Computer Number Systems - Revision history

Marc: /* Video Resources */

2022-07-04T21:23:50Z

Video Resources

@@ Line 289: / Line 289: @@
 == Video Resources ==
-There are many YouTube videos about computer number systems. Here are a handful that cover the topic nicely, without too many ads:
+===ACSL Videos===
 {|

Marc Brown: /* Converting from Decimal */

2021-11-15T21:43:11Z

Converting from Decimal

← Older revision		Revision as of 14:43, 15 November 2021
Line 169:		Line 169:
	== Converting from Decimal ==		== Converting from Decimal ==

	The algorithm to convert a number from an arbitrary base requires finding how many times successive powers of the base go into the number, starting with the largest power of the base less than the starting number. For example, converting ~~3206~~ from base 10 to octal proceeds as follows:		The algorithm to convert a number from an arbitrary base requires finding how many times successive powers of the base go into the number, starting with the largest power of the base less than the starting number. For example, converting 3306 from base 10 to octal proceeds as follows:

	1) The largest power of 8 that is less than or equal to ~~3206~~ is 512 ($8^3$). Divide ~~3206~~ by 512:		1) The largest power of 8 that is less than or equal to 3306 is 512 ($8^3$). Divide 3306 by 512:
	:$~~3206~~ = 6*8^3 + 234$		:$3306 = 6*8^3 + 234$

	2) The next power of 8 is 64 ($8^2$). Divide 234 by 64:		2) The next power of 8 is 64 ($8^2$). Divide 234 by 64:

Marc: /* Converting to Decimal */

2020-09-02T23:38:40Z

Converting to Decimal

← Older revision		Revision as of 16:38, 2 September 2020
Line 155:		Line 155:
	As we learned in the first or second grade, the decimal value of a decimal number is simply sum of each digit multiplied by its place value:		As we learned in the first or second grade, the decimal value of a decimal number is simply sum of each digit multiplied by its place value:

	:$12345 = 1×10^{4}+ 2×10^{3}+ 3×10^{2}+ 4×10^{1}+ 5×10{0} = ~~1×10000 + 2×1000 + 3×100 + 4×10 + 5×1 =~~ 10000 + 2000+ 300 +40+5 = 12345$		:$12345 = 1×10^{4}+ 2×10^{3}+ 3×10^{2}+ 4×10^{1}+ 5×10^{0} = 10000 + 2000+ 300 +40+5 = 12345$

	:$3079 = 3×10^{3}+ 0×10^{2}+ 7×10^{1}+ 9×10^{0} = 3000 + 70 + 9 = 3079$		:$3079 = 3×10^{3}+ 0×10^{2}+ 7×10^{1}+ 9×10^{0} = 3000 + 70 + 9 = 3079$

Marc: /* Converting to Decimal */

2020-09-02T23:33:20Z

Converting to Decimal

@@ Line 153: / Line 153: @@
 == Converting to Decimal ==
-The decimal value of a number in an arbitrary base is the sum of each digit multiplied by its place value.  Here are some examples of converting from one base into base 10:
+As we learned in the first or second grade, the decimal value of a decimal number is simply sum of each digit multiplied by its place value:
-:$12345 = 1×10000 + 2×1000 + 3×100 + 4×10 + 5×1 = 1×10^{4}+ 2×10^{3}+ 3×10^{2}+ 4×10^{1}+ 5×10{0}$
+:$12345 = 1×10^{4}+ 2×10^{3}+ 3×10^{2}+ 4×10^{1}+ 5×10{0} = 1×10000 + 2×1000 + 3×100 + 4×10 + 5×1 = 10000 + 2000+ 300 +40+5 = 12345$
 :$1101_{2}  = 1 × 2^3  + 1 × 2^2  + 0 × 2^1  + 1 × 2^0 = 8 + 4  + 0  + 1  = 13_{10}$
@@ Line 162: / Line 166: @@
 :$A5E_{16}  = 10 × 16^2 + 5 × 16^1 + 14 × 16^0 = 10 × 256 + 5 × 16 + 14 × 1 = 2560 + 80 + 14 = 2654_{10}$
 == Converting between Binary, Octal, and Hexadecimal ==

