Now that we’ve taken a trip to the past, let’s finish up back in the present. In this section’s reading, you saw how different cultures used number systems with different bases than the base-10 system we use in everyday modern life.

Other bases are still in common use today in our computers. Computers only know “On” and “Off”, which are represented with 1’s and 0’s. This creates a base-2 number system called binary.

To store your English papers, emails, and other documents containing letters, the computer first must convert the letter into a number. To do this, a system was created in the 1960’s called ASCII (American Standard Code for Information Interchange) which assigns a decimal value to each letter. Here are the ASCII codes for uppercase letters A – Z:

A | B | C | D | E | F | G | H | I | J | K | L | M |

65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | 74 | 75 | 76 | 77 |

N | O | P | Q | R | S | T | U | V | W | X | Y | Z |

78 | 79 | 80 | 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |

For example, to encode the letter M, it would first be converted to the decimal value 77. The decimal value 77 is then converted to binary (base-2) in the following way:

[latex]\begin{array}{c}2^{6} = 64\\2^{7} =12826= 6^{4}\\ 2^{7}=12^{8}\end{array}[/latex]

Therefore, the biggest power of 2 that divides into 77 is [latex]2^{6} = 6426 = 6^{4}[/latex].

Following the “new method” for converting from base-10 to other bases:

[latex]77\div6477\div64=1.203125[/latex]

[latex]0.203125\times20.203125\times2=0.40625[/latex]

[latex]0.40625\times20.40625\times2=0.8125[/latex]

[latex]0.8125\times20.8125\times2=1.625[/latex]

[latex].625\times20.625\times2=1.25[/latex]

[latex].25\times20.25\times2=0.5[/latex]

[latex].5\times20.5\times2=1.0[/latex]

So 77 (in base 10) is equivalent to 1001101 in binary. Since binary numbers usually come in groups of 8 digits, this number would commonly be written with a leading zero, as 01001101. We can check this conversion by converting back to base-10:

[latex]0\times27+1\times26+0\times25+0\times24+1\times23+1\times22+0\times21+1\times20=[/latex]

[latex]64+8+4+1=[/latex]

[latex]770\times27+1\times26+0\times25+0\times24+1\times23+1\times22+0\times21+1\times20=[/latex]

[latex]64+8+4+1=77[/latex]

Therefore, starting from base 10 with the number 77, it is equivalent to 1001101in binary. Since binary numbers are usually expressed using 8 digits, this number would commonly be written with a leading zero, as 01001101. We can check this conversion by converting back to base 10 in the following way:

[latex]0\times{2^{7}} + 1 \times{2^{6}} +0\times {2^{5}}+0\times{2^4}+1\times{2^3}+1\times{2^2}+0\times{2^1}+1\times{2^0}=64+8+4+1=77[/latex]

Now you can decode the following binary digits into letters:

01010100 01001000 01000101 01001100 01000001 01010011 01010100

Download the assignment from one of the links below (.docx or .rtf):