- This is Jorgensen chapter 3 as well as Carter chapter 1
- In base 10, we use the digits 0-9
- And the position of the digit lets us know the weight of the digit.
- $582 = 5\times 10^2 + 8\times10^1 + 2\times10^0$

- Ok, I got this but really?
- Let's discuss simple addition
- You have memorized that 3 + 1 = 4
- 0 + 1 = 1 , 1 + 1 = 2, 2 + 1 = 3, ... ,8 + 1 = 9
- Note each time we add one to the symbol, we know which symbol to go to next.
- But when we get to 9 + 1 we are out of digits.
- So we add a new rule, 9 + 1 = 10
- Ten 10 is constructed from the digits 1 and 0

- Again in base 10, we have 10 digits (or symbols) 0-9, and a base or radix of 10
- We do not have a special digit for 10, we construct it out of our other digits and position.
- We probably use base 10 because we have 10 fingers.

- why is 25 + 39 = 64?
- 25 is $2\times10^1 + 5 $
- 39 is $3\times10^1 + 9$
- When we add 5 + 9 we get $1\times10^1$ + 4 or a ones digit of 4
- When we add $2\times10^1 + 3\times10^1 + 1\times10^1$
- we get $6\times10^1$
- So the answer is 64

- In general $d_5d_4d_3d_2d_1d_0 = d_5\times10^5+d_4\times10^4+d_3\times1^3+d_2\times10^2+d_1\times10^1+d_0\times10^0$
- The digit determines the position multiplier
- The position of the digit, from the right starting at 0, determines the weight of the value.

- There is no reason to use base 10.
- The Mayans used base 20 and the Sumerians used base 60
- In computing we use base 2 or binary.

- Binary
- The digits are {0,1}
- The base or radix is 2.
- $101101_2 = 1\times2^5+0\times2^4+1\times2^3+1\times2^2+0\times2^1+1\times2^0$

- At this point, it is probably reasonable to memorize the powers of 2 up to $2^{10}$
- And the binary patters for 0-15
- Converting From Binary to Decimal
- To convert $101101_2$ to base 10
- Write the expansion and work it out.
- $101101_2 = 1\times2^5+0\times2^4+1\times2^3+1\times2^2+0\times2^1+1\times2^0$
- $= 1\times32 + 0\times16+1\times8+1\times4+0\times2+1\times1$
- $= 32+8+4+1 = 45$

- It is reasonable to check with an on-line calculator

- To convert decimal to binary
- Constantly divide by 2, noting the remainder.
- Convert 243 to binary
2|243 2|121 R 1 2|60 R 1 2|30 R 0 2|15 R 0 2|7 R 1 2|3 R 1 1 R 1

- Read the remainders backwards
- $243 = 11110011_2$