Binary

1
Binary Decimal Conversions

• Binary to Decimal Back Again
• Mr. Akuna 2004

2
Introduction to Binary Numbers

• Binary numbers are made up of binary digits, each
  digit is called a BIT, taken from the words
  BI-nary digi-T. Binary digits are either 0 or 1.
• Eight bits are grouped together to form a BYTE.
• Binary numbers are numbers just like decimal
  numbers but are in base 2 instead of base 10. The
  base tells you how many digits can be used.
• In base 10 (decimal) there are 10 0, 1, 2, 3,
  4, 5, 6, 7, 8, 9
• In base 2 (binary) there are 2 0, 1
•  

3
Binary to Decimal Conversion
Remember 8 bits a BYTE and each bit has a positional value so the first position is 20 1, the second position is 21 2, the third position is 22 4, and so on Take the binary number 11101 place each in position. Remember 8 bits a BYTE and each bit has a positional value so the first position is 20 1, the second position is 21 2, the third position is 22 4, and so on Take the binary number 11101 place each in position. Remember 8 bits a BYTE and each bit has a positional value so the first position is 20 1, the second position is 21 2, the third position is 22 4, and so on Take the binary number 11101 place each in position. Remember 8 bits a BYTE and each bit has a positional value so the first position is 20 1, the second position is 21 2, the third position is 22 4, and so on Take the binary number 11101 place each in position. Remember 8 bits a BYTE and each bit has a positional value so the first position is 20 1, the second position is 21 2, the third position is 22 4, and so on Take the binary number 11101 place each in position. Remember 8 bits a BYTE and each bit has a positional value so the first position is 20 1, the second position is 21 2, the third position is 22 4, and so on Take the binary number 11101 place each in position. Remember 8 bits a BYTE and each bit has a positional value so the first position is 20 1, the second position is 21 2, the third position is 22 4, and so on Take the binary number 11101 place each in position. Remember 8 bits a BYTE and each bit has a positional value so the first position is 20 1, the second position is 21 2, the third position is 22 4, and so on Take the binary number 11101 place each in position.
27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1
1 1 1 0 1
Next take the value for that position. Add 1684129notice 2 is not added in since its position has a 0, so binary 11101 is equal to decimal 29 Next take the value for that position. Add 1684129notice 2 is not added in since its position has a 0, so binary 11101 is equal to decimal 29 Next take the value for that position. Add 1684129notice 2 is not added in since its position has a 0, so binary 11101 is equal to decimal 29 Next take the value for that position. Add 1684129notice 2 is not added in since its position has a 0, so binary 11101 is equal to decimal 29 Next take the value for that position. Add 1684129notice 2 is not added in since its position has a 0, so binary 11101 is equal to decimal 29 Next take the value for that position. Add 1684129notice 2 is not added in since its position has a 0, so binary 11101 is equal to decimal 29 Next take the value for that position. Add 1684129notice 2 is not added in since its position has a 0, so binary 11101 is equal to decimal 29 Next take the value for that position. Add 1684129notice 2 is not added in since its position has a 0, so binary 11101 is equal to decimal 29
4
Check for Understanding

• What is the value of 00100101 ?
• What is the value of 11111110 ?
• What is the value of 10000001 ?

