Binary Numbers : An Extended Summary

Binary Value

1011011011

Paired Value

Octal Equivalent

Octal Value

1. Number System

Number System is a writing system that exists for the purpose of expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. Here, we discuss about positional number systems with positive radix (also known as base). In general, a number system with base ‘r’ contains ‘r’ number of digits, that go from 0,1, 2, …. r-1.

When we type some letters or words, the computer translates them in numbers as computers can understand only numbers. A computer can understand the positional number system where there are only a few symbols, called digits and these digits represent different values depending on the position they occupy in the number.

The value of each digit in a number can be determined using-

i) The digit

ii) The position of the digit in the number

iii) And. The base of the number system

The computer architecture supports the following number systems,

· Binary Number System

· Decimal Number System

· Octal Number System

· Hexadecimal Number System

1.1 Binary Number System

Binary Number System is the number system with base 2 that is, there are only two digits. This is the very reason that makes them to be known as base 2 number system.

Binary numbers are expressed with a combination of 1’s and 0’s and possibly with a binary point in it. The decimal equivalent of a binary number can be found by expanding the number into a power series with a base of 2.

Digits in a binary number are called bits (Binary digITs). When a bit is equal to 0, it does not contribute to the sum during conversion. Therefore, the conversion to decimal can be obtained by adding the numbers with powers of 2 corresponding to the bits that are equal to 1.

In computer system,

· 2¹⁰ is referred to as K (Kilo)

· 2²⁰ as M (Mega)

· 2³⁰ as G (giga)

· 2⁴⁰ as T (Tera) and so on.

1.2 Decimal Number System

Decimal Number System is the number system with base-10. The digits range from 0, 1, 2, 3, …. 9.

This is the number system that we use in our everyday life. The numbers are represented by combinations of digits defined above, possibly with a decimal point. Depending on its position in the string, each digit has an associated value of an integer raised to the power of 10.

1.3 Octal Number System

Octal Number System is the number system with base-8. The digits range from 0, 1, 2, …. 7.

The octal numbers are expressed with the combination of digits defined above, possibly with an octal point in it. The decimal equivalent of a octal number can be found by expanding the number into a power series with a base of 8.

1.4 Hexadecimal Number System

Hexadecimal Number System is the number system with base-16. The digits range from 0 to 9 and also comprise of alphabets A to F. Thus, it is an alphanumeric number system.

Here, A is equivalent to decimal 10, B is equivalent to decimal 11 and so on. The hexadecimal numbers are expressed with the combination of digits defined above, possibly with a hexadecimal point in it. The decimal equivalent of a hexadecimal number can be found by expanding the number into a power series with a base of 16.

1.5 Importance of Number System

Number Systems is crucial for understanding the processing of digital system. Digital system takes binary, octal & hexadecimal number as input and process it and generates output. Thus, in the field of information technology or embedded system everywhere one need to be well aware of number system in order to understand its operation.

The easiest way to vary instructions through electric signals is two-state system – on and off. On is represented as 1 and off as 0, though 0 is not actually no signal but signal at a lower voltage. The number system having just these two digits – 0 and 1 – is called binary number system and that is how computer function in general.

1.6 Numeric Equivalence Chart

Here, is an equivalence chart for numeric values in different number systems.

Decimal (Base 10)	Binary (Base 2)	Octal (Base 8)	Hexadecimal (Base 16)
00	0000	00	0
01	0001	01	1
02	0010	02	2
03	0011	03	3
04	0100	04	4
05	0101	05	5
06	0110	06	6
07	0111	07	7
08	1000	10	8
09	1001	11	9
10	1010	12	A
11	1011	13	B
12	1100	14	C
13	1101	15	D
14	1110	16	E
15	1111	17	F

Table 1: Equivalence Chart

2. Number Base Conversions

Decimal, Binary, Octal and Hexadecimal number systems are positional number systems. These number systems co-exist in the same environment. Thus, we can convert a number that exists in a number system to another and so on. Here, we study about how we can convert these numerical values from one number system to another with vice versa.

2.1 Binary to Decimal

For this conversion, we expand the number into a power series with a base of 2 and calculate the sum that exists in decimal system. Here, when a binary number is to be converted, the individual digits are multiplied from Most Significant Digit to Least Significant Digit with decrement in the power to the base.

For example:

· Convert (1111)₂ into a decimal number.

Solution: Given binary number is (1111)₂.

Here, multiplying each digit from MSD to LSD, with decrement in power to the base, we get,

= 1 x 2³ + 1 x 2² + 1 x 2¹ + 1 x 2⁰

= 1 x 8 + 1 x 4 + 1 x 2 + 1 x 1

= 8 + 4 + 2 + 1

= 15

Thus, (1111)₂ = (15)₁₀.

