Numbers and symbols of inequality etc.

The conceptualization of Numbers and Number Line is important in order to understand the concepts of Limits, Functions and Calculus. Those who want to develop a clear concept of Calculus must assimilate the basics of Numbers and Number line etc. discussed in the previous two post along with this one.

Understanding Equality and Inequalities:

Symbols:

=      >      <      ≤     ≥

In the previous post, the basics of Numbers and Number Line were discussed, continuing from where we left of, let us bring back the diagram, Fig. 9.01, of the Number Line and try to understand a few more points. 

Real Number Line Fig 9.01

Talking about real numbers, we say the real numbers are 'ordered'. It means they follow certain principles, in the sense that, when we say a is less than b, which is generally written, using symbols, as a < b ,then by subtracting a from b  i.e. b - a we would get a positive number.
The above statement can also be also expressed as:  b is greater than a , written as b > a

Now let's understand how and where can we use the symbols   ≤      ≥  

The symbol  a ≤  b  or ( b ≥  a) means that either a < b or a = b , which is read as " a is less than or equal to b"

Some of the example of inequalities are as follows:

7 < 7.4 < 7.5     - π <  π  < 3    √2 < 2     2 ≤  2

If we show these numbers on these numbers on the number line they will look like this, refer to Fig. 9.02:  

The positioning of various numbers according to the numerical values Fig. 9.02

Conventionally the  set of real numbers is denoted by ℝ .

Also conventionally, by using the word number ,we mean the real number. The various types of numbers, discussed this post, and the previous two posts can be represented in the from the Diagram, 
Fig. 9.03 posted here under:

Diagram showing various Numbers Fig 9.03

Some of the properties of the real numbers are;

Each real number has a decimal representation. In case the number is rational it has repeating decimals, for example:

1/4  = 0.2500..... = 0.250 (zero keeps repeating)

 9/7 = 1,285714285714..... where .258714 ( repeats itself)

The irrational numbers such as  √2  or π  have decimal representation which is non-repeating, shown below:

√2 = 1.414213562373095........... 

The numbers have other properties which they exhibit during multiplication, division etc. 

Number Line and representation of various numbers on it!

In the previous post, you have seen there are various types of Numbers namely Natural Number, Rational Numbers, Irrational Numbers...etc. 
In order to understand, the meaning of each term we need know how to identify them and also how do they differ from one another. We shall be examining the Number line, posted below: 

Fig. 1.7 Real Numbers Line 

Natural Numbers: These are 

1, 2, 3, 4, 5, 6.................

Integers: You will notice there is a "0" zero  at the centre of the Number Line, there are Numbers with (+) sign on the right side and Numbers with (-) sign on the left, since these are Numbers with (+ve) or (-ve) signs attached these are known as Integers. The integers are written as;

....,-3, -2, -1, 0, 1, 2, 3,......

Rational Number: The rational numbers are expressed as ratios of Integers and are expressed as:

r = p/q

where p and q are integers and   q ≠ 0. The examples are:

   1/2      -3/5      35 = 35/1      0.15 = 15/100      etc.

Note: The division by 0 is not possible, as such ex[pressions such as  3/0  and 0/0 are undefined.

Irrational Numbers: There are also some Real Numbers such as √2, which cannot be expressed as a ratio of integers and are, therefore, called Irrational Numbers. It is possible to show that the following are also irrational numbers;

     √3      √5     ∛2      π       3/π²

Having decoded all the above issues let's go back to the Real Line. You can see that the Real Numbers can be represented by points on the Real Line, The positive direction (towards the right of the zero) is being indicated by the arrow. An arbitrary point, on the line, '0' is referred to as the Origin and it represents the real number "0". 

You can choose any unit of measurement..... every positive number, say x, is represented by the point (on the real line) at a distance of x-units to the right of the '0', similarly each negative integer -x is represented on the left of '0' as per its value. 

Having understood all the points above, you may have noticed that any Number associated with a value "P" can be referred to as co-ordinate of P and also the line can be called a coordinate line  OR a Real Number Line or in a more simple expression a Real Line. 

For all practical purposes one can identify the point with its coordinate and consider the number as a point on the Real Line. 

Engineering Mathematics: Basics

Mathematics is the foundation of Engineering , it is not possible to develop a clear concept of subjects such as Physics, Strength of Materials, Machine tools, Vibration and controls, Thermodynamics, Electricity, Fluid Mechanics etc. unless you know Mathematics like the back of your hand. 

In order to conceptualize mathematics you must get back to the basics. Although these are the topics you must have learnt at your school levels but you need to revisit the same.

To start with we will take up the concept of  Number System so that you can get the hang of Analytical Geometry, Trigonometry, Calculus etc. which will be posted subsequently.

The Number System 

The Number System, in brief, consists of the the following:

1. Natural Number: N = {1,2,3,4,5,....} 
2. Whole Numbers: W= {0.1,2,3,4.....}
3. Integers: Z = {......-3,-2.-1,0,1,2,3...}
4. Rational Numbers: Q= { p/q } Rational Numbers are ratios of Integers (provided the denominator p  is not zero!)
5. Irrational Numbers: P , An Irrational Number is a decimal which never repeats nor terminates (will be explained in details later)
6. Real Numbers: R= {Possess a decimal representation}, will also be explained later.
7. Complex Numbers: C = Are also called Imaginary Numbers.

