BINARY ADDITION
BINARY ADDITION RULES
BINARY ADDITION RULES
Example 1:
Add the binary numbers 1.1, 011.1, and 101.1.
BINARY ADDITION RULES
Solution:
STEP 1:
Write down the numbers aligning the radix point
then start evaluating from the rightmost column.
BINARY ADDITION RULES
Solution:
STEP 2:
Cross out each pair of 1’s and generate a carry
every time a pair is crossed out.
BINARY ADDITION RULES
Solution:
STEP 3:
Two carries cannot be represented by the numeral
2. However, it can be represented as shown below:
BINARY ADDITION RULES
Solution:
STEP 4:
Continue evaluating until all the columns have
been evaluated.
BINARY ADDITION RULES
Solution:
STEP 5:
The final carry can be immediately carried down
as the most significant bit.
BINARY
SUBTRACTION
BINARY SUBTRACTION RULES
BINARY SUBTRACTION RULES
Example 1:
Subtract the binary number 1011.1 and 10010.
BINARY SUBTRACTION RULES
Solution:
STEP 1:
Write the numbers aligning the radix point
then start evaluating from the rightmost column.
NOTE: You need to borrow to evaluate the rightmost column.
BINARY SUBTRACTION RULES
Solution:
STEP 2:
If borrowing is required, borrow from the nearest 1
at the left digit. Add this to the next digit to its right until we
arrive at the column you are evaluating.
BINARY SUBTRACTION RULES
Solution:
STEP 3:
Perform rules 1 to 3 since you have already
borrowed from 1.
BINARY SUBTRACTION RULES
Solution:
STEP 4:
Perform borrowing from the nearest 1 at the left
digit. Repeat Step 2.
BINARY SUBTRACTION RULES
Solution:
STEP 5:
Perform rules 1 to 3 until you arrive at the
leftmost column.
BINARY
MULTIPLICATION
BINARY MULTIPLICATION RULES
BINARY MULTIPLICATION RULES
Example 1:
Multiply 1101 and 1011.01.
BINARY MULTIPLICATION RULES
Solution:
STEP 1:
Align the mutiplicand with the multiplier.
BINARY MULTIPLICATION RULES
Solution:
STEP 2:
Remember the two equations in performing binary
multiplications.
BINARY MULTIPLICATION RULES
NOTE:
Align the product with the multiplier digit to which you are
multiplying the multiplicand.
BINARY MULTIPLICATION RULES
Solution:
STEP 3:
Add the resulting products as aligned above. The
decimal point is placed on the fourth place from the right.
BINARY MULTIPLICATION RULES
Example:
Multiply 1011.1 and 1001.01.
BINARY DIVISION
BINARY DIVISION RULES
Example 1:
Divide 101101 and 101.
BINARY DIVISION RULES
Solution:
STEP 1:
Determine the number of digits (n) of the divisor.
The first n digits of the dividend is equal to the divisor.
BINARY DIVISION RULES
Solution:
STEP 2:
Write a 1 aligned with the resulting number’s
place value. Subtract the divisor from the corresponding
number.
BINARY DIVISION RULES
Step 3:
Bring down the next bit from the dividend.
NOTE: Place 0 if the divisor is > the dividend.
Place 1 if the divisor is < the dividend.
BINARY DIVISION RULES
STEP 4:
Repeat step 3 until all the difference is 0 and/or
the bits of the dividend has been brought down.
BINARY DIVISION RULES
Example:
Divide: 100101.1 by 111.
BOOLEAN
ALGEBRA
BOOLEAN ALGEBRA
- an algebra that is applied only to binary numbers
- has a list of operations that it can use to simplify its
equations
POSTULATES
- a set of rules defined on how a set of numbers can be
manipulated
- also called Huntington’s Postulates
- used to derive certain theorems applicable assuming certain
conditions
Huntington’s Postulates
Huntington’s Postulates
POSTULATE 1
- defines that variables can be defined using a (Boolean)
algebraic expression by using the symbol “=“
POSTULATE 11A
- defines the addition of binary numbers
- whenever two binary numbers are added, their sum is also a binary
number
POSTULATE 11B
- defines the multiplication of binary numbers
- whenever two binary numbers are multiplied, their product is also a
binary number
POSTULATE 111A
- defines the zero element in the binary numbering system
- whenever zero is added to any binary number, the value of the number to
which zero was added will not change
POSTULATE 111A
- zero is defined such that when it is added to a binary number, the binary
number will retain its value
POSTULATE 111B
- Defines the one element in the binary numbering system
- Whenever one is multiplied to any binary number, the value of the number
to which one was multiplied will not change
POSTULATE 111B
- One is defined such that when it is multiplied to a binary number, the
binary will retain its value
POSTULATE IV A
- Reversing the order of two binary numbers in performing addition is
shown as not to effect the resulting sum
POSTULATE IV B
- Reversing the order of two binary numbers in performing multiplication is
shown as not to effect the resulting product
POSTULATE VA
POSTULATE VB
POSTULATE VI
- defines the complement (A’) of any variable A such that when A’ is
multiplied to A, the resulting product is zero
POSTULATE VII
- defines at least two elements in K that are not equal
- assign the following expressions to variable X and Y:
BOOLEAN ALGEBRA vs. ALGEBRA
BOOLEAN ALGEBRA:
ALGEBRA:
1. Valid only for a set having two
elements (0 and 1)
1. Ordinary algebra is
applicable to a set of having
infinite number of elements
2. Although this property could
be derived or proven to hold
true the Huntington’s
postulates do not include
Associative Property
BOOLEAN ALGEBRA vs. ALGEBRA
BOOLEAN ALGEBRA:
ALGEBRA:
3. The distributive property of +
over · is valid since its set have
only two elements (0 and 1)
Not applicable
4. Additive and Multiplicative
Inverses are not defined
5. Subtraction and Division
operations are also undefined.
BOOLEAN ALGEBRA vs. ALGEBRA
BOOLEAN ALGEBRA:
ALGEBRA:
6. The definition of a
complement is defined
Not applicable
7. Especially defined to have
the property of duality
THEOREMS
Theorems 3 is sometimes called INVOLUTION and Theorems
6a&b are forms of ABSORPTION.
THEOREMS
Example
Given the Boolean Functions,
let us implement the
corresponding truth table
associated with Functions 1 to
5.
Solution:
F0 = 0
F0 is a constant equal to zero
F1 = XY
Example
F2 = XY’
Example
F3 = X
Example
F4 = X’Y
THEOREMS
THEOREMS
THEOREMS
THEOREMS
THEOREMS
THEOREMS
THEOREMS
THEOREMS
THEOREMS
THEOREMS
THEOREMS
EXAMPLES
Using the Postulates and Theorems presented previously, prove the
following Theorems and write down their respective truth tables: