1. What is this? How did you know its numerical value?

2. How did you know? How do you know that the symbols “1021” meant the number one thousand twenty-one? Base 10: –To represent all numbers less than ten (ie - one, two, three, etc) we choose special symbols (ie - “1”, “2”, “3”, etc) –For quantities greater than ten, we combine several symbols and determine the value of the entire string by looking at the positioning of each symbol.

3. The action starts at 10 … “42” uses two symbols to represent the quantity forty-two The position of the “4” symbol (to the left of the “2” symbol) tells us that “4” does not merely represent the quantity four Instead, these two symbols are read as “four groups of ten’s and two groups of one’s.” If we sum these quantities, we get the number forty-two Note that our “group sizes” were multiples of ten - which is why this system is called “base 10” “42” 2 * 1 == 2 + 4 * 10 == 40 forty-two Number of groups Size of group

4. A larger example “1021” 1 * 1 == 1 2 * 10 == 20 One thousand twenty one 0 * 100 == 000 + 1 * 1000 == 1000But all of these group sizes are powers of 10!

5. A larger example - rewritten “1021” 1 * 10 0 == 1 2 * 10 1 == 20 One thousand twenty one 0 * 10 2 == 000 + 1 * 10 3 == 1000This is why it’s called “base 10” Sometimes, a subscript is written below a number to show what its base is: 1021 10

6. Alternative Paradigms We can use other numbers for bases. “Binary” is “base two”, while “hexadecimal” means “base 16” “1021 16 ” 1 * 16 0 == 1 2 * 16 1 == 32 Four thousand one hundred twenty-nine 0 * 16 2 == 000 + 1 * 16 3 == 4096

7. A need for more symbols How do you represent the quantity ten in hexadecimal? “10 16 ” 0 * 16 0 == 0 + 1 * 16 1 == 16 sixteen “9 16 ” 9 * 16 0 == 9 nine Solution: Invent a new symbol for ten!

8. More symbols What symbols should we use to represent numbers like ten, eleven, twelve, thirteen, fourteen, and fifteen in hexidecimal? –And why don’t we need a symbol for sixteen? Computer scientists are lazy … so they reused some other commonly used symbols: the English alphabet! Ten“A” Eleven“B” Twelve“C” Thirteen“D” Fourteen“E” Fifteen“F”

9. All your base 36 “3FD 16 ” 13 * 16 0 == 13 15 * 16 1 == 240 One thousand twenty one + 3 * 16 2 == 768 “27EA 16 ” 10 * 16 0 == 10 14 * 16 1 == 224 Ten thousand two hundred eighteen 7 * 16 2 == 1792 + 2 * 16 3 == 8192