The Number Line, for better understanding of integers:

Number Line

The Number Line can also be used for showing the decimal, the way it is shown below:

Number Line Showing Decimal 

The Pythagorean Theorem: Its application and discovery of Irrational Numbers!

The Pythagorean Theorem,  is probably one of the earliest theorems known to our civilizations. The theorem is named after Greek mathematician and philosopher, Pythagoras. He is credited with many a contributions to mathematics, it is quite likely some of the work may have, actually, been the work of his students.

The Pythagorean Theorem is a statement about triangles containing a right angle. The Pythagorean Theorem states that:

"The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides." 
                                      

The point which needs to be understood clearly is the fact that during the era of Pythagoras the concept of Algebra didn't exist. The theorem was proved by using Geometry, the Geometrical representation used was something similar to the one shown above.

Applying the Pythagorean Theorem some of the problems solved are: 

If A= 3;  B = 4;   then C= 5, similarly, some more are
5 12 13
7 24 25
9 40 41
11 60 61

  (These are known as Pythagorean Triples all of them are whole Numbers, positive integers)

Pythagoras later discovered that the Square Root of 2 cannot be expressed as a ratio of two integers. This fact disturbed Pythagoras and his followers to a great extent. For them mathematics was a sacred issue, they were possessed , in a way, with the belief that any two lengths were integral multiple of some Unit-Length. They considered this as the threat to their belief and made many attempts to suppress the fact that the Square Root of 2 was an Irrational Number.

The importance of Pythagorean Theorem cannot be overlooked. We must realise that during that era, when it was discovered,  it must have enabled people solve problems of heights and distance.  The development of the concept of Trigonometry, later took over, as a tool for solving heights & Distance problems thus making Pythagorean Theorem redundant, so to say. 

The discoveries of Pythagoras laid the foundation for development of the concept of Rational and Irrational Numbers to start with and then moving on to development of the concept of Imaginary Numbers. The students will appreciate the importance of these developments as we go forward and unravel the utility of the above concepts and principles.

Angle and its importance in the field of Engineering

Continuing from the previous post ..........

The definition of an Angle, its various theorems and corollaries must have been taught to you before you reached the High School level. The purpose of this post is to help you grasp the importance of "Angles" vis a vis its vast ranging Engineering  Applications. There are innumerable machine components which couldn't have been designed or manufacture without the knowledge or use of Angle. For example even the most basic tool such as an Axe can neither be designed nor can it be manufactured  unless you specify the sharpness of its cutting edge, which, in engineering term, is called the Angle at which one side is inclined to the other side. 

As an Engineer ,you would be making use of  Angle, along with Straight Lines, virtually in everything you design . The sketch of the Ramp from the previous post is being reproduced, for reference sake, hereunder: 

This is a  Right Angled Triangle. You may recall the concept of  Trigonometry is built around the  Right Angled Triangle.  For the present Trigonometry is being put on hold to discuss the Pythagoras Theorem. 

Pythagoras
Much before the advent of Trigonometry, the Right Angled Triangle was used by the Greek  Philosopher and Mathematician Pythagoras (Era 500 BC). He proved that the three sides of a Right Angled Triangle hold a relationship with one another,  which was proved by him using Geometry. The proof of the Pythagorean Theorem that was inspired by a figure in the book  Vijaganita, (Root Calculations), by the Hindu mathematician Bhaskara. Bhaskara.

The Images of the relic of pythagoras's work, for those who would be curious to see, are depicted below: 

Relic showing how the theorem Graphics

The graphics of the Pythagoras theorem is also being depicted here-under, the graphic presents a picture similar to the one used in the Relic.

How Inventions of the Stone-Age became the Foundation of the Principles of Engineering

In the previous posts,  the the history of Mechanical Engineering had been touched upon. There is a need to revisit the invention of the first ever machine, known as a tool now a days, to get the depth of basic principles upon which the foundation of Mechanical Engineering has been built and such a tremendous growth has come about, 

You have already seen this picture of a tool made from stone. The same is being reproduced here with a white boundary  drawn along its borders indicating its resemblance to an axe. This is an axle without a customary handle.  

This Axe like tool must have been made by using some stones, which had been harder, in order to cut this particular stone.  The key to the success of this invention was the sharpness  of the cutting edges. This invention was not a result of well planned experiments or any kind of research behind it. This must have been achieved after so many trials and errors. The basic purpose behind this invention was to produce a Sharp-Edge. 

The very idea of Sharp-Edge bring us to the concept of Angles. The invention, therefore, becomes immensely significant. You will appreciate, without the concept of Angle,  the evolution of initially geometry and subsequently trigonometry would not have been possible. 

