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Number system |
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Decimal number system |
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Binary number system |
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Hexadecimal number system |
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Octal number system (not required for this
course) |
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How to convert from one number system to
another? |
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Convert decimal number to binary number |
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Convert binary number to decimal number |
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Convert hexadecimal number to binary number |
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Convert binary number to hexadecimal number |
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The decimal number system is the most popular
number system and it is the one people are most familiar with. |
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Its origin is probably related to the fact that
human beings have 10 fingers. |
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It uses the 10 symbols 0, 1, 2, 3, 4, 5, 6, 7,
8, 9 and the base is 10. |
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The lowest digit is 0 and the highest digit is 9
– one less than the base of 10. |
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A decimal number can thus be expressed as: |
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dn * 10 n + dn-1
* 10 n-1 + …+ d1 * 101 + d0 *
10 0 |
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where, dj is one of the ten decimal digits 0 ~ 9 |
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10i is the power
of 10 of each digit dj that defines its positional significance,
or simply the power of that digit. |
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For example, |
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( 845 )10 = 8 * 102 + 4 * 101 + 5 * 100 |
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= 8 * 100 + 4 * 10 + 5 * 1 |
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We will use ( 845 )10 to represent a
decimal number. |
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In the binary number system the base is 2. |
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There are only two symbols, 0 and 1, called
binary digits or bits in short. |
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Its lowest digit is 0 and its highest digit is 1
which is one less than the base of 2. |
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A binary number can thus be expressed as: |
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dn * 2 n + dn-1
* 2 n-1 + …+ d1 * 21 + d0 * 2 0 |
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where, dj is one of the two binary digits 0 ~ 1 |
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2i is the power of 2 of each digit dj that
defines its positional significance, or simply the power of that digit. |
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We will use (110011)2 to represent a
binary number. |
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Least significant bit (LSB, the right-most bit) |
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Most significant bit (MSB, the left-most bit) |
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The hexadecimal number system has base 16 and
therefore it requires 16 symbols – a lowest digit of 0 and a highest digit
with a value equivalent to decimal 15 (one less than the base of 16). |
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By convention, we use letter A through F to
represent the hexadecimal digits corresponding to decimal values 10 through
15. |
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In other words, hexadecimal number system uses
the 16 symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. |
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In hexadecimal number system, we can have : |
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numbers like 876 consisting solely of
decimal-like digits, |
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numbers like 8A55F consisting of digits and
letters, |
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and numbers like FFE consisting solely of
letters. |
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Occasionally, a hexadecimal number spells a
common word such as FACE or FEED which can appear strange to programmers. |
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A hexadecimal number can thus be expressed as: |
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dn * 16 n + dn-1
* 16 n-1 + …+ d1 * 161 + d0 *
16 0 |
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where, dj is one of the 16
hexadecimal digits 0 ~ 9 and A ~ F |
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16i is the power of 16 of
each digit dj that defines its positional significance, or
simply the power of that digit. |
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For example, |
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( 3DA )16 = 3 * 162 + 13 * 161 + 10 * 160 |
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= 3 *
256 + 208 + 10 |
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= 768 + 208
+10 |
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= 986 |
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We will use ( 3DA )16 to represent a
hexadecimal number. |
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Divide the decimal number by 2. |
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The remainder is the binary bit, beginning with
the least significant bit(LSB). |
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Continue dividing by 2 until you are left with
0, and a remainder of either 1 or 0. |
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The Last remainder is the most significant
bit(MSB) of the binary number. |
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(123)10 = ( ? )2 |
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Show ˝ Show the remainder |
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123 1 (LSB) |
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61 1 |
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30 0 |
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15 1 |
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7 1 |
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3 1 |
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1 1 |
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0 0 (MSB) |
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So, (123)10 = ( 0111 1011 )2 |
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Binary number Hexadecimal number |
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0000 0 |
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0001 1 |
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0010 2 |
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0011 3 |
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0100 4 |
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0101 5 |
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0110 6 |
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0111 7 |
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1000 8 |
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1001 9 |
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1010 A |
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1011 B |
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1100 C |
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1101 D |
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1110 E |
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1111 F |
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Group bits into sets of 4 |
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Add leading 0 if needed |
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Check the table in slide no.13 |
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For example, |
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( 111 1011 )2 = ( 0111 1011 )2 |
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= ( 7
B )16 |
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Check the table in slide no.13 |
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May remove leading 0 |
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For example, |
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(
7 B )16 = ( 0111
1011 )2 |
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= (
111 1011 )2 |
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Notes