on hexadecimal