Mathematics that I learned – By Angela

NATURAL NUMBERS

Natural numbers – We know , we use 1, 2, 3, 4,……when we begin to count. They come naturally when we start counting . Hence, mathematicians call the counting numbers as natural numbers.

How to add natural numbers

To add a natural number of  any digit write the number below one another. We can write the number in any order.  Place the numbers , in such a way that the ones place is matching.Then add till the number of places.

Look at the picture below to add a natural number.

How to subtract natural numbers

To subtract a natural number of 1 digit , 2 digit , 3 digit or any digit write the biggest number first then the smallest number. Write the number below one another .Place the numbers , in such a way that the ones place is matching. Now subtract till the number of places.

FACTORS

WHAT IS FACTOR

If we divide a number with any number and if we get the reminder 0 that number is the factor of the first number.

HOW TO FIND A FACTOR OF A NUMBER

To find a factor of a number we should divide the number with 1 . Then we should divide that number with 2 and if the remainder is 0 , that number is the factor of 2. Continue this process till the half of the number.

WHAT IS PRIME NUMBER

Prime number are numbers that have 2 factors. Those factors are 1 and the number itself.

WHAT IS PRIME FACTOR

The factors of a number  with the prime number is called the prime factors of the number.

WHAT IS PRIME FACTORIZATION ? HOW TO FIND OUT ?

The method using to find the prime factors of a number is called prime factorization of the prime factors of a number .

There are two methods to do prime factorization . They are                                           1 . Factor tree                                                                                                                             2. Prime factorization

  FACTOR TREE                                                                                                                       

PRIME FACTORIZATION 

HOW TO FIND FACTORS OF A NUMBER USING PRIME FACTORS

To find all factors of a number multiply all possible combination.

Now look at the examples shown below.

 

LCM – LOWEST COMMON MULTIPLE

Method 1 –  In this method, first we have to write the numbers as shown in the fig. . Now we can select any smallest number which should be at least divisible with any one of the number. Now we can divide each number with the divisor. If the number is divisible with divisor, then we can write the quotient bellow that number. If the number is not divisible then we can directly bring down the number itself. Now repeat the process as shown in the figure until all the numbers reach 1. Then by multiply all the divisors we will get the LCM.

HCF – HIGHEST COMMON FACTOR

Method 1 – In this method first write the numbers as shown in the given fig.Now we should write the least possible number which should be divisible with all the numbers. Repeat this process till there is no more possible number that should be divisible with all the number.Then multiply all the divisor to get the HCF. If there is only one divisor , that divisor is the HCF of all the number.

CO – PRIME NUMBER

Two numbers that have 1 as the only common factor is known as co – prime numbers.

TWIN PRIME NUMBERS

Two consecutive prime numbers with only 1 composite number is called twin prime numbers

Example : 11 , 13

PRIME TRIPLE

Three consecutive prime numbers with only 1 composite prime number is called prime triple.

Example : 3, 5, 7

Divisibility rule of numbers 1 to 11

Divisibility rule of numbers 1

Number 1 is divisible with all numbers.

Divisibility rule of numbers 2

If a number is an even number then it is divisible by 2. Or if the last digit (ones place) of the number is 0,2,4,6,8, then it is divisible by 2.

Example : 24 , 32 , 50 .

Divisibility rule of numbers 3

The given number is divisible by 3, then the sum of the all digits if the number will be the multiple of 3.

Example : 54 = 5 + 4 =9

Divisibility rule of numbers 4

A number with 3 or more digits is divisible with 4, then the number formed by it last 2 digits (once and tens place) will be divisible with 4.

Example : 524

Divisibility rule of numbers 5

The number which has either 0 or 5 in its once place is divisible with 5.

Example : 235 , 350

Divisibility rule of numbers 6

If a number is divisible with 2 and 3 both then the number is divisible with 6 also.

Example : 24                                                                                                                               Last digit is 4 . It is divisible by 2.                                                                                        2+4 = 6 . 6 is divisible with 3.                                                                                                therefore , 24 is divisible by 6

Divisibility rule of numbers 8

A number with 4 or more digit is divisible by 8, if the number formed by its last 3 digit is divisible by 8.

