Converting between Bases
Look at the place value charts and information below for base 10, base 2, base 5, base 8, base 12, and base 16. In
particular, find what is the same about the bases and what is different about them.
10
Ones
10
Tens
4
Hundreds
Ten
Thousands
5
Thousands
Hundred
Thousands
Base 10, a.k.a. decimal and Hindu-Arabic:
3
2
1
0
10
10
10
10
Number of digits: 10
Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Thirty-Two’s
Sixteen’s
Eight’s
Four’s
Two’s
Ones
Base 2, a.k.a. binary:
5
4
3
2
1
0
2
2
2
2
2
2
Number of digits: 2
Digits: 0, 1
3125’s
625’s
125’s
Twenty-Five’s
Fives
Ones
Base 5:
5
4
3
2
1
0
5
5
5
5
5
5
Number of digits: 5
Digits: 0, 1, 2, 3, 4
32,768’s
4096’s
512’s
Sixty-Four’s
Eights
Ones
Base 8, a.k.a. octal:
5
4
3
2
1
0
8
8
8
8
8
8
Number of digits: 8
Digits: 0, 1, 2, 3, 4, 5, 6, 7
248,832’s
20,736’s
1728’s
144’s or
Grosses
Twelves or
Dozens
Ones
Base 12, a.k.a. duodecimal (not be confused with the Dewey
Decimal system found in most public and school libraries. ☺ )
125
124
123
122
121
120
Number of digits: 12
Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T, E
(“T” represents the value “ten” and “E” represents the value
“eleven”. Each place in the chart contains only one digit. In
base 12, we cannot use “10” for the value “ten” because
10twelve equals 12 in base 10.)
Base 16, a.k.a. hexadecimal
1,048,576’s
65,536’s
4096’s
256’s
Sixteen’s
Ones
165
164
163
162
161
160
Number of digits: 16
Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
(A = 10, B = 11, C = 12, D = 13, E = 14, F = 15)
List what is the same and what is different among these bases.
Same:
Different:
These place values charts will be helpful for filling in the chart that is the second part of this handout.
Complete the following table by writing the base 10 numbers given in each of the indicated bases. Remember that
when you write a number in another base that you write the base as a subscript at the end of the number. For
example, the base 10 number 11 is written 1011two in base 2, 21five in base 5, 13eight in base 8, Etwelve in base 12, and
Bsixteen in base 16.
B
Base 10
1
2
3
4
5
6
7
8
9
10
13
15
19
23
28
30
33
35
40
Base 2
Base 5
Base 8
Base 12
Base 15
MTH – 151 Computer Logic and Chapter 4 Review
1.
The computer logic part of this test contains problems like those in the exercises and in Quiz 9: binary
addition, evaluation of logic gates, half-adders, full-adders, and parallel-adders.
2.
For each of the following historical numeration systems:
a) What is the base?
b) What type of system is it? (Simple grouping, Multiplicative grouping, Positional)
c) Is there a symbol for zero?
Ancient Egyptian
Traditional Chinese
Hindu-Arabic
2. Write the following number in the Traditional Chinese System.
3,874
3. Write the following number in our system.
4. Subtract the following numbers in expanded form. Show your work.
7103 – 2367
5. Multiply the following numbers using the Egyptian Algorithm. Show your work.
19 × 53
6. Write the following base 10 numbers in the base indicated:
a) 2376 in base eight.
b) 37,762 in base sixteen.
c) 579 in base two.
7. Write the following numbers in base 10.
a) 21305six
b) 2TE3twelve
c) A9DCsixteen
8. Convert each number in the given base to the base indicated.
a) 1101001101101binary to octal
b) 4032seven to base twelve
Key for “Converting between Bases”
Same: (These can be in any order.)
The name of the base is the number used as the base in the second row of the place value chart. For
example, base 2 uses “2” as the base in its place value chart.
The exponents in the second row are the same.
The name of the base equals the number of digits.
All of the bases have a “Ones” place.
All of the bases have a “0” digit.
The next place left of the “Ones” place equals the number of the base. For example, base 10 has a “Tens”
place and base 8 has an “Eights” place.
You may have come up with other correct answers.
Different: (These can be in any order.)
The places, other than the “Ones” place, have different values. For example, for base 5, the places are
“Ones”, “Fives”, “Twenty-Five’s”, “125’s”, etc., but for base 12, the places are “Ones”, “Twelves”,
“144’s”, etc.
Some of the bases have letters for digits.
The number of digits is different for each base.
The number used as the base in the second row of the place value chart is different for each base.
You may have come up with other correct answers.
Base 10
1
2
3
4
5
6
7
8
9
10
13
15
19
23
28
30
33
35
40
Base 2
1two
10two
11two
100two
101two
110two
111two
1000two
1001two
1010two
1101two
1111two
10011two
10111two
11100two
11110two
100001two
100011two
101000two
Base 5
1five
2five
3five
4five
10five
11five
12five
13five
14five
20five
23five
30five
34five
43five
103five
110five
113five
120five
130five
Base 8
1eight
2eight
3eight
4eight
5eight
6eight
7eight
10eight
11eight
12eight
15eight
17eight
23eight
27eight
34eight
36eight
41eight
43eight
50eight
Base 12
1twelve
2twelve
3twelve
4twelve
5twelve
6twelve
7twelve
8twelve
9twelve
Ttwelve
11twelve
13twelve
17twelve
1Etwelve
24twelve
26twelve
29twelve
2Etwelve
34twelve
Base 16
1sixteen
2sixteen
3sixteen
4sixteen
5sixteen
6sixteen
7sixteen
8sixteen
9sixteen
Asixteen
Dsixteen
Fsixteen
13sixteen
17sixteen
1Csixteen
1Esixteen
21sixteen
23sixteen
28sixteen
Chapter 4 Review KEY