2.2 Binary to Octal

For this conversion, we group 3 individual bits from the least significant bits and move along towards the most significant ones. If there is only a single or double bit remaining in the pairing made from the most significant side, 0s can be added to make up a group of three. Then, the groups are paired up from their equivalence chart and the corresponding octal value is written, which can be seen in equivalence chart. At last, the equivalent octal digits are paired from the least significant ones to the most significant value.

For example:

· Convert (1011011011)₂ into an octal number.

Solution: Given binary number is (1011011011)_2.

Here, a pair of 3 bits is made from the LSD to MSDs. Then, their equivalent octal value is written.

Thus, (1011011011)₂= (1333)₈.

2.3 Binary to Hexadecimal

For this conversion, we group 4 individual bits from the least significant bits and move along towards the most significant ones. If there is only a single-double or triple bit remaining in the pairing made from the most significant side, 0s can be added to make up a group of four. Then, the groups are paired up from their equivalence chart and the corresponding hexadecimal value is written, which can be seen in equivalence chart. At last, the equivalent hexadecimal digits are paired from the least significant ones to the most significant value.

For example:

· Convert (110101111)₂ into a hexadecimal number.

Solution: Given binary number is (110101111)_2.

Here, a pair of 4 bits is made from the LSD to MSDs. Then, their equivalent hexadecimal value is written.

Binary Value	110101111
Paired Value	0001	1010	1111
Hexadecimal Equivalent	1	A	F
Hexadecimal Value	1AF

Thus, (110101111)₂= (1AF)₁₆.

2.4 Decimal to Binary

Decimal numbers can be converted to binary by repetitive division of the number by 2 while the remainders are noted down. Then, the remainders are written collectively from the last division to the direction of first.

For Example:

· Convert (55)₁₀ into a binary number.

Solution: Here, (55)₁₀ is a decimal number.

The number is repetitively divided by 2 and the remainders are collectively further written from the last division to the first to obtain the binary number.

2	55	Remainder
2	27	1
2	13	1
2	6	1
2	3	0
2	1	1
	0	1

Thus, (55)₁₀= (110111)₂.

2.5 Octal to Binary

Octal Numbers can be converted with the simple method of writing up the equivalent values of the individual octal values in the binary form and alas, pairing them up to get the corresponding binary value. Each octal digit gives 3 binary digits which can be seen in equivalence chart.

Example:

· Convert (42076)₈ into a binary number.

Solution: Here, (42076)₈ is an octal number.

Individual octal digits share an equivalent binary value and they are written together to get the binary value.

Octal Value	42076
Individual Value	4	2	0	7	6
Equivalent Binary Value	100	010	000	111	110
Binary Value	100010000111110

Thus, (42076)₈ = (100010000111110)₂.

2.6 Hexadecimal to Binary

Hexadecimal Numbers can be converted with the simple method of writing up the equivalent values of the individual hexadecimal values in the binary form and alas, pairing them up to get the corresponding binary value. Each hexadecimal digit gives 4 binary digits which can be seen in equivalence chart.

Example:

· Convert (A23BC)₁₆ into a binary number.

Solution: Here, (A23BC)₁₆ is a hexadecimal number.

Individual octal digits share an equivalent binary value and they are written together to get the binary value.

Hexadecimal Value	A23BC
Individual Value	A	2	3	B	C
Equivalent Binary Value	1010	0010	0011	1011	1100
Binary Value	10100010001110111100

Thus, (A23BC)₁₆ = (10100010001110111100)₂.

3. Complements

Complements are used in digital computers for simplifying the subtraction operation and for logical manipulation. There are two types of complements for each base-r system: r's complement and the second as the (r - 1)'s complement. When the value of the base r is substituted, the two types are referred to as the 2's complement and 1's complement for binary numbers, the 10's complement and 9's complement for decimal numbers etc. The 1’s complement and the 2’s complement of a binary number is important because they permit the representation of negative numbers. The method of 2’s complement arithmetic is commonly used in computers to handle negative numbers.

3.1 Diminished Radix Complement

(r-1)’s complement of a number N is defined as (rⁿ-1) – N. This complement is also known as diminished radix complement.

Here, N is the given number

r is the base of number system

n is the number of digits in the given number

For example, in the decimal number system; r = 10 and (r-1) = 9, so the 9’s complement of N is (10ⁿ-1) – N.

