1. Binary
“There are 10 types of people in the world: Those who understand binary, and those who don't.”

2. What is Binary? Binary is the base-2 numeral system.
We normally use the base-10 system.
The only symbols in binary are “0” and “1”.
“0” represents no electron flow, aka “OFF”.
“1 “ represents electrons flowing, aka “ON”.

3. Bits and Bytes Each digit in a binary number is a bit.
Eight bits make up one byte.

4. Base-10 versus Base-2 BASE 10 100 = 1 101 = 10 102 = 100 103 = 1000
Base-10 uses ten as the base of the exponent, while Base-2 uses a two for the base.
BASE 10
100 = 1
101 = 10
102 = 100
103 = 1000
BASE 2
20 = 1
21 = 2
22 = 4
23 = 8

5. Converting Binary Values
Add up all of the place values that hold a “1”.
Place Values: 8’s, 4’s, 2’s, 1’s
0101 = = 5
1110 = = 14
0111 = = 7

6. Adding Binary Numbers 0 1 0 1 . . . . . . 5 + 0 0 0 1 . . . . . . +1
0 + 1 = 1
1 + 0 = 1
1 + 1 = carry the left “1”
= carry the left “1”
=================

7. What is an Overflow? 1 1 1 1 = 15 + 0 0 0 1 = + 1 ============
Overflow occurs when there is a carry into a column that doesn’t exist!
= 15
= + 1
============
= 0?

8. Unsigned versus Signed
An unsigned binary number can only be positive and has all of it’s bits calculated into the size of the value.
A signed binary number can be positive or negative. The left most bit determines the sign. “0” denotes positive, while “1” denotes negative.
Unsigned
= 5
= 10
= 13
Sign&Mag
= 5
= -2
= -5

9. Problems with Sign&Mag
There are two zeros: 0000 (+0) and 1000 (-0).
It is possible to have a carry into the “sign bit” by accident, causing errors in math!
= 7
= 2
=============
= -1? Should = 9

10. One’s Compliment 0 0 0 1 = 1 1 1 1 0 = -1 0 1 1 0 = 6 1 0 0 1 = -6
Flip all of the bits and call that negative!
If you have a carry out from sign bit during addition, do an end-around carry!
This still has two zeroes: 0000 and 1111.
= 1
(flip for negative)
= -1
= 6
(flip for negative)
= -6

11. Two’s Compliment 0 0 0 = 0 flip+1 maps to 0 0 0
Flip all of the bits, then add one, call it negative!
Do not do an end-around-carry!
= 0 flip+1 maps to 0 0 0
= 1 flip+1 maps to 1 1 1
= 2 flip+1 maps to 1 1 0
= 3 flip+1 maps to 1 0 1
= -4 given to negative side
= -3 from mapped +3
= -2 from mapped +2
= -1 from mapped +1

12. Range of a Binary Number
It depends on:
Number of Bits
System: Unsigned, Sign&Mag, One’s, Two’s
4-bit Unsigned can go from:
= “0”
= “15”
4-bit Signed in Two’s Compliment can go from:
= “-8”
= “7”