Marc: /* Decimal Values */

2020-09-02T23:05:07Z

Decimal Values

← Older revision		Revision as of 16:05, 2 September 2020
Line 151:		Line 151:
	\|}		\|}

	== Decimal ~~Values~~ ==		== Converting to Decimal ==

	The decimal value of a number in an arbitrary base is the sum of each digit multiplied by its place value. Here are some examples of converting from one base into base 10:		The decimal value of a number in an arbitrary base is the sum of each digit multiplied by its place value. Here are some examples of converting from one base into base 10:

	:$12345 = 1×10000 + 2×1000 + 3×100 + 4×10 + 5×1 = 1×10^{4}+ 2×10^{3}+ 3×10^{2}+ 4×10^{1}+ 5×10{0}.$		:$12345 = 1×10000 + 2×1000 + 3×100 + 4×10 + 5×1 = 1×10^{4}+ 2×10^{3}+ 3×10^{2}+ 4×10^{1}+ 5×10{0}$

	:$1101_{2} = 1 × 2^3 + 1 × 2^2 + 0 × 2^1 + 1 × 2^0 = 8 + 4 + 0 + 1 = 13_{10}$		:$1101_{2} = 1 × 2^3 + 1 × 2^2 + 0 × 2^1 + 1 × 2^0 = 8 + 4 + 0 + 1 = 13_{10}$

	:$175_{8} = 1 × 8^2 + 7 × 8^1 + 5 × 8^0 = 1 × 64 + 7 × 8 + 5 × 1 = 64 + 56 + 5 = 125_{10}$.		:$175_{8} = 1 × 8^2 + 7 × 8^1 + 5 × 8^0 = 1 × 64 + 7 × 8 + 5 × 1 = 64 + 56 + 5 = 125_{10}$

	:$A5E_{16} = 10 × 16^2 + 5 × 16^1 + 14 × 16^0 = 10 × 256 + 5 × 16 + 14 × 1 = 2560 + 80 + 14 = 2654_{10}$.		:$A5E_{16} = 10 × 16^2 + 5 × 16^1 + 14 × 16^0 = 10 × 256 + 5 × 16 + 14 × 1 = 2560 + 80 + 14 = 2654_{10}$

	== Converting between Binary, Octal, and Hexadecimal ==		== Converting between Binary, Octal, and Hexadecimal ==

Marc: /* Converting between Binary, Octal, and Hexadecimal */

2020-09-02T18:40:15Z

Converting between Binary, Octal, and Hexadecimal

← Older revision		Revision as of 11:40, 2 September 2020
Line 165:		Line 165:
	== Converting between Binary, Octal, and Hexadecimal ==		== Converting between Binary, Octal, and Hexadecimal ==

	Converting from octal to ~~base 2~~ is simple: replace each octal digit by its corresponding 3 binary bits. For example:		Converting from octal to binary is simple: replace each octal digit by its corresponding 3 binary bits. For example:

	:$375_{8} = \ \ 011\ \ \ 111\ \ \ 101_{2} = 11111101_2$		:$375_{8} = \ \ 011\ \ \ 111\ \ \ 101_{2} = 11111101_2$

	Converting from hex to ~~base~~ is also simple: replace each hex digit by its corresponding 4 binary bits. For example:		Converting from hex to binary is also simple: replace each hex digit by its corresponding 4 binary bits. For example:

	:$FD_{16} = \ \ 1111\ \ \ 1101_{2} = 11111101_2$		:$FD_{16} = \ \ 1111\ \ \ 1101_{2} = 11111101_2$

	Converting from binary to either octal or hex is pretty simple as well: group the ~~bit~~ by 3s or 4s (starting at the right), and convert each group:		Converting from binary to either octal or hex is pretty simple as well: group the bits by 3s or 4s (starting at the right), and convert each group:

