Number Systems | What Is A Number System In A Computer?

We all know that a computer or any other machine can communicate in machine language only. So how come you can type in any language and a computer gives the output in human-readable format?

It means when we enter any data it’s being converted into something which only a computer/machine can understand and then the response from the computer is converted into a human-readable format so that humans can interact with the computer.

In digital systems, instructions are given through electric signals; variation is done by varying the voltage of the signal. Having 10 different voltages to implement a decimal number system in digital equipment is difficult. So, many number systems that are easier to implement digitally have been developed.

Number System

The Number System in a computer are:

Binary number system
Octal number system
Decimal number system
Hexadecimal number system

Binary number system

It has only two digits ‘0’ and ‘1’ so its base is 2.

In this number system, there are only two types of electronic pulses; the absence of an electronic pulse which represents ‘0’and the presence of an electronic pulse which represents ‘1’. Each digit is called a bit.

A group of four bits (1101) is called a nibble and a group of eight bits (11001010) is called a byte. The position of each digit in a binary number represents a specific power of the base (2) of the number system.

The binary number system is also a positional value system, where each digit has a value expressed in powers of 2.

In any binary number, the rightmost digit is called the least significant bit (LSB) and the leftmost digit is called the most significant bit (MSB).

For example, 10011 represents in the following ways

(1 X 2⁴) + (0 X 2³) + (0 X 2²) + (1 X 2¹) + (1 X 2⁰)

16 + 0 + 0 + 2 + 1

Computer memory is measured in terms of how many bits it can store. Here is a chart for memory capacity conversion.

1 byte (B) = 8 bits
1 Kilobytes (KB) = 1024 bytes
1 Megabyte (MB) = 1024 KB
1 Gigabyte (GB) = 1024 MB
1 Terabyte (TB) = 1024 GB
1 Exabyte (EB) = 1024 PB
1 Zettabyte = 1024 EB
1 Yottabyte (YB) = 1024 ZB

Octal number system

It has eight digits (0, 1, 2, 3, 4, 5, 6, 7) so its base is 8. Each digit in an octal number represents a specific power of its base (8).

As there are only eight digits, three bits (2³=8) of a binary number system can convert any octal number into a binary number. This number system is also used to shorten long binary numbers.

The three binary digits can be represented with a single octal digit.

The decimal equivalent of an octal number is the sum of the product of each digit with its positional value.

7268 = 7×82 + 2×81 + 6×80

= 448 + 16 + 6

= 47010

Decimal number system

The decimal number system has ten digits starting from 0-9 so its base is 10.

The position of each digit in a decimal number represents a specific power of the base (10) of the number system.

For example, let’s say we have three numbers – 734, 971 and 207. The value of 7 in all three numbers is different−

In 7314, the value of 7 is 7 thousand or 7000 or 7 × 1000 or 7 × 103
In 9701, the value of 7 is 7 hundred or 700 or 7 × 100 or 7 × 102
In 2007, value 0f 7 is 7 units or 7 or 7 × 1 or 7 × 100

The weightage of each position can be represented as follows −

Hexadecimal number system

In this number system, there are 16 digits that range from 0 to 9 and A to F. So, its base is 16.

The A to F alphabet represents 10 to 15 decimal numbers. The position of each digit in a hexadecimal number represents a specific power of the base (16) of the number system.

As there are only sixteen digits, four bits (24=16) of a binary number system can convert any hexadecimal number into a binary number.

It is also known as an alphanumeric number system as it uses both numeric digits and alphabets.

The decimal equivalent of any hexadecimal number is the sum of the product of each digit with its positional value.

27FB16 = 2×163 + 7×162 + 15×161 + 10×160

= 8192 + 1792 + 240 +10

= 1023410

Number System Relationship

The following table depicts the relationship between decimal, binary, octal and hexadecimal number systems.

