Non-integer numbers conversion
Example:
$ 17.375_{10} = 1*10^1 + 7*10^0 + 3*10^{-1} + 7*10^{-2} + 5*10^{-3} = $
$ = 10 + 7 + \frac{3}{10} + \frac{7}{100} + \frac{5}{1000} $
How do you convert decimal fractions to binary?
You convert the whole integer numbers separately:
$ 17_{10} = 10001_2 $
Then you convert fractional part 0.375_{10}:
Multiply by 2, see if you have a whole number 1:
0.375*2 = 0.75 = 0 + 0.75
0.75*2 = 1.5 = 1 + 0.5 Multiply the remainder of 0.75
0.5 *2 = 1.0 = 1 + 0. STOP
$ 0.375_{10} = 0. \color{red} {011}_2 $
$ 17.375_{10} = 10001. \color{red} {011}_2 $
How to convert a binary fractional number to a base 10 number?
$ 10001. \color{red} {011}_2 = 1*2^4 + 0*2^3 + 0*2^2 + 0*2^1 + 1*2^0 + \color{red} { 0*2^{-1} + 1*2^{-2} + 1*2^{-3} }$
$ 10001. \color{red} {011}_2 = 1*16+ 0*8 \ + 0*4 \ + 0*2 \ + 1*1 + \color{red} { 0* \frac{1}{2} + 1*\frac{1}{4} + 1*\frac{1}{8} }$
$ 10001. \color{red} {011}_2 = 16 + 1 + \color{red} { \frac{0}{2} + \frac{1}{4} + 1*\frac{1}{8} }$
$ 10001. \color{red} {011}_2 = 17 + \color{red} { \frac{3}{8} } $
$ 10001. \color{red} {011}_2 = 17. \color{red} { 375}_{10} $
In general:
$$ A_n A_{n-1} A_{n-2} ... A_0 . C_{-1} C_{-2} ... C_{-k} $$
for base b, and position n and k,
In decimal units corresponds to:
$$ A_n * b^n + A_{n-1} * b^{n-1} + ... + A_0 * b^0 + C_{-1} * b^{-1} + C_{-2} * b^{-2} + ... + C_{-k} * b^{-k} $$
see reference 2
Convert the decimal number 11.625 into binary
$ 11 \color{red} {.625}_{10} $
$ 11_{10} = 1011_2 $
$ \color{red} {.625}_{10} = $
Multiply by 2 and carry over the remainder:
2*0.625 = 1.25 = 1 + 0.25
2*0.250 = 0.50 = 0 + 0.50
2*0.500 = 1.00 = 1 + 0.00 // STOP
What is the decimal number 0.03125 in binary?
$ 0.03125_{10} = ? $ in binary
Multiply by 2 and carry over the remainder:
2 * 0.03125 = 0.0625 = 0 + 0.0625
2 * 0.06250 = 0.1250 = 0 + 0.1250
2 * 0.12500 = 0.2500 = 0 + 0.2500
2 * 0.25000 = 0.5000 = 0 + 0.5000
2 * 0.50000 = 1.0000 = 1 + 0.0000 // STOP
$ 0.03125_{10} = 0.00001_2 $
Operations with binary numbers
See reference 3Addition of binary numbers
$ 101_2 + 111_2 = ? $
$ 1110 $ \\ carry-over numbers
$ 0101_2 + $
$ 0111_2 = $
$ \overline{1100}_2 $
$ 1110 $ \\ carry-over numbers
$ 0101_2 + $
$ 0111_2 = $
$ \overline{1100}_2 $
double-check in decimal: 5 + 7 = 12
Q.E.D.
Subtraction of binary numbers
$ 110 - $ \\ 6 decimal
$ 101 = $ \\ 5 decimal
$ \overline{0 0 1}_2 $ \\ 1 decimal
$ \overline{0 0 1}_2 $ \\ 1 decimal
Multiplication of binary numbers
0*0 = 0
1*1 = 1
1*0 = 1
Exercise:
1111 * 11 = 101101
15 * 3 = 45
Division of binary numbers
Number bases webinar video
See reference 4
"The only time you are getting better is when you are stuck.- So, embrace being stuck! "
Extra video: numbers 1
See reference 5
- prime numbers
- highest known prime number $ 2^{77232917} - 1$
- exponentials
- $ x^0 = 1 $
- $ x^1 = x $
- $ x^2 = x * x $
- $ x^{-1} = \frac{1}{x} $
- $ x^{-2} = \frac{1}{x*x} = \frac{1}{x^2} $
- $ x^{-n} = \frac{1}{x^n} $
Extra video: numbers 2
- see reference 6
- base ten fractional
- what is the lowest base you can have
- base 2
- conversion exercises
see reference 7
See reference 8
References
- https://www.coursera.org/learn/uol-cm1015-computational-mathematics/lecture/wrj47/non-integer-numbers-conversion
- https://www.coursera.org/learn/uol-cm1015-computational-mathematics/quiz/QW8GL/topic-1-lesson-3
- https://www.coursera.org/learn/uol-cm1015-computational-mathematics/lecture/0qco4/operations-with-binary-numbers
- https://www.coursera.org/learn/uol-cm1015-computational-mathematics/lecture/y71es/number-bases-webinar-video
- https://www.coursera.org/learn/uol-cm1015-computational-mathematics/lecture/K02pr/extra-video-numbers-1
- https://www.coursera.org/learn/uol-cm1015-computational-mathematics/lecture/lHwNt/extra-video-numbers-2
- https://docs.google.com/spreadsheets/d/19zWX5qNcfVwG37VUSi0aeCMfNSOwhq1MMNtdwguyk5k/edit#gid=0
- https://calculator.name/baseconvert/quinary/decimal/1212
