Chapter iii: Number System
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Transcript of Chapter iii: Number System
A set of values used to represent different quantities.
Examples:
Number of students in a Class or number of viewers watching a certain TV program.
It includes audio, graphics, video, text, and numbers.
Base or Radix are the total number of digits used in a number system.
Some important number systems are as follows:
DECIMAL number system
BINARY number system
OCTAL number system
HEXADECIMAL number system
The decimal number system is used in general. However, the computers use binary number , octal, and hexadecimal number systems.
BINARY NUMBER SYSTEM
DECIMAL NUMBER SYSTEM
It is the most widely used number system.
It consists of ten numbers from 0 to 9.
It’s base is 10.
Examples:
1. 145010
2. 24210
3. 1000002410
OCTAL NUMBER SYSTEM
It is the shorthand representation of binary numbers.
Any digit in this system is always less than 8.
It consists of eight digits from 0 to 7.
It’s base is 8.
Examples:
1. 56568
2. 1248
3. 3788
HEXADECIMAL NUMBER SYSTEM
It consists of 16 digits from 0 to 9 and A to F.
The alphabets A to F represent decimal numbers 10 to 15.
It’s base is 16.
Examples:
1. 29716
2. BA5916
3. BACA16
CONVERSION FOR HEXADECIMALDECIMAL HEXADECIMAL
1 0001
2 0010
3 0011
4 0100
5 0101
6 0110
7 0111
8 1000
9 1001
10-A 1001
11-B 1011
12-C 1100
13-D 1101
14-E 1110
15-F 1111
STEP BINARY NUMBER DECIMAL NUMBER
Step 1 111012 ((1x24)+(1x23)+(1x22)+(0x21)+(1x20)) 10
Step 2 111012 (16+8+4+0+1) 10
Step 3 111012 2910
1.Determine the column (positional) value of each digit (this depends on the position of the digit and the base of the number system.)2.Multiply the obtained column values (in step 1) by the digits in the corresponding columns.3.Sum the products calculated in step 2. The total is the equivalent value in decimal.
BINARY TO OCTALSTEPS:
1.Divide the binary digits into groups of three (starting left to right).
2.Convert each group of three binary digits to one octal digit.
STEP BINARY NUMBER OCTAL NUMBER
Step 1 101012 010 101
Step 2 101012 28 58
Step 3 101012 258
Add a (0) Zero digit to complete
the 3 digits group.
BINARY TO HEXADECIMALSTEPS:
1.Divide the binary digits into groups of four (starting from the right).
2.Convert each group of four binary digits to one hexadecimal symbol.
STEP BINARY NUMBERH10101EXADECIMAL
NUMBER
Step 1 101012 0001 0101
Step 2 101012 110 510
Step 3 101012 1516
Add two(2) Zero(0) digits to complete the four
(4) digits group.
DECIMAL TO BINARY
STEP OPERATION RESULT REMAINDER
Step 1 29/2 14 1
Step 2 14/2 7 0
Step 3 7/2 3 1
Step 4 3/2 1 1
Step 5 1/2 0 1
1.Divide the decimal number to be converted by the value of the new base.
2.Get the remainder from step 1 as the rightmost digit (least significant digit) of new base number.
3.Divide the quotient of the previous divide by the new base.
4.Record the remainder from step 3 as the next digit (to the left) of the new base number.
DECIMAL TO OCTAL
DIVISION RESULT REMAINDER
250/8 31 2
31/8 3 7
3/8 0 3
Steps:1.Divide decimal number by 8. Treat the division as an integer division.2.Write down the remainder (in octal). To get the remainder, multiply the result by 8 and subtract it to the decimal number/result.3.Repeat step 1-3 until the result is zero.4.The octal value is the digit sequence of the remainders from the lastto first.
25010 = 3728
DECIMAL TO HEXADECIMALSteps:
1.Divide decimal number by 16. Treat the division as an integer division.
2.Write down the remainder (in hexadecimal).
3.Repeat step 1-3 until the result is zero.
4.The hex value is the digit sequence of the remainders from the last to first.
DIVISION RESULT REMAINDER (in HEX)
256/16 16 0
16/16 1 0
1/16 0 1
25610 =10016
OCTAL TO BINARYSteps:
1.Convert each octal digit to a 3 digit binary number (the octal digits may be treated as decimal for this conversion).
2.Combine all the resulting binary groups (of 3 digits each) into a single binary number.
STEP OCTAL NUMBER BINARY NUMBER
Step 1 258 210 510
Step 2 258 0102 1012
Step 3 258 0101012
258 = 0101012
OCTAL TO DECIMALSteps:
1.Start the decimal result at 0.
2.Remove the most significant octal digit (leftmost) and add it to the result.
3.If all octal digits have been removed, you’re done. Stop.
4.Otherwise, multiply the result by 8.
5.Go to step 2.
Octal Digits OperationDecimal Result
OperationDecimal Result
345 +3 3 x8 24
45 +4 28 x8 224
5 +5 229 done
3458= (3*82)+(4*81)+(5*80) = (3*64)+(4*8)+(5*1) = 22910
HEXADECIMAL TO BINARYSteps:
1.Convert each hexadecimal digit to a 4 digit binary number (the hexadecimal digits may be treated as decimal for this conversion).
2.Combine all the resulting binary groups (of 4 digits each) into a single binary number.
STEPHEXADECIMAL
NUMBERBINARY NUMBER
Step 1 15 12 52
Step 2 15 00012 01012
Step 3 15 000101012
1516 = 000101012
HEXADECIMAL TO DECIMALSteps:
1.Get the last digit of the hex number, call this digit the Current Digit.
2.Make a variable, let’s call it power. Set the value to Zero.
3.Multiply the current digit with (16^power). Store the result.
4.Increment power by one.
5.Set the current digit to the previous digit of the Hex Number.
6.Repeat from step 3 until all digits have been multiplied.
7.Sum the result of step 3 to get the answer Number.
MULTIPLICATION RESULT
9x(16^0) 9
8x(16^1) 128
5x(16^2) 1280
Answer 1417
58916 = 141710
CRISTINA FABROS
MICAH HADASSAH GUILLERMO
DANILO PALTENG
JANUEL ANTONIO
CHERRY MARIE GALAUSROSEANN FORONDACRISTINA FABROS MICAH HADASSAH GUILLERMODANILO PALTENGJANUEL ANTONIO
WORKING COMMITTEE
BS ACCOUNTANCY 1-2