	:$10000001111000101100 = 10\ 000\ 001\ 111\ 000\ 101 \ 100 = 2017054_8$		:$10000001111000101100 = 10\ 000\ 001\ 111\ 000\ 101 \ 100 = 2017054_8$
Line 179:		Line 179:
	:$10000001111000101100 = 1000\ 0001\ 1110\ 0010\ 1100 = 81E2C_{16}$		:$10000001111000101100 = 1000\ 0001\ 1110\ 0010\ 1100 = 81E2C_{16}$

	~~You can convert~~ between base 8 and 16 by expressing the number in base 2 (easy to do!) and then converting that number from base 2 (another easy operation)! This is shown below in Sample Problem #1.		Converting between base 8 and 16 is easy by expressing the number in base 2 (easy to do!) and then converting that number from base 2 (another easy operation)! This is shown below in Sample Problem #1.

	== Using Hexadecimal Numbers to Represent Colors ==		== Using Hexadecimal Numbers to Represent Colors ==

Marc: /* Different Number Systems */

2020-09-02T16:33:15Z

Different Number Systems

← Older revision		Revision as of 09:33, 2 September 2020
Line 5:		Line 5:
	== Different Number Systems ==		== Different Number Systems ==

	A ''number system'' is the way we name and represent numbers. The number system we use in our daily life is known as '''decimal number system''' or '''base 10''' because it is based on 10 different digits: 0, 1, 2, ..., 8, and 9. The base of number is written as subscript to a number. For example, the decimal number "twelve thousand three hundred and forty-five" is written as $12345_{10}$. Without a base, it's assumed that a number is in base 10.		A ''number system'' is the way we name and represent numbers. The number system we use in our daily life is known as '''decimal number system''' or '''base 10''' because it is based on 10 different digits: 0, 1, 2, ..., 8, and 9. The base of number is written as subscript to a number. For example, the decimal number "twelve thousand three hundred and forty-five" is written as $12345_{10}$. Without a base or explicit context, it's assumed that a number is in base 10.

	In computer science, apart from the decimal system, three additional number systems are commonly used: '''binary''' (base-2), '''octal''' (base-8), and '''hexadecimal''' or just '''hex''' (base-16). Binary numbers are important because that is how ~~number~~ are stored in the computer. Octal and hexadecimal are used to represent binary numbers is a user-friendly way. Each octal symbol represents 3 binary bits and each hex digit represents 4 binary bits. For example, the decimal number 53201, stored as in the computer as the binary number 10000001111000101100, is represented by the octal number 2017054, and the hex number 81E2C. The table below compares the number systems:		In computer science, apart from the decimal system, three additional number systems are commonly used: '''binary''' (base-2), '''octal''' (base-8), and '''hexadecimal''' or just '''hex''' (base-16). Binary numbers are important because that is how numbers are stored in the computer. Octal and hexadecimal are used to represent binary numbers in a user-friendly way. Each octal symbol represents 3 binary bits and each hex digit represents 4 binary bits. For example, the decimal number 53201, stored as in the computer as the binary number 10000001111000101100, is represented by the octal number 2017054, and the hex number 81E2C. The table below compares the number systems:
	:{\| class="wikitable"		:{\| class="wikitable"
	!Number System		!Number System
Line 150:		Line 150:
	\|14		\|14
	\|}		\|}

	== Decimal Values ==		== Decimal Values ==

Marc: /* Decimal Values */

2020-09-02T16:19:31Z

Decimal Values

← Older revision		Revision as of 09:19, 2 September 2020
Line 153:		Line 153:
	== Decimal Values ==		== Decimal Values ==

	The value of a ~~any~~ number is the sum of each digit multiplied by its place value. Here are some examples of converting from one base into base 10:		The decimal value of a number in an arbitrary base is the sum of each digit multiplied by its place value. Here are some examples of converting from one base into base 10:

	:$12345 = 1×10000 + 2×1000 + 3×100 + 4×10 + 5×1 = 1×10^{4}+ 2×10^{3}+ 3×10^{2}+ 4×10^{1}+ 5×10{0}.$		:$12345 = 1×10000 + 2×1000 + 3×100 + 4×10 + 5×1 = 1×10^{4}+ 2×10^{3}+ 3×10^{2}+ 4×10^{1}+ 5×10{0}.$