HEXADECIMAL	DECIMAL	OCTAL	BINARY
0	0	0	0
1	1	1	1
2	2	2	10
3	3	3	11
4	4	4	100
5	5	5	101
6	6	6	110
7	7	7	111
8	8	10	1000
9	9	11	1001
A	10	12	1010
B	11	13	1011
C	12	14	1100
D	13	15	1101
E	14	16	1110
F	15	17	1111

ASCII(American Standard Code for the Information Interchange)

Besides numerical data, the computer must be able to handle alphabets, punctuation marks, mathematical operators, special symbols, etc. that form the complete character set of the English language. The complete set of characters or symbols is called alphanumeric code. The complete alphanumeric code typically includes −

26 upper-case letters
26 lowercase letters
10 digits
7 punctuation marks
20 to 40 special characters

Now a computer understands only numeric values, whatever the number system used. So all characters must have a numeric equivalent called the alphanumeric code. The most widely used alphanumeric code is American Standard Code for Information Interchange (ASCII). ASCII is a 7-bit code that has 128 (27) possible codes.

Standard ASCII Characters

Dec	Hex	Oct	Char	Description
0	0	0	^@	Null (NUL)
1	1	1	^A	Start of heading (SOH)
2	2	2	^B	Start of text (STX)
3	3	3	^C	End of text (ETX)
4	4	4	^D	End of transmission (EOT)
5	5	5	^E	Enquiry (ENQ)
6	6	6	^F	Acknowledge (ACK)
7	7	7	^G	Bell (BEL)
8	8	10	^H	Backspace (BS)
9	9	11	^I	Horizontal tab (HT)
10	0A	12	^J	Line feed (LF)
11	0B	13	^K	Vertical tab (VT)
12	0C	14	^L	New page/form feed (FF)
13	0D	15	^M	Carriage return (CR)
14	0E	16	^N	Shift out (SO)
15	0F	17	^O	Shift in (SI)
16	10	20	^P	Data link escape (DLE)
17	11	21	^Q	Device control 1 (DC1)
18	12	22	^R	Device control 2 (DC2)
19	13	23	^S	Device control 3 (DC3)
20	14	24	^T	Device control 4 (DC4)
21	15	25	^U	Negative acknowledgement (NAK)
22	16	26	^V	Synchronous idle (SYN)
23	17	27	^W	End of transmission block (ETB)
24	18	30	^X	Cancel (CAN)
25	19	31	^Y	End of medium (EM)
26	1A	32	^Z	Substitute (SUB)
27	1B	33	^[	Escape (ESC)
28	1C	34	^	File separator (FS)
29	1D	35	^]	Group separator (GS)
30	1E	36	^^	Record separator (RS)
31	1F	37	^_	Unit separator (US)
32	20	40		Space
33	21	41	!	Exclamation mark
34	22	42	“	Quotation mark/Double quote
35	23	43	#	Number sign
36	24	44	$	Dollar sign
37	25	45	%	Per cent sign
38	26	46	&	Ampersand
39	27	47	‘	Apostrophe/Single quote
40	28	50	(	Left parenthesis
41	29	51	)	Right parenthesis
42	2A	52	*	Asterisk
43	2B	53	+	Plus sign
44	2C	54	,	Comma
45	2D	55	–	Hyphen/Minus
46	2E	56	.	Full stop/Period
47	2F	57	/	Solidus/Slash
48	30	60	0	Digit zero
49	31	61	1	Digit one
