Fraction

A fraction is part of a whole. If we divide a whole thing, it will be divided into several pieces. Each piece can be seen as fractional part of the original whole thing.

In the above picture a circle is cut into 2 pieces and another circle is cut into 8 pieces. Each part can be said as fraction of the original circle and it can be represented as 1/2 and 1/8. Here 1 is called numerator and 2 and 8 is called denominator.

The numerator represents a number of equal parts while the denominator indicates how many of those parts make up a unit or a whole. For example, in the fraction 4/8, the numerator, 4, tells us that the fraction represents 4 equal parts, and the denominator, 8, tells us that 8 parts make up a whole. The denominator cannot be zero here. So, relation between can represented as :

Numerator

-------------------

Denominator

Proper Fraction and Improper Fraction:

A proper fraction is a fraction where the numerator is smaller than denominator.

Below are examples of improper fractions:
2/4, 3/5, 3/7

An improper fraction is a fraction where the numerator is greater than or equal to the denominator. Below are examples of improper fractions:
9/4, 5/5, 7/3

Mixed number:
A mixed number sometimes also called mixed fraction has a whole number and proper fraction. It is actually sum of a  whole number(not zero) and a proper fraction. This sum (+) symbol is never expressed.
3 4⁄5
It is actually  3 + 4⁄5 and is equivalent to 19/5.

Prime Number and Composite Number

What are prime numbers?
It is a whole number which has exactly two factors. The factors are 1 and the number itself. So, a prime number can be divided evenly by 1 and the number itself.

Example:
2,3,5,7 etc.
Pick a number 3 or 7. 3 can be divided by 1 and 3 itself. It is not evenly divisible by any other whole numbers.

What are composite numbers?
Composite numbers have more than two factors. A composite number has factors in addition to 1 and the itself.

Example:
4,9,18,77 etc.
Pick a number 18. it can be divided by 1,3,6,9 and 18 itself. So, it has more factors.

What is Multiple?

Multiples are numbers that result when we multiply one whole number by another whole number. The first four multiples of 3 are 3,6,9,12
3 × 1 = 3
3 × 2= 6
3 × 3= 9
3 × 4 = 12

We can write multiples of 5 as below:
5 × 1 = 5
5 × 2 = 10
5 × 3 = 15
5 × 8 = 40 

 We can never list all of the multiples of a number. In our example, 3 could be multiplied by an infinite number of numbers to find new multiples.

What is Factor in mathematics?

Factors are whole numbers that can evenly divide another number. Factors give us a way to break down a number into smaller pieces.
Example:
1×12=12
2× 6 =12
3×4 = 12
So, we see that the number 12 can be formed by multiplying these number. All these number(1,12,2,6,3,4) are factor of 12.
A number can have MANY factors!
Finding Factors
We can find the factors for 16 by thinking about numbers that will divide into 16 evenly.
1 is a factor of 16 because 1 can be divided into 16 with no remainder.
16÷1=16
The quotient 16 is also a factor of 16.
2 is a factor of 16 because 2 can be divided into 16 with no remainder.
16÷2=8
The quotient 8 is also a factor of 16.
4 is a factor of 16 because 4 can be divided into 16 with no remainder.
16÷4=4
The quotient 4 is also a factor of 16.
So, the 'Factor' of 16 are 1,16,2,8 and 4.

Googol Number

Hey, it's not Google, it is Googol.

In mathematics, some large number has naming and googol is such one. You may be confused with the word 'Google' - the giant search engine name but Googol, not Google is a number that describe a huge number.

A googol is the large number that equals 1 followed by 100 zeros. In exponent form it written as 10¹⁰⁰.

The term was used in 1920 by a 9 years old boy Milton Sirotta (1911–1981), nephew of American mathematician Edward Kasner. May be it was silly when he named it as googol.

Googol, a quantity that surpasses even the number of hydrogen atoms in the observable universe.

It is a belief that Larry Page and Sergey Brin, the founders of Google, named their search engine after the term googol.

What is Scientific notation?

Scientists and engineers often have to deal with super huge (like 6,000,000,000,000,000,000,000) and super small numbers (like 0.0000000000532) . How can they do this without tiring their hands out? How can they look at a number and understand how large or small it is without counting the digits? The answer is to use scientific notation. If you come to this tutorial with a basic understanding of positive and negative exponents, it should leave you with a new appreciation for representing really huge and really small numbers!

