Name: ____________
Computer Number Systems Packet
Number Systems are used to express numbers using digits or symbols. Most number
systems are fairly straight forward. But there are a couple, like the Roman Numerals
that can a bit more complex. Understanding how they work and how they relate to each
other is important for computers. This is because computers work using the binary
number system since a zero and a one can be easily represented using ground and
positive voltage. There are others like octal and hexadecimal that are also used
frequently.
The key aspect of the number systems that we will look at is the base. The base
indicates the number of digits for each place. The one that we use is base 10 (decimal).
The 10 digits that we use in this are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Why do we use this
over something else like base 8?..... It is for no other reason than the fact that we are
born with 10 fingers and that is what all humans learn to count with.
Names
Most number systems can be referred to using their base like base 5, base 7, and base 9.
Some are so common that they are referred to using their actual name. They are:
base 2 – binary
base 8 – octal
base 10 – decimal
base 16 – hexadecimal
When we write out the numbers, we can indicate what base they are in by using a
subscript. (Example – 359010)
Counting
Counting in the systems that we will look at will all function the same way for each one.
You already learned counting in the decimal number system in elementary school. Lets
review!
We will start counting with our first digit which is 0. Then we run through all our other
digits from 1 all the way to 9. Then when we run out of digits, we change the 9 back to
our first digit 0. But at the same time we add a 1 to the next position. As we progress to
9 each position, it will revert to 0 and a 1 is added to the next position. So a 9,999 will
transition to 10,000 as each 9 goes to 0 and 1 gets added to the next position.
The other thing to keep in mind is that the last digit in each set is one less than the base.
So octal, which is base 8, has digits 0, 1, 2, 3, 4, 5, 6, 7. Notice that, since we start
counting at 0, that gives us a total of 8 digits.
Hexadecimal
Hexadecimal has 16 digits but we only are familiar with 10. So we use the letters A
through F to replace the numbers 10 through 15. Why? For no other reason other than
we are familiar with those symbols and they are already on a standard keyboard.
Page 1
Counting Chart
Decimal
Octal
Binary
Hexadecimal
Base 5
Base 6
Base 9
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
Page 2
Adding Numbers
Adding again works that same way that it did in elementary school. We add the
numbers in each position and if the sum goes over the base, we subtract the base and
carry a 1 to the next position. When we add 48 with 78, we first add the 8 with 8 and
get a 16. The 6 (16-10) stays at the ones place and we add a 1 to the tens place. For the
tens place we add 1 + 4 + 7 and get a 12. We leave the 2 (12-10) and carry a 1. So the
result will be 126.
Examples
111 1
11101102
+ 1101002
10101010
1 11
517418
+326548
104615
1 1
CA97116
+E295416
1AD2C5
1st : 0 + 0 = 0
2nd : 1 + 0 = 1
3rd : 1 + 1 = 2 → 2 - 2 = 0, carry 1
4th : 1 + 0 + 0 = 1
5th : 1 + 1 = 2 → 2 - 2 = 0, carry 1
6th : 1 + 1 + 1 = 3 → 3 - 2 = 1, carry 1
7th : 1 + 1 = 2 → 2 - 2 = 0, carry 1
8th : 1
1st : 1 + 4 = 5
2nd : 4 + 5 = 9 → 9 - 8 = 1, carry 1
3rd : 1 + 7 + 6 = 14 → 14 - 8 = 6, carry 1
4th : 1 + 1 + 2 = 4
5th : 5 + 3 = 8 → 8 - 8 = 0, carry 1
6th : 1
1st : 1 + 4 = 5
2nd : 7 + 5 = 12(C)
3rd : 9 + 9 = 18 → 18 - 16 = 2, carry 1
4th : 1 + 10(A) + 2 = 13(D)
5th : 12(C) + 14(E) = 26
→ 26 - 16 = 10(A), carry 1
6th : 1
Adding Binary Numbers
102
+ 12
1012
+ 12
1012
+ 1102
10102
+ 1112
10112
+ 111012
101112
+ 10012
1010112
+ 1110102
101110102
+ 111111012
1101102
+ 1010112
1110112
+ 1001112
10101112
+ 10101102
111111112
+ 111111112
Page 3
Adding Numbers in Different Bases
7528
+5538
1023
+1113
4025
+3115
1678
+4178
2123
+1203
2045
+3415
75216
+55316
65B16
+AA316
E3A16
+85B16
4559
+5839
3234
+1114
1536
+5436
1057
+6157
181010
+921510
7248
+1578
Page 4
Converting Numbers to Decimal Equivalents
A number in base b can be converted to its equivalent in base 10 by using place values.