50	32	62	2	Digit two
51	33	63	3	Digit three
52	34	64	4	Digit four
53	35	65	5	Digit five
54	36	66	6	Digit six
55	37	67	7	Digit seven
56	38	70	8	Digit eight
57	39	71	9	Digit nine
58	3A	72	:	Colon
59	3B	73	;	Semicolon
60	3C	74	<	Less-than sign
61	3D	75	=	Equal/Equality sign
62	3E	76	>	Greater-than sign
63	3F	77	?	Question mark
64	40	100	@	Commercial at/At sign
65	41	101	A	Latin capital letter A
66	42	102	B	Latin capital letter B
67	43	103	C	Latin capital letter C
68	44	104	D	Latin capital letter D
69	45	105	E	Latin capital letter E
70	46	106	F	Latin capital letter F
71	47	107	G	Latin capital letter G
72	48	110	H	Latin capital letter H
73	49	111	I	Latin capital letter I
74	4A	112	J	Latin capital letter J
75	4B	113	K	Latin capital letter K
76	4C	114	L	Latin capital letter L
77	4D	115	M	Latin capital letter M
78	4E	116	N	Latin capital letter N
79	4F	117	O	Latin capital letter O
80	50	120	P	Latin capital letter P
81	51	121	Q	Latin capital letter Q
82	52	122	R	Latin capital letter R
83	53	123	S	Latin capital letter S
84	54	124	T	Latin capital letter T
85	55	125	U	Latin capital letter U
86	56	126	V	Latin capital letter V
87	57	127	W	Latin capital letter W
88	58	130	X	Latin capital letter X
89	59	131	Y	Latin capital letter Y
90	5A	132	Z	Latin capital letter Z
91	5B	133	[	Left square bracket
92	5C	134		Reverse solidus/Backslash
93	5D	135	]	Right square bracket
94	5E	136	^	Circumflex accent/Caret
95	5F	137	_	Underscore/Low line
96	60	140	`	Grave accent
97	61	141	a	Latin small letter a
98	62	142	b	Latin small letter b
99	63	143	c	Latin small letter c
100	64	144	d	Latin small letter d
101	65	145	e	Latin small letter e
102	66	146	f	Latin small letter f
103	67	147	g	Latin small letter g
104	68	150	h	Latin small letter h
105	69	151	i	Latin small letter i
106	6A	152	j	Latin small letter j
107	6B	153	k	Latin small letter k
108	6C	154	l	Latin small letter l
109	6D	155	m	Latin small letter m
110	6E	156	n	Latin small letter n
111	6F	157	o	Latin small letter o
112	70	160	p	Latin small letter p
113	71	161	q	Latin small letter q
114	72	162	r	Latin small letter r
115	73	163	s	Latin small letter s
116	74	164	t	Latin small letter t
117	75	165	u	Latin small letter u
118	76	166	v	Latin small letter v
119	77	167	w	Latin small letter w
120	78	170	x	Latin small letter x
121	79	171	y	Latin small letter y
122	7A	172	z	Latin small letter z
123	7B	173	{	Left curly bracket
124	7C	174	\|	Vertical line/Vertical bar
125	7D	175	}	Right curly bracket
126	7E	176	~	Tilde
127	7F	177	DEL	Delete (DEL)

