Roman Number

Roman number was introduced and used by Roman people.

There is no zero(0) symbol in roman number.

The chart of roman numerals :

I -1        II -2 III - 3 IV - 4 V - 5 VI - 6 VII - 7 VIII - 8 IX - 9 X - 10

XI - 11 XII - 12 XIII - 13 XIV - 14 XV - 15 XVI - 16 XVII - 17 XVIII - 18 XIX - 19 XX - 20

XXX - 30

XL - 40

L - 50

LX - 60

LXX - 70

LXXX - 80

XC - 90

C - 100

D - 500

M - 1000

Proportion

We know that ratio is comparison of two numbers and if two ratios are equal then it is called Proportion. So, Proportion says that two ratios are equal.
Though ratio and proportion are different terms for the same mathematical concept but they are often used together as a phrase, "ratio and proportion". 8/10 = 12/15
Here, the common ratio is 4/5.
Do the ratios 18/9 and 2/1 form a proportion? (Answer it by yourself)

#Take another example of word problem so that you can understand it more.
Rob can eat 10 cakes in 60 minutes. In this way, how much time will it take to eat 35 cakes?
Let's think, it will take M minutes. Then
10/60=35/M
10M= 60 × 35
10M=2100
M= 2100/10

M=210

So, it will take 210 minutes to eat all the 35 cakes.

#You are making lemonade. The amount of sugar you need depends on the amount of lemonade mix you use. Is the amount of sugar you need proportional to the amount of lemonade mix you use?

Cups of sugar	1/3	1	3
Packets of lemonade mix	1	3	9

For each column in the table, let's write the ratio (in simplest form) of cups of sugar to packets of lemonade mix.
1/(1/3) =3 3/1=3 9/3=3
So, the answer is yes.

Ratios

We often compare two things and this comparison is called 'Ratio' in mathematics. A ratio compares two different quantities. So, it also a relation between two quantities. 

For example, there are 10 instructors and 30 girls in a hall room. So the ration between instructor and girl is 10 to 30 and in simplification it is 1 to 3 which means for every 3 girls there is one instructor.

Here are a few different ways we can describe the ratio of instructors to girls:

• There is 1 instructor for every 3 girls.
• The ratio of instructor to girls 1 to 3.
• The ratio of instructor to girls 1/3
• The ratio of instructor to girls 1:3

Ways to write ratios:

Here are several ways to describe the same ratio:

-  6 to 11

-  6:11

-  6/11

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.

Functions and Equations

An equation is a statement where two expressions are equal. Here left and right side will be equal. 

Suppose x + 3 = 10. For a particular value x, the equation will be satisfied. If x=7 then 7+3=10. The left and right side are equal.

Function is a bit different. It is a relationship between two variables. Function is written as f(x)=x+3. It takes a set of value as input and return a particular output. The input variable f(x) is called Independent variable because it takes input as f(0),f(1),f(2) etc. and returns corresponding output as 3,4,5 etc. x + 3 is dependent on f(x) that's why x+3 is called as Dependent variable.

In equation, we need a particular value of variable to make it equal where function takes a set of  permissible input and returns regarding output.

What is Function in Mathematics?

A function in mathematics is just like relation between input and output. So a function takes input and returns an output. It is denoted as f.  Most of the time, you see it in below form.

f(x) = x² + 1 

For a given value of x, you will get the result of  x² + 1.

Value of x	Output
0 1 2 3 4	1 2 5 10 17

We see that for a particular value of variable x, we get a corresponding output. This phenomena is called function in mathematics.

We shall denote function of x as f(x), F(x), φ(x) etc.

The number x that we use for the input of the function is called the 'Argument' of the function. It may seems to us that we can pick any number as argument  and for this reason we can call it 'Independent Variable' and the output of the function, e.g. f(x), f(3), etc. depends upon the argument, can be called the 'Dependent  variable'.
We can draw a picture of a function on graph by taking argument-value pairs of the function and describing each by a point in the plane, with x coordinate given by the argument and y coordinate given by the value for that pair.

Word Problem of System of Equation

Q. Jimmy is 12 years older than Brandon. 17 years ago, Jimmy was 4 times as old as Brandon. How old is Brandon now?

Suppose Jimmy's current age is J and Brandon's age is B. Jimmy is 12 years older than Brandon then
J = B + 12
17 years ago Jimmy was 4 times as old as Brandon.
So, 17 years ago Jimmy's age was J-12 and Brandon's age was B-17 then
J-17 = 4(B - 17)
Put J = B + 12 in above equation.
B + 12 - 17 = 4( B - 17 )
B - 5 = 4B - 68
3B = 63
B = 21
Brandon is 21 years old.

Q. Your class has 40 students and some want to watch movie and some want to watch stage drama. The cost of movie ticket is 20$ and stage drama is 10$. Total cost of ticket is 500$. How many students went to watch movie and how many want to watch stage drama.