Marc: /* Converting between Binary, Octal, and Hexadecimal */

2020-09-02T16:18:44Z

Converting between Binary, Octal, and Hexadecimal

← Older revision		Revision as of 09:18, 2 September 2020
Line 166:		Line 166:

	Converting from octal to base 2 is simple: replace each octal digit by its corresponding 3 binary bits. For example:		Converting from octal to base 2 is simple: replace each octal digit by its corresponding 3 binary bits. For example:
	:$375_{8} = \ \ 011\ \ \ 111\ \ \ 101_{2}$
			:$375_{8} = \ \ 011\ \ \ 111\ \ \ 101_{2} = 11111101_2$

	Converting from hex to base is also simple: replace each hex digit by its corresponding 4 binary bits. For example:		Converting from hex to base is also simple: replace each hex digit by its corresponding 4 binary bits. For example:
	:$FD_{16} = \ \ 1111\ \ \ 1101_{2}$
			:$FD_{16} = \ \ 1111\ \ \ 1101_{2} = 11111101_2$

	Converting from binary to either octal or hex is pretty simple as well: group the bit by 3s or 4s (starting at the right), and convert each group:		Converting from binary to either octal or hex is pretty simple as well: group the bit by 3s or 4s (starting at the right), and convert each group:

	:$10000001111000101100 = 10\ 000\ 001\ 111\ 000\ 101 \ 100 = 2017054_8$		:$10000001111000101100 = 10\ 000\ 001\ 111\ 000\ 101 \ 100 = 2017054_8$

	:$10000001111000101100 = 1000\ 0001\ 1110\ 0010\ 1100 = 81E2C_{16}$		:$10000001111000101100 = 1000\ 0001\ 1110\ 0010\ 1100 = 81E2C_{16}$

	You can convert between base 8 and 16 by expressing the number in base 2 (easy to do!) and then converting that number from base 2 (another easy operation)! This is shown below in Sample Problem #1.		You can convert between base 8 and 16 by expressing the number in base 2 (easy to do!) and then converting that number from base 2 (another easy operation)! This is shown below in Sample Problem #1.

Marc: /* Decimal Values */

2020-09-02T16:16:34Z

Decimal Values

← Older revision		Revision as of 09:16, 2 September 2020
Line 153:		Line 153:
	== Decimal Values ==		== Decimal Values ==

	The value of a any number is the sum of each digit multiplied by its place value. Here are some examples:		The value of a any number is the sum of each digit multiplied by its place value. Here are some examples of converting from one base into base 10:

	:$12345 = 1×10000 + 2×1000 + 3×100 + 4×10 + 5×1 = 1×10^{4}+ 2×10^{3}+ 3×10^{2}+ 4×10^{1}+ 5×10{0}.$		:$12345 = 1×10000 + 2×1000 + 3×100 + 4×10 + 5×1 = 1×10^{4}+ 2×10^{3}+ 3×10^{2}+ 4×10^{1}+ 5×10{0}.$
Line 161:		Line 161:
	:$175_{8} = 1 × 8^2 + 7 × 8^1 + 5 × 8^0 = 1 × 64 + 7 × 8 + 5 × 1 = 64 + 56 + 5 = 125_{10}$.		:$175_{8} = 1 × 8^2 + 7 × 8^1 + 5 × 8^0 = 1 × 64 + 7 × 8 + 5 × 1 = 64 + 56 + 5 = 125_{10}$.

	:$A5E_{16} = 10 × 16^2 + 5 × 16^1 + 14 × 16^0 = 10 × 256 + 5 × 16 + 14 × 1 = 2560 + 80 + 14 = 2654_{10}$.		:$A5E_{16} = 10 × 16^2 + 5 × 16^1 + 14 × 16^0 = 10 × 256 + 5 × 16 + 14 × 1 = 2560 + 80 + 14 = 2654_{10}$.


	== Converting between Binary, Octal, and Hexadecimal ==		== Converting between Binary, Octal, and Hexadecimal ==