Now that we've taken a trip to the past, let's finish up back in the present. In this week'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):

$\displaystyle {2}^{{6}}={64},\quad{2}^{{7}}={128}$, so the biggest power of 2 that divides into 77 is $\displaystyle {2}^{{6}}={64}$. Following the "new method" for converting from base-10 to other bases:

$\displaystyle {77}\div{64}$ = 1.203125
$\displaystyle {0.203125}\times{2}$ = 0.40625
$\displaystyle {0.40625}\times{2}$ = 0.8125
$\displaystyle {0.8125}\times{2}$ = 1.625
$\displaystyle {0.625}\times{2}$ = 1.25
$\displaystyle {0.25}\times{2}$ = 0.5
$\displaystyle {0.5}\times{2}$ = 1.0

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:

$\displaystyle {0}\times{2}^{{7}}+{1}\times{2}^{{6}}+{0}\times{2}^{{5}}+{0}\times{2}^{{4}}+{1}\times{2}^{{3}}+{1}\times{2}^{{2}}+{0}\times{2}^{{1}}+{1}\times{2}^{{0}}={64}+{8}+{4}+{1}={77}$

For this week's project, I want you to decode the following binary codes into letters:

01000001 01001100 01001100 01000100 01001111 01001110 01000101

In letters, this means: