Basic operation on number base
Rules of Binary Addition
• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 0, and carry 1 to the next more significant figure
For instance,
00011010 + 00001100 = 00100110
0 0 1 1 carries
0 0 0 1 1 0 1 0 = 26(base 10)
+ 0 0 0 0 1 1 0 0
________________________________________ = 12(base 10)
0 0 1 0 0 1 1 0 = 38(base 10)
00010011 + 00111110 = 01010001
0 1 1 1 1 1 Carries
0 0 0 1 0 0 1 1 = 19(base 10)
+ 0 0 1 1 1 1 1 0
________________________________________ = 62(base 10)
0 1 0 1 0 0 0 1 = 81(base 10)
Note: The rules of binary addition when you remove the “carries” following it are the same as the truths of the XOR gate.
Rules of Binary Subtraction
• 0 - 0 = 0
• 0 - 1 = 1, and borrow 1 from the next more significant figure
• 1 - 0 = 1
• 1 - 1 = 0
For instance,
00100101 - 00010001 = 00010100
0 0 0 borrows
0 0 1 10 0 1 0 1 = 37(base 10)
- 0 0 0 1 0 0 0 1
________________________________________ = 17(base 10)
0 0 0 1 0 1 0 0 = 20(base 10)
00110011 - 00010110 = 00011101
0 10 1 borrows
0 0 1 1 0 10 1 1 = 51(base 10)
- 0 0 0 1 0 1 1 0
________________________________________ = 22(base 10)
0 0 0 1 1 1 0 1 = 29(base 10)
Rules of Binary Multiplication
• 0 x 0 = 0
• 0 x 1 = 0
• 1 x 0 = 0
• 1 x 1 = 1, with no carry or borrow parts
For instance,
00101001 × 00000110 = 11110110
0 0 1 0 1 0 0 1 = 41(base 10)
× 0 0 0 0 0 1 1 0
________________________________________ = 6(base 10)
0 0 0 0 0 0 0 0
0 0 1 0 1 0 0 1
0 0 1 0 1 0 0 1
________________________________________
0 0 1 1 1 1 0 1 1 0 = 246(base 10)
00010111 × 00000011 =01000101
0 0 0 1 0 1 1 1 = 23(base 10)
× 0 0 0 0 0 0 1 1
________________________________________ = 3(base 10)
1 1 1 1 1 carries
0 0 0 1 0 1 1 1
0 0 0 1 0 1 1 1
________________________________________
0 0 1 0 0 0 1 0 1 = 69(base 10)
Note: The rules of binary multiplication are equivalent to the truths of the AND gate.
Another Method:
Binary multiplication is equivalent to repeated binary addition; add the multicand to itself the multiplier number of times.
For instance,
00001000 × 00000011 = 00011000
1 carries
0 0 0 0 1 0 0 0 = 8(base 10)
0 0 0 0 1 0 0 0 = 8(base 10)
+ 0 0 0 0 1 0 0 0
________________________________________ = 8(base 10)
0 0 0 1 1 0 0 0 = 24(base 10)
Binary Division
Binary division is the repeated process of subtraction, just as in decimal division.
For example,
00101010 ÷ 00000110 = 00000111
1 1 1 = 7(base 10)
________________________________________
1 1 0 ) 0 0 1 10 1 0 1 0 = 42(base 10)
- 1 1 0 = 6(base 10)
________________________________________
1 borrows
1 0 10 1
- 1 1 0
________________________________________
1 1 0
- 1 1 0
________________________________________
0
10000111 ÷ 00000101 = 00011011
1 1 0 1 1 = 27(base 10)
________________________________________ 1 0 1 ) 1 0 0 10 0 1 1 1 = 135(base 10)
- 1 0 1 = 5(base 10)
________________________________________
1 1 10
- 1 0 1
________________________________________
1 1
- 0
________________________________________
1 1 1
- 1 0 1
________________________________________
1 0 1
- 1 0 1
________________________________________
0
Explanatory Notes
Binary Number System
System Digits: 0 and 1
Bit (short for binary digit): A single binary digit
LSB (least significant bit): The rightmost bit
MSB (most significant bit): The leftmost bit
Upper Byte (or nybble): The right-hand byte (or nybble) of a pair
Lower Byte (or nybble): The left-hand byte (or nybble) of a pair
In every base N, there are N digits. In the decimal system, for instance, we have 10 of them: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. In base 7, there are seven digits: 0, 1, 2, 3, 4, 5, 6. When N exceeds 10 we begin adding English letters as required. There is no distinction made between capital and lower case letters. Base 36 makes use of all decimal digits and all the letters of the English alphabet.
You need practice to get accustomed with different bases; for the rules are the same as in the decimal system. The sum or product of two digits may only produce one or two digit numbers. In the 2 digit numbers case, if required, the first digit is carried over to the next operation on the left. For instance, in base 7, 36 + 144 = 213. Indeed, from right to left, 6 + 4 = 13. Then 3 + 4 + 1 = 11, and finally 1 + 1 = 2.
Also, 144 × 36 = 6243. Numbers equal in one base are equal in any other base. Conversion between bases does not go against arithmetic identities. In base 2, 2 + 2 = 4 appears as 10 + 10 = 100 - although this appears different but they have exactly the same meaning.
The same thing applies to 2 × 2 = 4 which is true in all bases beginning with 5. In bases 4, 3, and 2 it looks like:
2 × 2 = 10
2 × 2 = 11
10 × 10 = 100, in that order
A few more things you need to know about and check out are as follows:
1. For every N>1, (N)10 = (10)N.
2. For addition, in the lower right corner, there is always a 2-digit number present whose first digit is 1 while the second digit is the last but one digit of the system.
3. All tables are symmetric with regard to the diagonal from the upper left to the lower right corner.
4. Successive numbers on any north-west to south-east diagonal, in every addition tables, vary by 2.
5. For multiplication, the number in the lower right angle is constantly gotten from its "addition" equivalent by substituting the two digits.
6. In the last row of multiplication tables, last digits increase by 1 if you go from the right to the left. At the same time, the first digits are less by 1.
7. The numbers in the last line before the last are as well related to each other: if for addition we have a number 1a, then for multiplication its equivalent will be a2.
8. One can make use of addition tables to play the same game as with the Calendar tables.
9. For multiplication tables this is as well true so long as the selected entries are multiplied instead of being added up.
10. As a specific case, in multiplication tables determinants of any 2 x 2 square are 0.
Addition & Subtraction
Let's consider a decimal addition.
As an example, we have 26 plus 36,
26
+36
To add these two numbers, we first take into consideration the "ones" column and calculate 6 plus 6, which gives us 12. Since 12 is greater than 9 (bearing in mind that base 10 is made up of digits 0-9), we "carry" the 1 from the "ones" column to the "tens column" and leave the 2 in the "ones" column.
Working out the "tens" column, we calculate 1 + (2 + 3), which results in 6. Since 6 is less than 9, there is nothing to "carry" and we leave 6 in the "tens" column.
26
+36
62
Binary addition
The binary addition is being carried out in a similar manner , except that just 0's and 1's can be used, rather than the whole spectrum of 0-9. This in reality makes binary addition much simpler than decimal addition, as we are just required to remember the following:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10
An example of binary addition is shown below:
101
+101
a) To add these two numbers, we first take into consideration the "ones" column and calculate 1 + 1, which (in binary) gives 10. We "carry over" the 1 to the "tens" column, and the leave the 0 in the "ones" column.
b) Going over to the "tens" column, we calculate 1 + (0 + 0), which results in 1. Nothing is "carried over" to the "hundreds" column, and we leave the 1 in the "tens" column.
c) Going over to the "hundreds" column, we calculate 1 + 1, which results to 10. We "carry over" the 1 to the "thousands" column and leave the 0 in the "hundreds" column.
101
+101
1010
Another example of binary addition is provided below:
1011
+1011
10110
Observe that in the "tens" column, we have 1 + (1 + 1), where the first 1 is "carried" from the "ones" column. Remember that in binary,
1 + 1 + 1 = 10 + 1 = 11
Binary subtraction
Binary subtraction is also simplified in so far as we bear in mind the way subtraction and the base 2 number system is being done. Let's first have look at a simple example.