Example : 3248

Divisibility rule of numbers 9

If the sum of the digits of a number is divisible by 9 then the number itself is divisible by 9.

Example : 99 = 9+9=18

Some more divisibility rules
  • If a number is divisible by another number, then it is divisible by each of the factors of that number.

Ex : 24 is divisible by 8 . It is divisible by each of the factors of 8. Factors of 8 are 1 , 2 , 4 and 8. So 1,2,4 and 8 are also the factors of 24.

  •  If a number is divisible by two co-prime numbers, then it is divisible by it product also.

Ex :The number 80 is divisible by 4 and 5 . It is also divisible by 4 multiplied by 5 = 20 . 4 and 5 are co – prime numbers.

  • If two given number are divisible by a number, then their sum is also divisible by that number.

Ex : The numbers 16 and 20 are both divisible by 4 . The number 16 +20 = 36 is also divisible by 4.

  • If two given number are divisible by a number, then their difference is also divisible by that number.

Ex : The numbers 35 and 20 are both divisible by 5 . There difference is 35 – 20 = 15 . It is divisible by 5.

BASIC GEOMETRICAL IDEAS

Point

Point is the basic unit of geometry. it have no length breadth and no thickness. That means point have no dimensions. It can be represented by dot and can be called as dotA.

LINE 

a line is a collection of points going endlessly in both directions along a straight path.

Plane

A plane is a smooth flat surface which extends endlessly in all directions. It does not have boundaries. A plane have only length and breadth but no thickness.

Line segment

A line segment is the part of the line with 2 endpoints. It can also define as the shortest distance between 2 points. (line segment PQ).

Ray

Ray is the part of the line with one start point and extended endlessly in one direction.

Angle

Interior of an angle

The space with in the arms of an angle is called interior of the angle.

Exterior of an angle

The space outside the arms of an angle is called exterior of an angle.

Adjacent angles

Two angles which have a common arm,a common vertex, and the non common arm lie on either side of common arm are called adjacent angles.

curves

a curve is a collection of points going endlessly in both directions along a straight or curved path.

Simple curve

Any curve which does not cross itself and can be draw without taking the pencil is called simple curve.

Open curve

Any curve which have a start point and an end point then it is called open curve.

Closed curve

Any curve which have no start point and no end point then it is called closed curve.

Different spaces in a closed curve

  1. Interior of the curve (inside )
  2. Boundary (on the curve)
  3. Exterior of the curve (outside)

The interior and boundary together in a closed curve is called the region

Polygons

The simple closed curve made up with line segment is called polygon

Side, Vertex, Diagonals of a polygon

The line segments used to form a polygon is called side. 

The common points of two sides (line segments) in a polygon is called vertex.

The line segment between two opposite vertex of a polygon is called diagonal of the polygon.

Adjacent vertex

If two vertex of a polygon formed by at least one common side is called adjacent vertex.

Opposite vertex

If two vertex of a polygon formed by no common side is called opposite vertex.

Addition, multiplication, division and subtraction of rational numbers / fractional numbers

Division

If we have decimal places in the dividend, we can divide the dividend with divisor with out decimal places and move the decimal point from right to left as per the number of places in the dividend.

If we have the decimal places in the divisor, we can divide the dividend with divisor without decimal places and move the decimal point in the quentiont from left to right  as per the number of decimal places in the divisor.

example 652/3.2

Different types of angles



An Angle can formed when two rays originate from the same endpoint.  The rays making angle are called the arm of the angle. angle can also form by intersecting two lines or line segments. We can see the various type of angle in the fig-1. Angle can be represent by ∠ABC. Where  AB and BC are the arms of the angles and B is the vertex of the angle.

Complementary angle : If the sum of 2 angles are equal to 90°, then that angles are called complementary angles.

Supplementary angles : If the sum of 2 angles are equal to 180°, then that angles are called Supplementary angles.

Adjacent angles: If two angles are adjacent angles then they have a common vertex .a common arm and there non -common arm are the different side of the common arm. In the below figure-2 we can see a adjacent angles.