To express large number in short way, scientists have developed a method and this is called Scientific Notation. It is also known as Standard Form. It is generally expressed as

a × 10ⁿ

Here, 'a' is coefficient and a real number whereas n is a integer. The value of  'a' will be 1 ≤ a < 10. So 'a' going to be greater than or equal to 1, and it  will be less than 10.

So, let's see how can we represent large number in scientific notation whether it  is positive or negative.

Suppose the number is 0.0002077. It's scientific notation will be 2.077 x 10^-4. More examples are below.

 245,600,000,000  ---    2.456 × 10¹¹

 13,040,000           ---    1.304 × 10⁷

− 53,000                ---    − 5.3 × 10⁴ 

0.0000000003457      ---    3.457 ×  10^-10  

As you can see, the exponent of 10 is the number of places the decimal point must be shifted to give the number in long form. A positive exponent shows that the decimal point is shifted that number of places to the right. A negative exponent shows that the decimal point is shifted that number of places to the left. 

Most calculators and many computer programs present very large and very small results in scientific notation, typically invoked by a key labelled E.

Units and Conversion

Length
1 Foot = 12 Inch
1 Foot = 30.45 Centimeters
1 Foot = .3048 Meter
1 Yard =  3 Feet
1 Yard = .914 Meter
1 Inch =  2.54 Centimeters
1 Meter = 39.37 Inch
1 Mile =  1.609 Kilo-meters
1 Mile =  1609.34 Meters
1 Mile =  5280 Feet
1 Mile =  1760 Yard
1 Nautical Mile =  1.15078 Mile
1 Kilometers = 1000 Meters

Mass
1 Metric Ton = 1000 Kilogram
1 Kilogram =  1000 Gram
1 Kilogram =  2.20462 Pound
1 Kilogram =  35.274 Ounces
1 Pound = 16 Ounces
1 Pound =453.6 Grams

Volume
1 Gallon = 4 Quart (US)
1 Gallon = 8 Pint   (US)
1 Gallon = 15.77 Cups (US)
1 Gallon = 3.78 Liters

Area
1 Square Kilometer = 0.386 Mile
1 Square Foot = 144 Square inchs
1 Hectare =  2.47 Acre
1 Acre = 0.404686 Hectare
1 Acre = 4046.86 Square Meters
1 Acre = 43560.038 Square feet

Time
1 Hour = 60 Minutes
1 Hour =  3600 Seconds
1 Minute = 60 Seconds
1 Day = 24 Hours
1 Day = 1440 Minutes
1 Day = 86400 Seconds
1 Week = 168 Hours
1 Week = 10080 Minutes

System of Measurement

There various system of measurement in the world. To get an idea of anything in the world, we need to measure it and this measurement is always associated with number and quantities. According to Wikipedia, "A system of measurement is a collection of units of measurement and rules relating them to each other".

There were various measurement systems in various civilizations around the world but, at present, two systems are more stable or popular or agreed and these are Metric System and International System of Units(SI). The International System of Units (SI) is the modern form of the metric system, and is the most widely used system of measurement.

Metric System:
The Metric System was developed by French in 18th century. Previously in Europe, there were other system of measurements in different places but most of the European countries were formulating a unique measurement throughout the country and they were benefiting in trade and business. The inconsistency problem in the size of units of measure was prevalent in France that led them to adopt a completely new system of measurement. A Commission was set up and it submitted a report in the National Assembly and a decimal-based system of measurement  was finally accepted by the leaders. Gradually this system was introduced in other countries specially in non-english speaking countries and it got popularity. US is the only country that has not accepted it as its official system of measurement.
The whole system was derived from the properties of natural objects, namely the size of the Earth and the weight of water, and simple relations in between one unit and the other.

Name	The Number	Power	Prefix	Symbol
trillion	1,000,000,000,000	1012	tera	T
billion	1,000,000,000	109	giga	G
million	1,000,000	106	mega	M
thousand	1,000	103	kilo	k
hundred	100	102	hecto	h
ten	10	101	deka	da
unit	1	100
tenth	0.1	10-1	deci	d
hundredth	0.01	10-2	centi	c
thousandth	0.001	10-3	milli	m
millionth	0.000 001	10-6	micro	µ
billionth	0.000 000 001	10-9	nano	n
trillionth	0.000 000 000 001	10-12	pico	p