In the ASCII character set, the Decimal values 0 to 31, as well as the Decimal value 127, represent symbols that are non-printable or non-graphical characters.

It is possible to generate these non-printable characters using a key sequence where ^ represents the control key on your keyboard.

For example, you could generate a carriage return (Decimal value 13) by pressing the control key followed by the letter M on your keyboard (^M).

All other symbols in the character set can be printed or represented on the screen and they are known as graphical characters.

Extended ASCII Characters

Dec	Hex	Oct	Char	Description
128	80	200
129	81	201
130	82	202
131	83	203
132	84	204
133	85	205
134	86	206
135	87	207
136	88	210
137	89	211
138	8A	212
139	8B	213
140	8C	214
141	8D	215
142	8E	216
143	8F	217
144	90	220
145	91	221
146	92	222
147	93	223
148	94	224
149	95	225
150	96	226
151	97	227
152	98	230
153	99	231
154	9A	232
155	9B	233
156	9C	234
157	9D	235
158	9E	236
159	9F	237
160	A0	240		No-break space
161	A1	241	¡	Inverted exclamation mark
162	A2	242	¢	Cent sign
163	A3	243	£	Pound sign
164	A4	244	¤	Currency sign
165	A5	245	¥	Yen/Yuan sign
166	A6	246	¦	Broken bar
167	A7	247	§	Section sign
168	A8	250	¨	Diaeresis
169	A9	251	©	Copyright sign
170	AA	252	ª	Feminine ordinal indicator
171	AB	253	«	Left-pointing double-angle quotation mark
172	AC	254	¬	Not sign
173	AD	255		Soft hyphen
174	AE	256	®	Registered trademark sign
175	AF	257	¯	Macron
176	B0	260	°	Degree sign
177	B1	261	±	Plus-minus sign
178	B2	262	²	Superscript two
179	B3	263	³	Superscript three
180	B4	264	´	Acute accent
181	B5	265	µ	Micro sign (mu)
182	B6	266	¶	Pilcrow sign
183	B7	267	·	Middle dot
184	B8	270	¸	Cedilla
185	B9	271	¹	Superscript one
186	BA	272	º	Masculine ordinal indicator
187	BB	273	»	Right-pointing double-angle quotation mark
188	BC	274	¼	Vulgar fraction one quarter
189	BD	275	½	Vulgar fraction one half
190	BE	276	¾	Vulgar fraction three quarters
191	BF	277	¿	Inverted question mark
192	C0	300	À	Latin capital letter A with grave
193	C1	301	Á	Latin capital letter A with acute
194	C2	302	Â	Latin capital letter A with circumflex
195	C3	303	Ã	Latin capital letter A with tilde
196	C4	304	Ä	Latin capital letter A with diaeresis
197	C5	305	Å	Latin capital letter A with ring above
198	C6	306	Æ	Latin capital letter AE
199	C7	307	Ç	Latin capital letter C with cedilla
200	C8	310	È	Latin capital letter E with grave
201	C9	311	É	Latin capital letter E with acute
202	CA	312	Ê	Latin capital letter E with circumflex
203	CB	313	Ë	Latin capital letter E with diaeresis
204	CC	314	Ì	Latin capital letter I with grave
205	CD	315	Í	Latin capital letter I with acute
206	CE	316	Î	Latin capital letter I with circumflex
207	CF	317	Ï	Latin capital letter I with diaeresis
208	D0	320	Ð	Latin capital letter ETH
209	D1	321	Ñ	Latin capital letter N with a tilde
210	D2	322	Ò	Latin capital letter O with grave
211	D3	323	Ó	Latin capital letter O with acute
212	D4	324	Ô	Latin capital letter O with circumflex
213	D5	325	Õ	Latin capital letter O with tilde
214	D6	326	Ö	Latin capital letter O with diaeresis
215	D7	327	×	Multiplication sign
216	D8	330	Ø	Latin capital letter O with stroke
217	D9	331	Ù	Latin capital letter U with grave
218	DA	332	Ú	Latin capital letter U with acute
219	DB	333	Û	Latin capital letter U with circumflex
220	DC	334	Ü	Latin capital letter U with diaeresis
221	DD	335	Ý	Latin capital letter Y with acute
222	DE	336	Þ	Latin capital letter THORN
223	DF	337	ß	Latin small letter sharp s
224	E0	340	à	Latin small letter a with grave
225	E1	341	á	Latin small letter a with acute
226	E2	342	â	Latin small letter a with circumflex
227	E3	343	ã	Latin small letter a with tilde
228	E4	344	ä	Latin small letter a with diaeresis
229	E5	345	å	Latin small letter a with ring above
230	E6	346	æ	Latin small letter ae
231	E7	347	ç	Latin small letter c with cedilla
232	E8	350	è	Latin small letter e with grave
233	E9	351	é	Latin small letter e with acute
234	EA	352	ê	Latin small letter e with circumflex
235	EB	353	ë	Latin small letter e with diaeresis
236	EC	354	ì	Latin small letter i with grave
237	ED	355	í	Latin small letter i with acute
238	EE	356	î	Latin small letter i with circumflex
239	EF	357	ï	Latin small letter i with diaeresis
240	F0	360	ð	Latin small letter eth
241	F1	361	ñ	Latin small letter n with tilde
242	F2	362	ò	Latin small letter o with grave
243	F3	363	ó	Latin small letter o with acute
244	F4	364	ô	Latin small letter o with circumflex
245	F5	365	õ	Latin small letter o with tilde
246	F6	366	ö	Latin small letter o with diaeresis
247	F7	367	÷	Division sign/Obelus
248	F8	370	ø	Latin small letter o with stroke
249	F9	371	ù	Latin small letter u with grave
250	FA	372	ú	Latin small letter u with acute
251	FB	373	û	Latin small letter u with circumflex
252	FC	374	ü	Latin small letter u with diaeresis
253	FD	375	ý	Latin small letter y with acute
254	FE	376	þ	Latin small letter thorn
255	FF	377	ÿ	Latin small letter y with diaeresis

Number Systems | What is a number system in a computer?

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