Expression 
Power of 10 
Remainder 
Remainder/10^{n} expression 
Result 
Powers of 10 for emphasis! 
4825 % 10 
0 
5 
5 / 10^{0} 
5 
10^{0} = 1 
4825 % 100 
1 
25 
25 / 10^{1}  2 
10^{1} = 10 
4825 % 1,000 
2 
825 
825 / 10^{2}  8 
10^{2} = 100 
4825 % 10,000 
3 
4825 
4825 / 10^{3}  4 
10^{3} = 1,000 
4825 % 100,000 
4 
4825 
4825 / 10^{4}  0 
10^{4} = 10,000 
Remainder 
Division 
Power of 10 
Result 
Powers of 10 for emphasis! 
4825 % 10 
4825 / 10 = 482 
0 
5 
10^{0} = 1 
482 % 10 
482 / 10 = 48 
1 
2 
10^{1} = 10 
48 % 10 
48 / 10 = 4 
2 
8 
10^{2} = 100 
4 % 10 
4 / 10 = 0 
3 
4 
10^{3} = 1,000 
0 % 10 
0 / 10 = 0 
4 
0 
10^{4} = 10,000 
Number / 10 each step 
Remainder % 10 each step 
4825 
5 
482 
2 
48 
8 
4 
4 
0 
0 
Base 
Number 
Expression 
2 
100101 
1*2^{5} + 0*2^{4} + 0*2^{3} + 1*2^{2}
+ 0*2^{1} + 1*2^{0} 
8 
3702 
3*8^{3} + 7*8^{2} + 0*8^{1} + 2*8^{0} 
16 
F2A6 
15*16^{3} + 2*16^{2} + 10*16^{1} +
6*16^{0} 
23 
GM2K04 
16*23^{5} + 22*23^{4} + 2*23^{3} +
20*23^{2} + 0*23^{1} + 4*23^{0} 
Symbol 
A 
B 
C 
D 
E 
F 
G 
H 
I 
J 
K 
L 
M 
N 
O 
P 
Q 
R 
S 
T 
U 
V 
W 
X 
Y 
Value 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
Base 
Number 
Expression 
Base 10 
2 
100101 
1*2^{5} + 0*2^{4} + 0*2^{3} + 1*2^{2}
+ 0*2^{1} + 1*2^{0} 
37 
8 
3702 
3*8^{3} + 7*8^{2} + 0*8^{1} + 2*8^{0} 
1,986 
16 
F2A6 
15*16^{3} + 2*16^{2} + 10*16^{1} +
6*16^{0} 
62,118 
23 
GM2K04 
16*23^{5} + 22*23^{4} + 2*23^{3} +
20*23^{2} + 0*23^{1} + 4*23^{0} 
109,172,908 
Base 10 
EVERYTHING on this side is base 2, including the
exponents! 
638 
110*1010^{10} + 11*1010^{1} + 1000*1010^{0} 
Expression / 23 
Result % 23 
Base 23 digit 
(16*23^{5} + 22*23^{4} +
2*23^{3} + 20*23^{2} + 0*23^{1} +
4*23^{0}) / 23 
4 
4 
(16*23^{4} + 22*23^{3} + 2*23^{2} + 20*23^{1} + 0*23^{0}) / 23  0 
0 
(16*23^{3} + 22*23^{2} + 2*23^{1} + 20*23^{0}) / 23  20 
K 
(16*23^{2} + 22*23^{1} + 2*23^{0}) / 23  2 
2 
(16*23^{1} + 22*23^{0}) / 23  22 
M 
(16*23^{0}) / 23  16 
G 
0 / 23 
0 
0 
Number / 23 
Result % 23 
Base 23 digit 
109,172,908 / 23 
4 
4 
4,746,648 / 23  0 
0 
206,376 / 23  20 
K 
8,972 / 23  2 
2 
390 / 23  22 
M 
16 / 23  16 
G 
0 / 23 
0 
0 
base 10 / 2 
Remainder % 2 
48334 
0 
24167 
1 
12083 
1 
6041 
1 
3020 
0 
1510 
0 
755 
1 
377 
1 
188 
0 
94 
0 
47 
1 
23 
1 
11 
1 
5 
1 
2 
0 
1 
1 
0 
0 
base 10 / 8 
Remainder % 8 
48334 
6 
6,041 
1 
755 
3 
94 
6 
11 
3 
1 
1 
base 10 / 16 
Remainder % 16 
Hex digit 
48334 
14 
E 
3,020 
12 
C 
188 
12 
C 
11 
11 
B 
0 
0 
0 
Fraction 
Base 10 
Expression 
Comments 
3/10 
0.3 
3*10^{1} 
Easy, finite 
53/100 
0.53 
5*10^{1} + 3*10^{2} 
Pretty easy still 
8591/10000 
0.8591 
8*10^{1} + 5*10^{2} + 9*10^{3}
+ 1*10^{4} 
More numbers, same pattern 
3/5 
0.6 
6*10^{1} 
Make the denominator a multiple of 10 first 
7/25 
0.28 
2*10^{1} + 8*10^{2} 
Same pattern 
1/3 
0.3333... 
3*10^{1} + 3*10^{2} + 3*10^{3} +...  A finite fraction, but an infinite decimal expansion 
7/13 
0.538461,538461,538461... 
5*10^{1} + 3*10^{2} + 8*10^{3}
+ ... 
This repeats after a while  as will ALL quotients of 2
integers. 
3/4 
0.75 
7*10^{1} + 5*10^{2} 
A power of 2 as a denominator 
3/8 
0.375 
3*10^{1} + 7*10^{2} + 5*10^{3} 
2^{3} as a denominator 
Expression 
Power of 10 
Remainder 
Part of Expression 
Integer Part 
Powers of 10 for emphasis! 
0.8591 * 10 
1 
0.591 
8 / 10^{1} 
8 
10^{1} = 0.1 
0.591 * 10 
2 
0.91 
5 / 10^{2}  5 
10^{2} = 0.01 
0.91 * 10 
3 
0.1 
9 / 10^{3}  9 
10^{3} = 0.001 
0.1 * 10 
4 
0 
1 / 10^{4}  1 
10^{4} = 0.000,1 
0 
5 
0 
0 / 10^{5}  0 
10^{5} = 0.000,01 
Base 
Number 
Expression 
2 
0.100101 
1*2^{1} + 0*2^{2} + 0*2^{3} +
1*2^{4} + 1*2^{5} + 1*2^{6} 
8 
0.3702 
3*8^{1} + 7*8^{2} + 0*8^{3} +
2*8^{4} 
16 
0.F2A6 
15*16^{1} + 2*16^{2} + 10*16^{3}
+ 6*16^{4} 
23 
0.GM2K04 
16*23^{1} + 22*23^{2} + 2*23^{3}
+ 20*23^{4} + 0*23^{5} + 4*23^{6} 
Expression 
Power of 5 
Part of Expression 
Integer Part 
Powers of 5 for emphasis! 
0.28 * 5 