Place Value Formula
b8 b 7 b 6 b5 b 4 b3b 2 b1b 0 .b 1b 2 b 3b 4
Given 1394210, what do the individual digits represent?
13942 10 = 1(10)4 + 3(10)3 + 9(10)2 + 4(10)1 + 2(10)0
= 10,000 + 3000 + 900 + 40
+2
Likewise, the digits in the base 2 number 110101 represent powers of the base 2.
110101 2 = 1(2)5 + 1(2)4 + 0(2)3 + 1(2)2 + 0(2)1 + 1(2)0
= 1(32) + 1(16) + 0(8) + 1(4) + 0(2) + 1(1)
= 32 + 16 + 0 + 4
+0
+1
= 5310
The same can be done for any other base such as converting 13F16 to base ten.
13F16 = 1(16)2 + 3(16)1 + F(16)0
= 1(256) + 3(16) + 15(1)
= 256 + 48
+ 15
= 319 10
Or to converting 3758 to base ten.
3758 = 3(8)2 + 7(8)1 + 5(8)0
= 3(64) + 7(8) + 5(1)
= 192 + 56
+5
= 253 10
Page 5
Converting Numbers to Decimal Equivalents Practice
1012 =
10102 =
1011012 =
1101010.112 =
2716 =
13AD16 =
34178 =
110213 =
Page 6
Converting from Binary to Hexadecimal
Converting back and forth from binary to hexadecimal is straight forward. This is because 4 binary digits
matches exactly with one hexadecimal digit. To covert from decimal to hexadecimal, just look up each
hexadecimal digit and write out the equivalent set four binary digits. To go the other way, circle groups of four
binary digits and write out the equivalent hexadecimal digit. This should be done from right to left and fill in
any leading zero’s
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
Example 1:
8
1000
Example 2:
0100
4
8D316 =
?
2
D
1101
100010111002 =
3
0111
?
0101
5
16
1100
C
Converting from Binary to Hexadecimal Practice
1011011011101100010110112
1000111010111101011101100100101001012
A7F316
D239AE16
Page 7
Converting from Decimal to other Bases
One way to do this is to repeatedly divide the decimal number by the base in which it is
to be converted, until the quotient becomes zero. As the number is divided, the
remainders - in reverse order - form the digits of the number in the other base. This is
also referred to as the divide and conquer algorithm.
Examples
173410 → 110110001102
0 _ R _1
2 1 _ R _1
2 3_R_0
2 6 _ R _1
2 13 _ R _1
173410 → 6C616
0_ R_6
16 6 _ R _12
16 108 _ R _ 6
16 1734
173410 → 33068
0_ R_3
8 3_ R_3
8 27 _ R _ 0
8 216 _ R _ 6
8 1734
2 27 _ R _ 0
2 54 _ R _ 0
2 108 _ R _ 0
2 216 _ R _1
2 433 _ R _1
2 867 _ R _ 0
2 1734
Some things to note:
 The remainders go in reverse order. So the remainder left at the end is the most
significant digit. And the first remainder that you obtained is the least significant
digit.
 When we got a remainder of 12 for hexadecimal, we put a C when we wrote out
our hexadecimal number.
Page 8
Converting from Decimal to BaseX Practice
4710 =___________2
27410 =___________2
771210 = ___________2
4542510 =___________2
Page 9
4710 =___________8
27410 =___________8
771210 = ___________16
4542510 =___________16
771210 =___________16
4542510 =___________16
Page 10
Page 11