Here the angle ∠ABD and ∠DBC are adjacent angle. But also note that ∠ ABC is not adjacent angle. More over when there is two adjacent angle, then the sum is always equal to the angle formed by two non-common arms

that is  ∠ABD+∠CBD=∠ABC

Linear pair of Angle

If the non-common arm of an adjacent angle are opposite ray the angles are called linear pair of angle. As shown in the fig3, AB and BC are non common arm and the angle ABD and angle CBD are linear pair of angle.

From the fig3 we can see that ∠ABD+∠CBD=∠ABC (adjacent angle).  And ∠ABC =180° as it is straight angle. So from the above statement we can make the axiom 6.1 as follows

If a ray stand on a line, then the sum of two adjacent angle so formed is 180°

The opposite (converse of each other) of the axiom 6.1 can be state like bellow.

If sum of two adjacent  angle is 180° then the non common arm of the angle form a line.

This two axioms together called linear pair axiom.

Facts about lines / definitions of lines



Lines/(straight line)

Eventually distributed points in any direction can called  line.

Eventually distributed points in particular direction  can called   straight line.

In this section we are discussing about the straight line only.

As shown in the fig1-a A line has no end. It can be extended in either side endlessly. It can denoted by as shone in the fig.

As shown in the figure 1-b a line segment have a start and end point.  It can be denoted as shown in the figure.

A ray have a start point but don’t have an end point.  it can also be denoted by as shown in the figure 1-c.

But in normal case the line, line segment and the ray are denoted by AB only. And the  length (measure) of a line segment is also denoted by AB.

Parallel lines are two lines whose every point have equal perpendicular distance each other. it can be see in the figure 1-d.

Intersetting ling are two lines which a common point.  An intersecting line will also create angles on intersecting point. We can see an example of intersecting line as shown in the fig-e.

Line can also denoted by a single letter like m,n or l.

If three or more points lie on a same line, then this points are called collinear points. Otherwise they are called non collinear points.

Algebraic identities or equations



  1. (x+y)(x+y) = x²+2xy+y²
  2. (x-y)(x-y) = x²-2xy+y²
  3. (x+y)(x-y) = x²-y²
  4. (x+a)(x+b) = x²+(a+b)x+ab
  5. (x+y+z)² = x²+y²+z²+2xy+2xz+2yz
  6. (x+y)³ = x³+y³+3x²y+3y²x  or  x³+y³+3xy(x+y)
  7. (x-y)³ = x³-y³-3x²y+3y²x  or  x³-y³-3xy(x-y)
  8. x³+y³+z³-3xyz = (x+y+Z)(x²+y²+z²-xy-xz-yz)
  9. x³+y³ = (x+y)(x²-xy+y²)
  10. x³-y³ = (x-y)(x²+xy+y²)
  11. if x+y+z = 0 then x³+y³+z³ = 3xyz

Equations for surface area and volume of geometrical shapes



surface area

Area of rectangle = l x b -length x birth

Area of the circle = πr²

Surface area of cuboid = 2( lb+bh+hl) where l= length, b= breth h = height

Surface area of cube = 6a² where a= edge of the cube

Lateral Surface area of cube = 4a² where a= edge of the cube

Curved surface area of a cylinder = 2πrh where r= radius of the cynder, h= height of the cylinder. 

Total surface area of the cylinder =  2πr(r+h) where r= radius of the cynder, h= height of the cylinder.

Curved surface area of cone = πrl, where r= radius of the cone, l is slant height of cone.

Total surface area of a cone = πr(l+r), where r radius, l= slant height

if we know height and radius of the cone we can find the slant height (l) of the cone by  l= √(r²+h²)

Surface area of a sphere = 4πr², where r is the radius of the sphere.

Curved surface area of a hemisphere (half sphere) = 2πr² 

Total surface area of a  hemisphere (half sphere) = 3πr² 

Volume

volume of cuboid = l x b x h where l= length, b= breth, h= height

Volume of the cube = a³ where a is the edge of the cube

Volume of the cylinder = πr²h where  r= radius, h = height

Volume of the cone = 1/3(πr²h) where r= radius of the cone , h= height of the cone

Volume of the sphere = 4/3(πr³) where r = radius of the sphere