Then, for a number(N) that is 46920,

r = 10

r -1 = 9

n = 5

Thus, 9’s complement of the given number can be found by,

(10⁵– 1) – 46920 = 99999 – 46920 = 53079

This indicates that 9’s complement of a decimal number can be obtained by subtracting the given number from a number that is as long as the given number which comprises of all 9s.

Similarly, in the binary number system; r = 2 and (r-1) = 1, so the 2’s complement of N is (2ⁿ-1) – N.

Then, for a number(N) that is 11010,

r = 2

r -1 = 1

n = 5

Thus, 1’s complement of the given number can be found by,

(2⁵– 1) – 11010 = (100000 – 1) - 11010 = 101

The catch here is, the (2ⁿ-1) term is further converted into a binary number. This indicates that 1’s complement of a binary number can be obtained by subtracting the given number from a number that is as long as the given number which comprises of all 1s. However, when subtracting binary digits from 1, we can have either 1 – 0 = 1 or 1 – 1 = 0, which causes the bit to change from 0 to 1 or from 1 to 0, respectively. Therefore, the 1’s complement of a binary number is formed by changing 1’s to 0’s and 0’s to 1’s.

3.2 Radix Complement

r’s complement of a number N is defined as rⁿ-N for N ≠ 0. This complement is also known as radix complement.

Here, N is the given number

r is the base of number system

n is the number of digits in the given number

Comparing with the (r – 1)’s complement, we note that the r’s complement is obtained by adding 1 to the (r – 1)’s complement, since rⁿ – N = [(rⁿ – 1) – N] + 1. Thus, the 10’s complement of decimal 5946 is 4053 + 1 = 4054. This indicates that the 10’s complement can simply be found by adding 1 to the 9’s complement. Similarly, 2’s complement of binary 1110010 is 1101 + 1 = 1110, which is by adding 1 to the 1’s complement of the given number.

In the previous definitions, it was assumed that the numbers did not have a radix point. If the original number N contains a radix point, the point should be removed temporarily in order to form the r’s or (r – 1)’s complement. The radix point is then restored to the complemented number in the same relative position. It is also essential to mention that the complement of the complement restores the number to its initial state. To see this relationship, we can see that the r’s complement of N is rⁿ– N, so that the complement of the complement is rⁿ – (rⁿ– N) which is equals to N, N being the original number.

4. Complement Method of Subtraction

The direct method of subtraction inculcated in school level implements the borrowing concept. This method instructs us to borrow a 1 from a higher significant value when the minuend digit is smaller than the subtrahend digit. This method is all fine and dandy when performed with a pen and paper but the digital hardware has a hard time implementing this, thus the method that these systems depend upon is complement subtraction.

The subtraction of two n-digit unsigned numbers M – N in base-r can be done in the following ways:

i. The minuend M is added to the r’s complement of the subtrahend M.
This performs, M + (rⁿ – N) = M – N + rⁿ.

ii. Then, if M >= N, then the sum produced will generate an end carry, rⁿ which is discarded leaving the result M – N.

iii. In case M < N, the sum does not produce an end carry and is equal to rⁿ – (N – M), which is the r’s complement of (N – M). Further to gain the familiar form, we calculate the r’s complement of the sum and place a negative sign in front.

For Example:

a. Using 10’s complement, subtract 76250 – 95420.

M = 76250

10’s complement of N = +04580

Sum = 80830

Answer: -(10’s complement of 80830) = -19169

b. Using 10’s complement, subtract 56789 – 12345.

M = 56789

10’s complement of N = + 87655

Sum = 144444

Discard end carry 10⁵ = - 100000

Answer = 44444

c. Using 2’s complement, subtract 1100110 – 1001101.

M = 1100110

2’s complement of N = + 110011

Sum = 10011001

Discard end carry 2⁸= - 10000000

Answer = 11001

Subtraction of unsigned numbers can also be done by means of the (r-1)’s complement. We have to remember that the (r-1)’s complement is one less than the r’s complement. Because of this, the result of adding the minuend to the complement of the subtrahend produces a sum that is one less than the correct difference when an end carry occurs. Removing the end carry and adding 1 to the sum is referred to as an end-around carry. The procedure with end-around carry is also applicable to subtracting unsigned decimal numbers with 9’s complement.

References

I. Digital Fundamentals – Ninth Edition, Thomas L. Floyd and Jain

II. Digital Logic and Computer Design – Morris Mano

III. Digital Design – Mano and Ciletti

IV. What is Number System? - Types of Number System & Significance - Electronics Coach

V. What Are The Importance Of Computer Number System? - Blurtit

VI. Numeral system - Simple English Wikipedia, the free encyclopedia

VII. Number Systems: An Introduction to Binary, Hexadecimal, and More (tutsplus.com)