It is very important to clearly understand and ingest in our minds the  concept of angles or slopes or gradients, whatever you call it. Without the clear concept of an angle or a slope it is not possible to understand Mathematics, Physics and Engineering Drawing. The simplest example of the use of  the concept of an angle is a ramp. The figure of a ramp is shown below:  

The ramp essentially consists of a base, also called Run, then there is the height also called Rise and finally there is a line joining the two end which is known as the Slope.

Complex machines basic concept

Complex Machines: 

Complex machines  combination of two or more Simple Machines, they are also known as Compound Machines. Some of the examples of Complex Machines are:

• Bicycle
• Lawn Mower 
• Wheel Barrow 
• Screw Jack 
• Shopping Trolley

Few Examples of Complex Machines

Hand Cart: A combination of two Wheels mounted on an Axle mounted on a platform. A hand cart enables you to carry heavy loads even if you are alone thus making your work easy. This type of Carts are now a days used to carry materials to short distances only. 

Bullock Cart: These are similar to the Hand Cart but they can be pulled using two oxen or buffaloes. These are getting extinct due to the advent of engine driven Trolleys etc. You may still be able to see them in operation in developing countries e.g. India, Pakistan, China etc. it is a real fun to watch such simplistic mechanisms in action. 

Lawn Mower: These are more complex than the Hand Cart or the Bullock-Cart. The mechanism of motion is same as that of a hand cart but it has a rotating drum with blades, attached to the wheel with a Chain & Sprocket arrangement, which enables us to mow the lawn by pushing the Lawn Mover. The Blade-Drum rotates as the Lawn-Mover moves thereby cutting the grass. 

Screw Jack: This is the simple Screw & Nut arrangement mounted on a housing. The Nut is rotated with moves the Screw up. This linear movement of the Screw is utilized fof lifting very heavy loads. 

Machines - definition & concept

The moment you think about Mechanical Engineering the picture of some kind of a machine emerges in your mind, be it your own bicycle, your mother’s sewing machine or a washing machine or your father’s bike or a car. These are all machines.

Let’s try to find out what a machine is! If you get hold of a dictionary and see the definition of a machine you would find the following definitions in general:

•  A machine is a device consisting of fixed and moving parts that modifies mechanical energy and transmits it in a more useful form
• An assembly of interconnected components arranged to transmit or modify force in order to perform useful work

Machines can be broadly divided into two major categories  :   

1.  Simple Machines  
2. Complex Machines

Basic Simple machines are: 

1. An Incline: Enables us to push a heavy load to the desired height in absence of a Crane

2. Wedge: Enables us to split an object into two or create a gap between two bodies.

WEDGE

3. A Lever: A rigid bar enables us to lift or move heavy load with small effort.

Simple LEVER

4. Screw Thread: Enables conversion of  rotational movement to linear movement.

SCREW-THREADS

5. Wheel an Axle: combination of a disc (wheel) & rod (Axle) for obtaining easy motion.

WHEEL & AXLE

6. Pulley: A  Grooved Wheel, fitted on a Pin, enabling us to lift heavy loads with small effort.

SIMPLE PULLEY

About Mechanical Engineering

History of Mechanical Engineering:

The evolution of Mechanical Engineering can be traced back several thousand years.  The development of various machines might have happened throughout the world simultaneously.  Since there was no systematic documentation process in the earlier times, the records of these developments are were not maintained. 

The exact date or period of emergence of Mechanical Engineering is not known, one can only imagine it started during the industrial revolution in Europe in the 18th century. Moreover the developments in the field of Physics must have provided the impetus to the growth of Mechanical Engineering in the 19th century.

Evolution of Tools and machines: 

We all know "necessity is the mother of invention" , accordingly as the need arose the stoneage man must have invented the first ever machine (tool) which might have looked something like this:

Cutting tool made from Stone

After some more development tools(machines) may have come into existence. 

Some Tools made from Stones

Invention of WHEEL is considered as a landmark in the growth of Mechanical Engineering.

WHEEL - revolutionized development

Invention of  Boat must have  had a huge impact on the development. 

Primitive Boat

One of the later Landmarks in the development of Mechanical Engineering was development of an Aircraft! 

First flight of the First Aircraft

In due course of time the various aspects of Mechanical Engineering also started getting systematically documented, leading to the system of printing books.   You will appreciate if such a system wasn't developed and nurtured, we couldn't have books. The availability of the books brought about the concept of formal education leading to the building up  schools and then universities. The education system brought about the massive development in the whole world. There are so many Universities all over the world  offering good Engineering courses.

 There are various diploma and degree courses in many branches of Engineering these days. Mechanical Engineering being one of them. 

Formal courses of Mechanical engineering comprise:

• Mathematics (mainly calculus, differential equations, and linear algebra) 
• Physics and chemistry  
• Statics and dynamics 
• Strength of materials 
• Materials Science 
• Thermodynamics (heat transfer, energy conversion etc.) 
• Internal combustion engines 
• Fluid mechanics 
• Machine design (including kinematics and dynamics) 
• Metrology (measurement, Limits and Fits) 
• Manufacturing engineering i.e. Production Technology  
• Vibrations and controls   
• Hydraulics and Pneumatics 
• Engineering design and product design 
• Engineering Drawing including CAD (computer aided drawing) 
• Mechatronics and robotics