Z80:Binary, Decimal, and Hexadecimal - Revision history

KermMartian at 06:06, 5 February 2016

2016-02-05T06:06:55Z

← Older revision		Revision as of 06:06, 5 February 2016
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	* Go to view		* Go to view
	* Change to Scientific mode (as opposed to the default "Standard" mode)		* Change to Scientific mode (as opposed to the default "Standard" mode)
	* Choose the base of the number you want to convert by selecting either ''he'', ''de'', ''oc'', or ''bi''.		* Choose the base of the number you want to convert by selecting either ''hex'', ''dec'', ''oct'', or ''bin''.
	* Determine how large of a number you're going to input, and adjust the panel next to it accordingly:		* Determine how large of a number you're going to input, and adjust the panel next to it accordingly:
	** Byte: Fits in 8 bits		** Byte: Fits in 8 bits

KermMartian: Adjusted lowercase z80 and styling

2016-02-03T20:52:30Z

Adjusted lowercase z80 and styling

Show changes

KermMartian: Fixed math.

2016-02-03T06:26:38Z

Fixed math.

← Older revision		Revision as of 06:26, 3 February 2016
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	The easier of the two, what you do is take each digit times 2 to the power of the column it is in and add it all together. Please note that the farthest right digit is the 0th digit, not the 1st digit. Here's and example:		The easier of the two, what you do is take each digit times 2 to the power of the column it is in and add it all together. Please note that the farthest right digit is the 0th digit, not the 1st digit. Here's and example:
	'''Binary Number''' 1001		'''Binary Number''' 1001
	'''Conversion to Decimal''' ~~[[$~~ 12^3 + 02^2 + 02^1 + 12^0=9 ~~$]]~~		'''Conversion to Decimal''' <math>12^3 + 02^2 + 02^1 + 12^0=9</math>

	== Decimal to Binary ==		== Decimal to Binary ==

	Converting back to binary is slightly more complicated. What you have to do is figure out the largest power of 2 (~~[[$~~ 2^n ~~$]]~~) that is smaller or equal to than your number. Write the nth digit as a 1 and continue. You then subtract the power of 2 from the number and repeat until the decimal number equals 0 (take the result of the subtraction as the new number). If you skip over a power of two, don't forget to write a 0.		Converting back to binary is slightly more complicated. What you have to do is figure out the largest power of 2 (<math>2^n</math>) that is smaller or equal to than your number. Write the nth digit as a 1 and continue. You then subtract the power of 2 from the number and repeat until the decimal number equals 0 (take the result of the subtraction as the new number). If you skip over a power of two, don't forget to write a 0.
	'''Some Useful Powers of 2''' ~~[[$~~ 2^0=1 ~~$]]~~,~~[[$~~ 2^1=2 ~~$]]~~,~~[[$~~ 2^2=4 ~~$]]~~,~~[[$~~ 2^3=8 ~~$]]~~,~~[[$~~ 2^4=16 ~~$]]~~,~~[[$~~ 2^5=32 ~~$]]~~,~~[[$~~ 2^6=64 ~~$]]~~,~~[[$~~ 2^7=128 ~~$]]~~,~~[[$~~ 2^8=256 ~~$]]~~,~~[[$~~ 2^9=512 ~~$]]~~,~~[[$~~ 2^10=1024 ~~$]]~~,~~[[$~~ 2^11=2048 ~~$]]~~,~~[[$~~ 2^12=4096 ~~$]]~~,~~[[$~~ 2^13=8192 ~~$]]~~,~~[[$~~ 2^14=16384 ~~$]]~~,~~[[$~~ 2^15=32768 ~~$]]~~		'''Some Useful Powers of 2''' <math>2^0=1</math>,<math>2^1=2</math>,<math>2^2=4</math>,<math>2^3=8</math>,<math>2^4=16</math>,<math>2^5=32</math>,<math>2^6=64</math>,<math>2^7=128</math>,<math>2^8=256</math>,<math>2^9=512</math>,<math>2^{10}=1024</math>,<math>2^{11}=2048</math>,<math>2^{12}=4096</math>,<math>2^{13}=8192</math>,<math>2^{14}=16384</math>,<math>2^{15}=32768</math>
	'''Decimal Number''' 121		'''Decimal Number''' 121
	'''Conversion to Binary'''		'''Conversion to Binary'''
	1. The largest power of 2 that works is ~~[[$~~ 2^6=64 ~~$]]~~.		1. The largest power of 2 that works is <math>2^6=64</math>.
	2. ~~[[$~~ 121-64= 57 ~~$]]~~. '''Binary number so far: 10000000'''		2. <math>121-64= 57</math>. '''Binary number so far: 10000000'''
	3. Next Power of 2 that works is ~~[[$~~2^5=32 ~~$]]~~.		3. Next Power of 2 that works is <math>2^5=32</math>.
	4. ~~[[$~~ 57-32=25 ~~$]]~~. '''Binary number so far: 11000000'''		4. <math>57-32=25</math>. '''Binary number so far: 11000000'''
	5. Next Power of 2 that works is ~~[[$~~ 2^4=16 ~~$]]~~.		5. Next Power of 2 that works is <math>2^4=16</math>.
	6. ~~[[$~~ 25-16=9 ~~$]]~~. '''Binary number so far: 11100000'''		6. <math>25-16=9</math>. '''Binary number so far: 11100000'''
	7. Next Power of 2 is ~~[[$~~ 2^3=8 ~~$]]~~.		7. Next Power of 2 is <math>2^3=8</math>.
	8. ~~[[$~~ 9-8=1 ~~$]]~~. '''Binary number so far: 11110000'''		8. <math>9-8=1</math>. '''Binary number so far: 11110000'''
	9. Next Power of 2 smaller that works is ~~[[$~~ 2^0=1 ~~$]]~~.		9. Next Power of 2 smaller that works is <math>2^0=1</math>.
	10. ~~[[$~~ 1-1=0 ~~$]]~~. You're done. Don't forget to write your zeros for the powers of 2 you skipped.		10. <math>1-1=0</math>. You're done. Don't forget to write your zeros for the powers of 2 you skipped.
	'''Final answer:''' 11110001		'''Final answer:''' 11110001

