NAME:__________________________________________________ CLASS:____________________
NUMDER BASES
DID YOU KNOW? Computers work using different number systems from the
one based on powers of 10 that we use in daily life. One of these is the
hexadecimal system. Find out what you can about hexadecimal numbers and why
they are important.
Denary, also known as "decimal" or "base 10," is the standard number system used
around the world. It uses ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) to represent all
numbers. Denary is often contrasted with binary, the standard number system used
by computers and other electronic devices. Binary System uses two digits, which
are 0s and1s.
NAME:__________________________________________________ CLASS:____________________
PLACE VALUE CHARTS
Here are some of the bases place value charts below, showing the place value for
the first 6 digits for the bases.
Base Two - Binary
NAME:__________________________________________________ CLASS:____________________
25
32’s
24
16’s
23
8’s
22
4’s
21
2’s
20
1’s
33
27’s
32
9’s
31
3’s
30
1’s
43
64’s
42
16’s
41
4’s
40
1’s
53
125’s
52
25’s
51
5’s
50
1’s
Base Three - Ternary
35
243’s
34
81’s
Base Four - Quarternary
45
1024’s
44
256’s
Base Five - Quinary
55
3125’s
54
625’s
Try Creating place value charts for Base 6, 7 and 8 for yourselves.
Denary system (Base Ten)
Place Value
105
100,000’s
104
10,000’s
103
1,000’s
102
100’s
101
10’s
100
1’s
We have ten fingers. This is probably why we started to count in tens and
developed a system based on ten for recording large numbers.
For example
3125 = 3 thousands + 1 hundred + 2 tens + 5 units
= (3 × 103 ) + (1 × 102 ) +( 2 × 101 ) + (5 × 100 )
= (3 × 1000) + ( 1 × 100) + ( 2 × 10) + ( 5 × 1)
= 3000 + 100 + 20 + 5
NAME:__________________________________________________ CLASS:____________________
= 3125
Each place value is ten times the value of its right-hand neighbour. The base of this
number system is ten and it is called the denary system.
Write 3810 as a number to the base 5.
(To write a number to the base 5 we have to find out how many ... 125s, 25s, 5s
and units the number contains.). (Starting with the highest value column.)
38 contain no 125s.
38 ÷ 25 = 1 remainder 13 i.e. 38 = 1 × 25 + 13
13 ÷ 5 = 2 remainder 3 i.e. 13 = 2 × 5 + 3
∴ 38 = (1 × 𝟓𝟐 )+ (2 × 5 )+ (3 )
= 𝟏𝟐𝟑𝟓
Example 1: Convert 510 (read 5 base 10) into base 2.
The Process:
1. Divide the "desired" base (in
this case base 2) INTO the number
you are trying to convert.
2. Write the quotient (the answer)
with a remainder like you did in
elementary school.
3. Repeat this division process
using the whole number from the
previous quotient (the number in
front of the remainder).
4. Continue repeating this division
until the number in front of the
remainder is only zero.
5. The answer is the remainders
read from the bottom up.
510 = 1012
(a
binary conversion)
Example 2: Convert 14010 to base 8.
NAME:__________________________________________________ CLASS:____________________
The process is the same as in
example 1. The answer is:
14010 = 2148
(an octal conversion)
Write the following numbers to base ststed in the brackets.
1) 810 (base 2)
2) 1310 (base 3) 3) 1910 (base 5)
4) 2710 (base 4) 5) 3810 ( base 7) 6) 4710 ( base 6)
7) 10310 (base 8) 8) 13210 (base 5 ) 9) 12110 ( base 7)
10) 21110 (base 6)
Example of writing of 𝟐𝟎𝟑𝟓 as a number to base 10
2035
= (2 × 25) + (0 × 5) + (3×1)
= 5010 + 0 + 310
= 5310
(Although we do not normally write fifty-three as 5310 , it is sensible to
do so when dealing with other bases as well.)
Example 2: Convert 2358 into base 10.
The Process:
Above each of the digits in your number, list
the power of the base that the digit
represents. See the example on the left. It is
NAME:__________________________________________________ CLASS:____________________
now a simple process of multiplication and
addition to determine your base 10
number. In this example you have
Now simply add these
0
values together.
5x8 =5
3 x 81 = 24 5 + 24 + 128 = 157
2 x 82 = 128 Answer: 2358 =
15710
**Remember: any number to the zero
power equals one.
Example 3: Convert 10112 to base 10.
The process is the same as in example 1.
1
x20 =1
1 + 2 + 0 + 8 = 11
1
1 x 2 =2
0 x 22 =0 Answer: 10112 =1110
1 x23 =8
Write the following numbers as denary numbers, i.e. to base 10
1) 315
2) 1214
3) 325
NAME:__________________________________________________ CLASS:____________________
4) 4004
5) 243
6) 2046
7) 205
8) 2407
9) 405
10) 435
11) 43
12) 3007