1 + 0.4 * 5 
1 
1 / 5^{1} 
1  5^{1} = 0.1 
2 + 0.0 
2 
2 / 5^{2}  2 
5^{2} = 0.01 
0 
3 
0 / 5^{3}  0 
5^{3} = 0.001 
Expression 
Power of 6 
Part of Expression 
Integer Part 
Powers of 6 for emphasis! 
0.47222... * 6 

2 + 0.833... * 6 
1 
2 / 6^{1} 
2  6^{1} = 0.1 
4 + 0.999... = 5 + 0.0 * 6 
2 
5 / 6^{2}  5 
6^{2} = 0.01 
0 
3 
0 / 6^{3}  0 
6^{3} = 0.001 
Number 
Digit base 2 
0.67578125 
 
1.3515625 
1 
0.703125 
0 
1.40625 
1 
0.8125 
0 
1.625 
1 
1.25 
1 
0.5 
0 
1.0 
1 
Number 
Digit base 2 
0.377 
 
0.754 
0 
1.508 
1 
1.016 
1 
0.032 
0 
0.064 
0 
0.128 
0 
0.256 
0 
0.512 
0 
1.024 
1 
0.048 
0 
Number  *8 
Digit base 8 
0.67578125_{10}   
5.40625 
5 
3.25 
3 
2.0 
2 
Number  *16 
Digit base 16 
0.67578125_{10}   
10.8125 
A 
13.0 
D 
a.i 
a.ii 
b.i 
b.ii 
12 +17  5 
12 + 17  5 
17 +12  5 
17 + 12  5 
Signed base 10 
5digit 9's complement 
12 
00012 > 99987 
Base 10 
10digit 2's complement (10:2) 
12 + 12  0 
11 1111 0100 00 0000 1100  00 0000 0000 
Base 10 
10digit 2's complement (10:2) 
12 + 17  5 
11 1111 0100 00 0001 0001  00 0000 0101 
Reference:IEEE 754 is actually at least 2 different number representation systems, one using 32 bits and the other 64 bits. We will focus on the 32bit version.
Harald Schmidt
Euskirchener Str. 43
53121 Bonn  Germany
Email:web@hschmidt.net
Number 
Scientific Notation 
Characteristic = power 
Mantissa = fractional part = significand 
6.4323 
6.4323 x 10^{0} 
0 
6.4323 
5,6140.338 
5.6140338 x 10^{4} 
4 
5.6140338 
0.00002343 
2.343 x 10^{5} 
5 
2.343 
bit 31 
bits 3023 
bit 220 
sign bit 
exponential part excess 127 
mantissa bit 22 is the 2^{1} bit the 2^{0} bit is always 1 and not represented explicitly 
Expression / 2 
Result % 2 
27056 
0 
13528 
0 
6764 
0 
3382 
0 
1691 
1 
845 
1 
422 
0 
211 
1 
105 
1 
52 
0 
26 
0 
13 
1 
6 
0 
3 
1 
1 
1 
Expression * 2 
Result 
0.8125 

1.6250 
1 
1.250 
1 
0.5 
0 
1 
1 