This document provides an overview of binary numbers and related concepts in computing: - Binary numbers are made up of 0s and 1s and have a base-2 value. Mixed binary numbers like 10.1 represent decimals. - A bit is a single binary digit, and bytes group together 8 bits. File sizes are measured in bytes, kilobytes, megabytes, and larger units. - Converting between binary and decimal involves placing the binary digits in powers of two and adding them up. Several exercises are provided to practice conversions. - Fractional binary numbers represent decimals the same way. - Two's complement is a common way computers represent negative integers in binary by inverting and adding one to the positive

1. Mathematics for
Computing
Dr. M H B Ariyaratne
https://goo.gl/bmAhX6

2. Binary Numbers
● Bi means two
● Made up of 0s and 1s
● Base 2 values
● Ex = 101 (one-oh-one/one-zero-one)
● How to show - 1012
Reference

3. Binary Numbers (Mixed Numbers)
Example: 10.1
The "10" means 2 in decimal,
The ".1" means half,
So "10.1" in binary is 2.5 in decimal

4. Bit
● bit = “Binary Digit”
● Single Binary Digit -> bit
● Ex, 10010 is 5 bits long

5. Bytes
· 1 Bit = Binary Digit
· 8 Bits = 1 Byte
· 1024 Bytes = 1 Kilobyte
· 1024 Kilobytes = 1 Megabyte
· 1024 Megabytes = 1 Gigabyte
· 1024 Gigabytes = 1 Terabyte
· 1024 Terabytes = 1 Petabyte
· 1024 Petabytes = 1 Exabyte

6. Binary to Decimal (Whole Numbers)

7. Exercise 1
Convert binary to decimal values
Example (10011)2 = (19)10
1. 1001011
2. 110101
3. 10001011
4. 01000000
5. 00111111

8. Decimal to Binary (Whole Numbers)

9. Exercise 2
Convert decimal to binary values
Example (19)10 = (10011)2
1. 63
2. 33
3. 100
4. 120
5. 168

10. Decimal to Binary (Fraction Numbers)

11. Binary to Decimal (Fraction Numbers)

12. Exercise 3
Convert decimal to binary values
Example (0.435)10 = (0.01101)2
1. 0.63
2. 0.435
3. 4.1
4. 8.64
5. 2.531

13. Exercise 4
Convert binary to decimal values
Example (0.10010)2 = (0.5625)10
1. 0.1
2. 0.1001
3. 0.111
4. 1.001
5. 11.11

14. Two's Complement
● The way computers represent integers
● Common math problems are very simple to implement
● Ex. (nibble or ½ byte)
○ 0000 - Zero
○ 0001 - one
○ 1111 - minus one
● To get the two's complement negative notation of an integer
○ write out the number in binary
○ Then invert the digits
○ Add one to the result
Ex. find how -28 would be expressed in two's complement notation

15. Two’s Complement

16. Thank you