27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1



37
254
129
5
Decimal to Binary Conversion
Take the decimal number 75, look at each number starting with 128, if it can be subtracted place a 1 otherwise place a 0. The first number that can be subtracted is 64 so place a 0 below 128 a 1 below 64. Subtract 64 from 75, the remainder is 11. The next number that can be subtracted is 8 so place a 0 below 32, a 0 below 16, a 1 below 8. Subtract 8 from 11, the remainder is 3. The next number that can be subtracted is 2 so place a 0 below 4 a 1 below the 2. Subtract 2 from 3, the remainder is 1. Place a 1 below the 1 subtract 1, the remainder is 0 and you have the binary number 01001011 Take the decimal number 75, look at each number starting with 128, if it can be subtracted place a 1 otherwise place a 0. The first number that can be subtracted is 64 so place a 0 below 128 a 1 below 64. Subtract 64 from 75, the remainder is 11. The next number that can be subtracted is 8 so place a 0 below 32, a 0 below 16, a 1 below 8. Subtract 8 from 11, the remainder is 3. The next number that can be subtracted is 2 so place a 0 below 4 a 1 below the 2. Subtract 2 from 3, the remainder is 1. Place a 1 below the 1 subtract 1, the remainder is 0 and you have the binary number 01001011 Take the decimal number 75, look at each number starting with 128, if it can be subtracted place a 1 otherwise place a 0. The first number that can be subtracted is 64 so place a 0 below 128 a 1 below 64. Subtract 64 from 75, the remainder is 11. The next number that can be subtracted is 8 so place a 0 below 32, a 0 below 16, a 1 below 8. Subtract 8 from 11, the remainder is 3. The next number that can be subtracted is 2 so place a 0 below 4 a 1 below the 2. Subtract 2 from 3, the remainder is 1. Place a 1 below the 1 subtract 1, the remainder is 0 and you have the binary number 01001011 Take the decimal number 75, look at each number starting with 128, if it can be subtracted place a 1 otherwise place a 0. The first number that can be subtracted is 64 so place a 0 below 128 a 1 below 64. Subtract 64 from 75, the remainder is 11. The next number that can be subtracted is 8 so place a 0 below 32, a 0 below 16, a 1 below 8. Subtract 8 from 11, the remainder is 3. The next number that can be subtracted is 2 so place a 0 below 4 a 1 below the 2. Subtract 2 from 3, the remainder is 1. Place a 1 below the 1 subtract 1, the remainder is 0 and you have the binary number 01001011 Take the decimal number 75, look at each number starting with 128, if it can be subtracted place a 1 otherwise place a 0. The first number that can be subtracted is 64 so place a 0 below 128 a 1 below 64. Subtract 64 from 75, the remainder is 11. The next number that can be subtracted is 8 so place a 0 below 32, a 0 below 16, a 1 below 8. Subtract 8 from 11, the remainder is 3. The next number that can be subtracted is 2 so place a 0 below 4 a 1 below the 2. Subtract 2 from 3, the remainder is 1. Place a 1 below the 1 subtract 1, the remainder is 0 and you have the binary number 01001011 Take the decimal number 75, look at each number starting with 128, if it can be subtracted place a 1 otherwise place a 0. The first number that can be subtracted is 64 so place a 0 below 128 a 1 below 64. Subtract 64 from 75, the remainder is 11. The next number that can be subtracted is 8 so place a 0 below 32, a 0 below 16, a 1 below 8. Subtract 8 from 11, the remainder is 3. The next number that can be subtracted is 2 so place a 0 below 4 a 1 below the 2. Subtract 2 from 3, the remainder is 1. Place a 1 below the 1 subtract 1, the remainder is 0 and you have the binary number 01001011 Take the decimal number 75, look at each number starting with 128, if it can be subtracted place a 1 otherwise place a 0. The first number that can be subtracted is 64 so place a 0 below 128 a 1 below 64. Subtract 64 from 75, the remainder is 11. The next number that can be subtracted is 8 so place a 0 below 32, a 0 below 16, a 1 below 8. Subtract 8 from 11, the remainder is 3. The next number that can be subtracted is 2 so place a 0 below 4 a 1 below the 2. Subtract 2 from 3, the remainder is 1. Place a 1 below the 1 subtract 1, the remainder is 0 and you have the binary number 01001011 Take the decimal number 75, look at each number starting with 128, if it can be subtracted place a 1 otherwise place a 0. The first number that can be subtracted is 64 so place a 0 below 128 a 1 below 64. Subtract 64 from 75, the remainder is 11. The next number that can be subtracted is 8 so place a 0 below 32, a 0 below 16, a 1 below 8. Subtract 8 from 11, the remainder is 3. The next number that can be subtracted is 2 so place a 0 below 4 a 1 below the 2. Subtract 2 from 3, the remainder is 1. Place a 1 below the 1 subtract 1, the remainder is 0 and you have the binary number 01001011
27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1
0 1 0 0 1 0 1 1
75
-64
11
-8
3
-2
1
-1
0
6
Check for Understanding

• What is decimal 83 in binary?
• What is decimal 54 in binary?
• What is decimal 255 in binary?

27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1



83
54
255
7
Least Significant Bit (LSB)

• Lets take another look at the binary number
  01001011
• Notice we can drop the leading 0 and express the
  binary number as 1001011
• This is because the least significant bit is the
  leftmost 1 which is necessary to maintain the
  accuracy of the binary number
• Another way of expressing this is to ask what is
  the least number of bits needed to express the
  binary number 01001011 ? The answer is 7
  (1001011)
• Or we can ask what is the least number of bits
  needed to express 75? Again the answer is 7
  (Notice this is a trick question since we are
  asking about the number of bits.)

8
Check for Understanding

• What is the least number of bits needed to
  express 3? 
• What is the least number of bits needed to
  express the binary number 00011010?
• What is the least number of bits needed to
  express 67?
• BONUS Question
• How can you tell quickly if a binary number is
  odd or even?

2 (binary number 11)
5 (binary number 11010)
7 (binary number 1000011)
If the last bit is 0 the decimal number will be
even. If the last bit is 1 the decimal number
will be odd.
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Final Check for Understanding

• Click on this link to bring up the Binary-Decimal
  Worksheet.
• Print complete the worksheet
• Turn in this worksheet with your other
  assignments.