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	A rather simple task, though slightly harder than converting binary to decimal. You do the same thing except make sure that when adding you don't go onto the next digit until the previous digit is larger than F (15 in decimal).		A rather simple task, though slightly harder than converting binary to decimal. You do the same thing except make sure that when adding you don't go onto the next digit until the previous digit is larger than F (15 in decimal).
	'''Binary number''' 11110011		'''Binary number''' 11110011
	'''Conversion to Hexadecimal''' ~~[[$~~ 12^7+12^6+12^5+12^4+02^3+02^2+12^1+12^0=F3 ~~$]]~~		'''Conversion to Hexadecimal''' <math>12^7+12^6+12^5+12^4+02^3+02^2+12^1+12^0=F3</math>

	You'll probably be asking "how is that easier than converting binary to decimal?"		You'll probably be asking "how is that easier than converting binary to decimal?"
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	== Hexadecimal to Decimal ==		== Hexadecimal to Decimal ==

	Simple. Just take the decimal equivalence of the hexadecimal digit, times it by ~~[[$~~ 16^n ~~$]]~~ where n equals the location the digit is in (0 being the farthest right), and add them all up.		Simple. Just take the decimal equivalence of the hexadecimal digit, times it by <math>16^n</math> where n equals the location the digit is in (0 being the farthest right), and add them all up.
	'''Hexadecimal Number''' 3BA		'''Hexadecimal Number''' 3BA
	'''Decimal Conversion''' ~~[[$~~ 316^2+1116^1+10160=954 ~~$]]~~		'''Decimal Conversion''' <math>316^2+1116^1+10160=954</math>

	== Decimal to Hexadecimal ==		== Decimal to Hexadecimal ==
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	Similar to converting from binary to hexadecimal. Just don't forget you need to add up to F (15) before going onto the next digit.		Similar to converting from binary to hexadecimal. Just don't forget you need to add up to F (15) before going onto the next digit.
	'''Decimal Number''' 68		'''Decimal Number''' 68
	'''Hexadecimal Conversion''' ~~[[$~~ 6A^1+8A^0=44 ~~$]]~~		'''Hexadecimal Conversion''' <math>6A^1+8A^0=44</math>

	== Setting up Windows Calculator to convert for you ==		== Setting up Windows Calculator to convert for you ==
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	*What's wrong with this?		*What's wrong with this?

	For solutions, see this [[answers-1 \|page]].		For solutions, see this [[z80:answers-1\|page]].