M= Number of student to watch movie
S = Number of student to watch stage drama

We know that total student is 40. Then
M+S = 40 
We also know total cost of ticket is 500$. Then
20M + 10S = 500
Put M = 40 - S  in above equation, it will become
20(40 - S) + 10S = 500
800 - 20S + 10S= 500
800 - 10S = 500
10S= 300
S= 30

Put this value in M+S = 40 then M + 30 = 40.
M = 10
30 students went to watch stage drama and 10 students went to watch movie.

Various Forms of Linear Equations(Two Variables)

There are three major forms of linear equations: point-slope form, slope-intercept form and standard form.

Slope-intercept form: 
y = mx + b
where m is slope and b is y-intercept.
Example: y = 2x + 1
Slope: m = 2
Intercept: b = 1

Point-slope form:
y − y₁ = m(x − x₁)
where m is slope and (x₁,y₁) is a point on the line.
Example: y - 1= 2 (x - 3)
Slope: m = 2
x₁ = 1
y₁ = 3

Standard form:
Ax + By = C
Where A,B,C are constants. 
Example: y + 3x= -10
A = 1
B =3
C = -10

Number of solutions to a system of equations

System of Linear Equations:
A system of linear equations or system of equations means two or more linear equations that are being solved simultaneously.

What is the solution of a system of equations?

When two equations meet or intersect at a point then this point is called solution of the system.

A system of linear equations usually has a single solution, but sometimes it can have no solution (parallel lines) or infinite solutions (same line). This article reviews all three cases.

One solution:
This is the most common situation. Here, two lines meet at one point.
6x − 2y = 8​
4x + y = −1​​

Since the lines intersect exactly once, the system has exactly one solution.

No solution:
A system of linear equations has no solution when the graphs are parallel.
y = −x + 6​
5x + 5y = 15​​

We see two distinct parallel lines. We can confirm that the lines are indeed parallel, since the slopes of both lines are equal to −1. Since distinct parallel lines don't intersect, we conclude that the system has no solutions. 

Infinite solutions:
A system of linear equations has infinite solutions when the graphs are the exact same line.
​​​​​2x + y = 5​
14x + 7y = 35

The two lines are the same, they intersect infinitely many times. This means that the system has infinitely many solutions.

Solving Linear Equation by Substitution Method

The substitution method is another technique for solving systems of linear equations.

We're asked to solve this system of equations:
2y - 4x = 2
y -x = 4

Now we will find the value of x in second equation:
x = y - 4

Put the value of x in first equation.
2y - 4(y - 4) = 2
2y - 4y + 16 = 2
-2y = -16 +2
-2y = -14
y = 7

Now, put the value y=7 in second equation and we will get
7 - x = 4
x = 3


The solution of the linear system is (3, 7).

Solving Linear Equation by Elimination Method

 Elimination method is a way to manipulate systems of equations in order to solve them algebraically. This is actually very similar to the way we manipulate single equations in order to solve them.

The elimination method is a technique for solving systems of linear equations. Let's walk through a couple of examples.


We're asked to solve this system of equations:
3y + 11x = −10
4y − 11x​​​ = 17
We notice that the first equation has a 11x and the second equation has a −11x term. These terms will cancel if we add the equations together—that is, we'll eliminate the x terms:​​
3y + 11x = −10
4y − 11x​​​ = 17
-----------------
7y + 0 = 7

Solving for y, we get:
7y + 0 = 7
7y = 7
y=1
plugging this value back into our first equation, we solve for the other variable:
3y + 11x = −10
3.1 + 11x = -10
11x = -10-3
11x = -13
x = -13/11
The solution to the system is x=−13/11 and y=1

Not all time this system will work easily. Suppose a system like 4x +2y = 24 and −6x +2y = 4. If you add these two equations together, no variables are eliminated. This time we have to multiply any equation with -1.

4x   + 2y    =    24 

 − (− 6x + 2y ) = −(4)
or
4x   +  2y   = 24  

6x  −  2y    = −4
-----------------------
10x + 0y =20

x = 2

Put this value back into our first equation, we solve for the other variable:

4x + 2y =24

4.2 + 2y = 24

8 + 2y = 24 

2y = 24-8

2y = 16

y = 8

The solution to the system is x=2 and y=8

What is an equation?

What do you notice about each of the following?
11 + 3 = 15
25 - 5 = 20
The right hand side is equal to left hand side.
An equation is a statement that two expressions are equal. For example, the expression 12+3 is equal to the expression 16-1(because they both equal 15), so we can write the following equation:
12+3=16-1

When the left-hand side was equal to the expression on the right-hand side then the equation is called true equations. Let's make sure we understand what a true equation is.
8+2 = 5+5
7-3= 4+1
Here one thing must be clear to you that 5+3 is an expression and it is numerical expression but if the expression includes variable then it is algebraic equation. For example, the equation x+2=6 has a variable in it.

How can we solve algebraic equation?

3 + Z =  17

if Z=14, then 3 + 14 = 17.