111
- 10
101
Observe that the variation is the same if this was decimal subtraction. Again, similar to decimal subtraction is the concept of "borrowing." Observe that "borrowing" takes place when a bigger digit, like 8, is subtracted from a smaller digit, like 5, as illustrated below in decimal subtraction.
35
- 8
27
For 10 minus 1, 1 is borrowed from the "tens" column for utilization in the "ones" column, abandoning the "tens" column with only 2. The following examples illustrate "borrowing" in binary subtraction.
10 100 1010
- 1 - 10 - 110
1 10 100
A number written in scientific notation is written as the product of a number between 1 and 10 and a number that is a power of 10 . This means that it is written as a quantity whose coefficient is between 1 and 10 and whose base is 10.
Addition and Subtraction
One of the characteristics of quantities with exponents is that numbers with exponents can be added and subtracted only when they belong to the same base and have the same exponent. Since all numbers in scientific notation possess the same base (10), we are required to only worry about the exponents. To be added or subtracted, two numbers in scientific notation must be influenced to get their bases to carry the same exponent. This will make certain that equivalent digits in their coefficients possess the same place value.
Multiplying a number by another number with the same base is the same as multiplying their coefficients and adding their exponents. Thus, if we want to add two quantities written in scientific notation whose exponents do not match, we can merely write one of the powers of 10 as the product of two smaller powers of 10 , one which is in agreement with the other term.
Alternately, if we want to preserve the exponent of the term with the larger power of 10 , we can simultaneously multiply and divide the other term by a power of 10 , applying the rule for multiplication of exponents in one case and dividing the coefficient in the other. This procedure has been outlined below:
Once the numbers are made up of the same base and exponents, we can add or subtract their coefficients.
The steps to adding or subtracting numbers in scientific notation are as follows:
• Find out the number by which to increase the smaller exponent by to make it equal to the larger exponent.
• Increase the smaller exponent by this number and shift the decimal point of the number with the smaller exponent to the left the same number of places. Divide by the suitable power of 10.
• Add or subtract the fresh coefficients.
• If the answer is not in scientific notation -if the coefficient is not between 1 and 10, convert it to scientific notation.
The answer ought to be made up of coefficient, base, and exponent.
Note: If the numbers begin with equivalent exponents, their coefficients can be added, but be cautious, the answer might be required to be converted to scientific notation.
Example 1: 2.456×105 +6.0034×108 = ?
• 8 - 5 = 3. The smaller exponent ought to be raised by 3.
• 2.456×105 = 0.002456×108
• 0.002456×108 +6.0034×108 = 6.005856×108
• 6.005856×108 is in scientific notation.
Therefore, 2.456×105 +6.0034×108 = 6.005856×108
Example 2:
5.10802×103 -6.1×10-2 = ?
• 3 - (-2) = 5. The smaller exponent must be increased by 5.
• 6.1×10-2 = 0.000061×103
• 5.10802×103 -0.000061×103 = 5.107959×103
• 5.107959×103 is in scientific notation.
Therefore, 5.10802×103 -6.1×10-2 = 5.107959×103 .
Example 3:
3.5×1018 + 5.3×1012 = ?
• 18 - 12 = 6. The smaller exponent ought to be increased by 6.
• 5.3×1012 = 0.0000053×1018
• 3.5×1018 + 0.0000053×1018 = 3.5000053×1018
• 3.5000053×1018 is in scientific notation.
Therefore, 3.5×1018 + 5.3×1012 = 3.5000053×1018 .
Example 4:
4.801×103 -2.2×107 = ?
• 7 - 3 = 4. The smaller exponent ought to be increased by 4.
• 4.801×103 = 0.0004801×107
• 0.0004801×107 -2.2×107 = - 2.1995199×107
• -2.1995199×107 is in scientific notation.
Therefore, 4.801×103 -2.2×107 = - 2.1995199×107 .
Example 5.
1.4×10-5 -5.67×10-6 = ?
• -5 - (-6) = 1. The smaller exponent ought to be raised by 1.
• 5.67×10-6 = 0.567×10-5
• 1.4×10-5 -0.567×10-5 = 0.833×10-5
• 0.833×10-5 = 8.33×10-6 in scientific notation.
Thus, 1.4×10-5 -5.67×10-6 = 8.33×10-6 .
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