	= Conclusion =		= Conclusion =

KermMartian: Created page with "= Existing Tutorials = Again, there are existing tutorials that touch on this topic. * Sigma's [http://media.taricorp.net/83pa28d/lesson/day03.html Learn Assembly in 28 days..."

2016-02-03T00:45:55Z

Created page with "= Existing Tutorials = Again, there are existing tutorials that touch on this topic. * Sigma's [http://media.taricorp.net/83pa28d/lesson/day03.html Learn Assembly in 28 days..."

New page

= Existing Tutorials =

Again, there are existing tutorials that touch on this topic.

* Sigma's [http://media.taricorp.net/83pa28d/lesson/day03.html Learn Assembly in 28 days, Day 3].
* Cobb's [http://users.hszk.bme.hu/~pg429/z80guide/part1.htm#numbers z80 Assembly, Data and Numbers].

If you still don't understand number bases, read on.

= The Binary Base =

The most basic counting system you can get, it is based on just 1's and 0's. Each ##blue|1## or ##blue|0## is a bit, and 8 bits make up a byte. Counting in binary is related to counting in decimal. Here's how: You keep counting in the units digit (farthest right) until there are no more number symbols to count. Then you increment the next digit and start over again. Here's an example:
'''binary''' 0,1,10,11,100,101,110
'''decimal''' 0,1,2,3,4,5,6

= The Decimal Base =

Hopefully you know how this base works. It's what you use everyday. The digits are ##blue|0,1,2,3,4,5,6,7,8,## and ##blue|9##.

= The Hexadecimal Base =

Counting in ones and zeros can be a pain, and converting between decimal and binary is no better. So, to help sort ease the pain of programmers we'll use the hexadecimal base. This base uses the digits ##blue|0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,## and ##blue|F##. Again, counting similar binary and decimal.

= Conversions of Binary, Decimal, and hexadecimal =

== Binary to Decimal ==

The easier of the two, what you do is take each digit times 2 to the power of the column it is in and add it all together. Please note that the farthest right digit is the 0th digit, not the 1st digit. Here's and example:
'''Binary Number''' 1001
'''Conversion to Decimal''' [[$ 1*2^3 + 0*2^2 + 0*2^1 + 1*2^0=9 $]]

== Decimal to Binary ==

Converting back to binary is slightly more complicated. What you have to do is figure out the largest power of 2 ([[$ 2^n $]]) that is smaller or equal to than your number. Write the nth digit as a 1 and continue. You then subtract the power of 2 from the number and repeat until the decimal number equals 0 (take the result of the subtraction as the new number). If you skip over a power of two, don't forget to write a 0.
'''Some Useful Powers of 2''' [[$ 2^0=1 $]],[[$ 2^1=2 $]],[[$ 2^2=4 $]],[[$ 2^3=8 $]],[[$ 2^4=16 $]],[[$ 2^5=32 $]],[[$ 2^6=64 $]],[[$ 2^7=128 $]],[[$ 2^8=256 $]],[[$ 2^9=512 $]],[[$ 2^10=1024 $]],[[$ 2^11=2048 $]],[[$ 2^12=4096 $]],[[$ 2^13=8192 $]],[[$ 2^14=16384 $]],[[$ 2^15=32768 $]]
'''Decimal Number''' 121
'''Conversion to Binary'''
1. The largest power of 2 that works is [[$ 2^6=64 $]].
2. [[$ 121-64= 57 $]]. '''Binary number so far: 10000000'''
3. Next Power of 2 that works is [[$2^5=32 $]].
4. [[$ 57-32=25 $]]. '''Binary number so far: 11000000'''
5. Next Power of 2 that works is [[$ 2^4=16 $]].
6. [[$ 25-16=9 $]]. '''Binary number so far: 11100000'''
7. Next Power of 2 is [[$ 2^3=8 $]].
8. [[$ 9-8=1 $]]. '''Binary number so far: 11110000'''
9. Next Power of 2 smaller that works is [[$ 2^0=1 $]].
10. [[$ 1-1=0 $]]. You're done. Don't forget to write your zeros for the powers of 2 you skipped.
'''Final answer:''' 11110001

== Binary to Hexadecimal ==

A rather simple task, though slightly harder than converting binary to decimal. You do the same thing except make sure that when adding you don't go onto the next digit until the previous digit is larger than F (15 in decimal).
'''Binary number''' 11110011
'''Conversion to Hexadecimal''' [[$ 1*2^7+1*2^6+1*2^5+1*2^4+0*2^3+0*2^2+1*2^1+1*2^0=F3 $]]

You'll probably be asking "how is that easier than converting binary to decimal?"

