Compiled by : Mr. Mark Anthony P. Cezar ( Lecture )

Numbers(s)

A number is a mathematical object used to count and measure.

Classification of numbers

Natural NumbersIntegersRational NumbersComplex NumbersComputable Numbers

Chapter II

Introduction to Numeric and Binary System

Binary Numeral System or Binary Number System

Base-2 number systemNumeric values using two symbols 0 and 1Used internally by almost all modern computersIs a writing system of expressing numbers with symbolThe words binary comes from “Bi” MEANING TWO”

Example: Bicycle, Binocular

Example:

The number 10 means 2 in decimal

Why?

"There are 10 kinds of people in the world, those who understand binary numbers, and those who don't."

The BIT as a starting point . . . .

Bit is the smallest unit of data.Where as, Binary Number is made up of only 0s and 1s

Number System Conversion

As computer programmer, it is important to know the different numeral system.

Same as . . .

BINARY ADDITION

BINARY SUBTRACTION

QUIZ

Addition Subtraction

BINARY MULTIPLICATON

BINARY DIVISION

Binary Multiplication

1.10 x 10 =2.100 x 11 =3.101 x 10=4.1011 x 11 =5.11011 x 101 =

Binary Division

1.100 / 10 =2.111 /11 =3.1010/100 =4.1101/11 =5.101111 / 10 =

Answer Multiplication 1-5 Answer Division 1-5

Do not just think . . .Analyze.Know the logic . . …..

You have a fox, a chicken and a sack of grain. You must cross a river with only one of them at a time. If you leave the fox with the chicken he will eat it; if you leave the

chicken with the grain he will eat it. How can you get all three across safely?

MIND TEASER

Chapter II

Logic CircuitsVariables, functions, truth tables,gates and networks

Digital Logic Logic Circuit used to build computerLogic circuit perform operation on digital signalsThe general form of logic circuit is a switching networkIn binary Logic circuits are only two values 0 and 1Digital HardwareLate 1960s and early 1970s saw a revolutionary in digital capability – smaller transistors -larger chip sizeMore transistor/chip gives greater functionalityIntegrated circuits are fabricated on silicon wafersChips have tens to million transistorProvides motivation for computer based design techniques

15 Minutes

Review

quiz

1.z=tb+c2.wxyx wy not(x) + wy

Quiz

Chapter IIILogic CircuitsSynthesis using AND, OR, andNOT gates

Optimized Implementation ofLogic Functions:Karnaugh Maps and MinimumSum-of-Product Forms

Example logic circuit design

• Assume we want to design a logic circuit withthree inputs x, y, and z• The circuit output should be 1 only when x=1and either y or z (or both) is 1– Three possible combinations• x=1, y=0, z=1 => xy’z• x=1, y=1, z=0 => xyz’• x=1, y=1, z=1 => xyz• The function could be written as– f(x,y,z)=xy’z+xyz’+xyz

Example logic circuit design

• The Karnaugh map (K-map) provides asystematic (and graphical) way of performingthis operation• Minterms can be combined by 14a when theydiffer in only one variable– f(x,y,z) = xyz+xyz’ = xy(z+z’) = xy(1) = xy• The K-map illustrates this combinationgraphically

Karnaugh map

Programmable logic devices, what are they? Integrated circuit that you program using a hardware description language such as VHDL or Verilog.

-Languages that you may have heard of are CUPL or ADA. With the correct tools you can even generate a schematic that is used to build the file that will be used to program the device