Learn Binary Number System-2

Continued from Last Week's Post

This is the continuation of last week's Article Learn Binary Number System 

Last week https://www.msaccesstips.com/2010/09/octal-numbering-system.htmlwe have gone through the fundamentals of the Binary Number System and learned how to convert decimal number 10 to binary and saw different methods also.

I hope you have tried converting the sample number 255, that I have given at the end of last week's article, using both methods shown there.

If you could not do it, then let us do it here.

Method-1:

1. Find the highest Integer Value from the Binary Table that goes into the Decimal Number and subtract that value from it. Here, 128 is the highest value that can be taken.

255

-128

=127

2. Write Binary digit 1 at the 128 number position underneath the Binary Table.

215	214	213	212	211	210	29	28	27	26	25	24	23	22	21	20
32,768	16,384	8,192	4,096	2,048	1,024	512	256	128	64	32	16	8	4	2	1
								1

3. The next highest integer number from the Binary Table that goes into 127 is 64.

127

-64

=63

215	214	213	212	211	210	29	28	27	26	25	24	23	22	21	20
32,768	16,384	8,192	4,096	2,048	1,024	512	256	128	64	32	16	8	4	2	1
								1	1

4. Repeat this method till you get the value of the unit position.

215	214	213	212	211	210	29	28	27	26	25	24	23	22	21	20
32,768	16,384	8,192	4,096	2,048	1,024	512	256	128	64	32	16	8	4	2	1
								1	1	1	1	1	1	1	1

You can cross-check the result by adding up all values taken from the 1s bit (the name of the binary digit) position to arrive at the value you were trying to convert into Binary.

Method-2:

1. Divide the decimal number by 2 and take the remainder and write at the unit position in the Binary Table.

   255/2 = Quotient = 127, Remainder = 1

2. Next step, take the Quotient Value (127) of the previous calculation, divide it by 2, and find the remainder. Write the remainder value to the left of the earlier written binary digit (bit).  Repeat this process till nothing is left to divide and write the final remainder value to the binary table.

127/2 = Quotient = 63,  Remainder = 1

63/2   =  Quotient = 31,  Remainder = 1

31/2   =  Quotient = 15,  Remainder = 1

15/2   =  Quotient =   7,  Remainder = 1

7/2   =  Quotient =     3,  Remainder = 1

3/2   =  Quotient =     1,  Remainder = 1

1/2   =  Quotient =     0,  Remainder = 1

You will get the Binary Number 11111111 of Decimal Number 255.

You can experiment with this with bigger Decimal Values or write some unknown Binary Values with random 1s and 0s and try converting them back into Decimal Numbers.

Next, let us try some additions and subtractions with Binary Numbers.  If you know how to do additions and subtractions with Decimal Numbers then you will have no problems with Binary Numbers. 

Example: Addition

11101110	238
+1110111	119
101100101	357

Start adding the rightmost digits:

1.   0+1 = 1

2.   next 1+1 = 2, put 0 and carry 2 to the next position (like 5+5=10 we put 0 at the unit's position and carry 1 to the next position to add)

3.   next 1+1+1 carry = 3(binary 11), put 1 and carries 2 to the next position

4.   next 1+1 carry = 2(binary 10), put 0 and carries 2 to the next position

5.   Next 1+1 carry = 2(binary 10), put 0 and carries 2 to the next position

6.   Next 1+1+1 carry = 3(binary 11), put 1 and carries 2 to the next position

7.   Next 1+1+1 carry = 3(binary 11), put 1 and carries 2 to the next position

8.   Next 1+1 carry = 2(binary 10), put 0 and put 1 in the next position.

Example: Subtraction

11001110	206
-1111111	127
1001111	79

1.   0-1 cannot be done, so take 2 from the next position, now 2-1 = 1, but the next position on the first line becomes 0.

2.   0-1 cannot be done, so take 2 from the next position, now 2-1 = 1, but the next position on the first line becomes 0.

3.   0-1 cannot be done, so take 2 from the next position, now 2-1 = 1, now the next 3 positions become 0.

4.   Take Value from the 8 value position and move forward to the 4 positions and to the 2 value position, 2-1 = 1

5.   1-1 = 0

6.   1-1 = 0

7.    After moving the value forward from the 7th digit position on the top line, it is now 0.  So move 2 from the next position. 2-1 = 1

You can try it out yourself, starting with smaller binary values and progressively, with bigger ones.

For your information, there is no Multiplication or Division in computers.  These calculations are achieved by successive addition or successive subtraction of values.

Continued../-

Earlier Post Link References:

1. Learn Binary Numbering System
2. Learn Binary Numbering System-2
3. Octal Numbering System
4. Hexadecimal Numbering System
5. Colors 24-Bits And Binary Conversion.
6. Create Your Own Color Palette