Here's your answer: If you haven't notice yet, but each digit in hexadecimal is represented by 4 digits in binary. What does this mean? It means that you only have to compute hexadecimal a nibble (4 bits) at a time. So:
'''Previous number split up into nibbles''' 1111 0011
'''First nibble (right one)''' 0011=3
'''Second nibble''' 1111=F

== Hexadecimal to Binary ==

Another simple task, again we'll use the notion of nibbles to help us. Take each hexadecimal digit and convert it to decimal (A=10,B=11,C=12,D=13,E=14,F=15). Then convert that number to binary.
'''Hexadecimal Number''' E7
'''First digit''' 7
'''Decimal equivalence''' 7
'''Binary conversion''' 7->0111
'''Second digit''' E
'''Decimal equivalence''' 14
'''Binary conversion''' 14->1110
'''Final answer''' 11100111

== Hexadecimal to Decimal ==

Simple. Just take the decimal equivalence of the hexadecimal digit, times it by [[$ 16^n $]] where n equals the location the digit is in (0 being the farthest right), and add them all up.
'''Hexadecimal Number''' 3BA
'''Decimal Conversion''' [[$ 3*16^2+11*16^1+10*16*0=954 $]]

== Decimal to Hexadecimal ==

Similar to converting from binary to hexadecimal. Just don't forget you need to add up to F (15) before going onto the next digit.
'''Decimal Number''' 68
'''Hexadecimal Conversion''' [[$ 6*A^1+8*A^0=44 $]]

== Setting up Windows Calculator to convert for you ==

If you didn't already know, you can use Windows Calculator to convert to certain bases. Here's how you set it up:

* Go to view
* Change to Scientific mode (as opposed to the default "Standard" mode)
* Choose the base of the number you want to convert by selecting either ''he'', ''de'', ''oc'', or ''bi''.
* Determine how large of a number you're going to input, and adjust the panel next to it accordingly:
byte: Fits in 8 bits
word: Fits in 16 bits
Dword: Fits in 32 bits
Qword: Fits in 64 bits
* Type in your number
* Click on the base you want to convert to.
* The number will be changed to the equivalent number for that base

= Common notation =

How do you distinguish one number base from other base numbers? The compiler has 2 ways:

== Denote before ==

Use a symbol to denote the base of the number.
'''binary''' %
'''hexadecimal''' $

Note that decimal has no pre-denote form.

== Denote after ==

Use a symbol after to denote the base of the number.
'''binary''' b
'''decimal''' d
'''hexadecimal''' h

Note that if you're using this method with hexadecimal, it is important that the first digit is not a letter (A,B,C,D,E, or F). If necessary, add a 0 to the front.

Just to save you some trouble, if you don't put a denotation for a number the compiler will assume that it is a decimal number.

= Review =

Here are some questions to make sure that you really understand this. If not, perhaps you should re-read this section.

== Convert the following numbers ==

Don't use a base converter, but a calculator is alright if you need to multiply certain things.

== Binary to Decimal ==

1. 1110
2. 11001001
3. 110101011000
4. 1111
5. 10001000

== Binary to Hexadecimal ==

1. 1001
2. 10101010
3. 10011101
4. 1100
5. 10000110

== Decimal to Binary ==

1. 10
2. 36
3. 57
4. 184
5. 376

== Decimal to Hexadecimal ==

1. 91
2. 100
3. 852
4. 4533
5. 424

== Hexadecimal to Binary ==

1. 10
2. AB
3. 8F
4. 3E
5. B7

== Hexadecimal to Decimal ==

1. 4
2. F8
3. EB
4. D1
5. 97

== Denotations ==

Identify the following numbers as binary, decimal, or hexadecimal according the the denotation.

1. %10001110
2. 1011
3. $100101
4. 10h
5. 1111b
6. 11
*7. AEh
8. 54d
9. %10010010011
10. 100000000001h

*What's wrong with this?

For solutions, see this [[answers-1 |page]].

= Conclusion =

It takes a lot of practice to get good at converting between bases. Although you could just use Windows calculator to convert between bases, it's still useful to know how it's done by hand. If used correctly, bases and converting between bases will make programming as well as understanding other people's code a lot easier.

{{lowercase}}
[[Category:z80 Assembly]]
[[Category:z80 Heaven]]