The International System of Units (SI):
The International System of Units (SI) is new and it can be said that it is a modern form of Metric System.  It is universally abbreviated SI (from the French Le Système. International d'Unités). This system was published after second world war.
German mathematician Gauss, in 1932,  defined the second as a base units. He measured earth’s magnetic field by three units: millimeter, gram and second. Later in 1860, Maxwell and Thomson formulated the concept of a coherent system of units with base units and derived units.
Finally in 1960, at 11th CGPM(General Conference on Weights and Measures), The International System of Units (SI) was published.
7 base units are:

Quantity	Name	Symbol
Length	meter	m
Mass	kilogram	kg
Time	second	s
Electrical Current	ampere	A
Temperature	kelvin	K
Amount of substance	mole	mol
Luminous intensity	candela	cd

  

Divisibility Rules

If we divide a number by another number and it leaves no remainder then the number is divisible by that number. So if we divide 20 by 5, the result is 4 and the remainder is 0. So the number 20 is divisible by 5.

How can you tell whether a number is divisible by another number without actually doing the division? Is it possible?

Yes, quite possible. This is called Divisibility Rules. These rules let you test if one number is divisible by another, without having much calculation.

We will here show divisibility rules for number 2 to 10. It can be stretched out more but here we will limit it to number 10.

Divisibility by 2:

If the last digit is even (0, 2, 4, 6, or 8) then the number will be divisible by 2. 

168 is divisible by 2 since the last digit is 8. So, 168 ÷ 2 = 84

Divisibility by 3:

The sum of the digits is divisible by 3.

987 is divisible by 3 since the sum of the digit 9+8+7=24 and 24 is divisible by 3. So, 987 ÷ 3 = 329

Divisibility by 4:

The last 2 digits are divisible by 4.

316 is divisible by 4 since 16 is divisible by 4. A bit different is number 100,2500 etc. Here the last 2 digits are 0 and divisible by 4.   So, 316 ÷ 4 = 79 and 2500 ÷ 4 = 625

Divisibility by 5:

The last digit is 0 or 5. 

55 is divisible by 5. So, 55 ÷ 4 = 11. 

2500 ÷ 5 = 500

Divisibility by 6: 

It is divisible by 2 and by 3.

2826 is divisible by 6 since it is divisible by 2 and 3. The last digit is 6 and it is divisible by 2 and the sum of all digits are 2+8+2+6=18 that is divisible by 3. So, 2826 ÷ 3 = 942.

Divisibility by 7:

Make double the last digit and then subtract it from the rest of the other digits. If the result is divisible by 7 then number is divisible by 7.

Suppose 756. Here the last digit is 6. If we double this digit then it is 6x2= 12. Now subtract this 12 from rest of the digit. It is 75-12=63 and 63 is divisible by 7. So, 756 is divisible by 7.

756 ÷ 7 = 108.

Divisibility by 8:

The last three digits are divisible by 8. 

Take a 11296 and check whether the last 3 digits are divisible by 8 or not. The last 3 digits 296 is divisible by 8 (296 ÷ 8 = 37), So 11296 is divisible by 8. 11296 ÷ 8 = 1412.

Another way is add the last digit after doubling the rest other digits.

Suppose 816. We will separate the last digit 6 and double the rest other digit 81 then 81x2=162. Now add 6 with 162 and it will be 162+6=168 that is divisible by 8. So, 816 is divisible by 8. 816÷ 8 = 102.

Divisibility by 9:

The sum of the digits is divisible by 9.

162 is divisible by 9 since the sum of the digits is 18 1+6+2=9, and 9 is divisible by 9. So, 162 ÷ 9 = 18. 

Divisibility by 10:  

The number ends in 0 i.e the last digit will be 0.

330 is divisible by 10 since the last digit is 0. 330 ÷ 10 = 33.    

For practicing in computer programming, such rules can be a handy tools to find out factors or multipliers.

What is MVC?

If you are in software development sector and specially in web development, surely you have heard a common phrase 'MVC' which means Model-View-Controller.

MVC is a software architectural pattern where the development is divided into 3 parts - the Model, the View and the Controller. The total software application is separated in a way that data processing or manipulation is done in Model part, User interface to view  and modify data is done in View part and  logical operation on data is done in Controller part.

MVC Components

• The Model represents the data structures. Model section maintain data. Typically  model classes contain functions that retrieve, insert, and update information in database.
• The View presents data to the user. A View is a interface which help user to interact with the data.
• The Controller does all the logical operations. It serves as an intermediary between the model and view.

MVC was probably first used in Smalltalk(object-oriented, dynamically typed, reflective programming language.)