The post Number appeared first on TeachifyMe.

]]>The natural numbers include whole numbers except 0.

E.g. 1,2,3,4,5,6…

Positive natural numbers, negative natural numbers along with 0 are called integers.

-1,-2,-3,0,+1,+2,+3…

A prime number is a number with exactly two factors (i.e. 1 and itself (1×3))

E.g. 2,3,5,7,111,13,17,19,23,29,31…

Those numbers which have more than two factors.

E.g. 1,4,6,8,9,10,12,14,15,16,18,20…

The factors of a number are the natural numbers which divide exactly into that number (without a remainder).

**8:** 1×8 **12:** 1×12

2×4 2×6

3×4

Multiples of a number are numbers in its times table.

Multiples of 5 are 5, 10, 15, 20, 25, 30…

** **

It is the highest factor which is common to all of the given numbers.

E.g. **8:** 1×8 **12:** 1×12

2×__4__ 2×6 In this example the common factors are 1, 2, 4 and the

3×__4__ Highest factor is 4; therefore the HCF of 12 and 8 is 4.__ __

In this example the common factors are 1, 2, 4 and the highest factor is 4; therefore the HCF of 12 and 8 is 4.__ __

A rational number is one which can be expressed in the form where a and b are integers and b is not equals to 0.

Numbers that can’t be expressed as as a fraction or a ratio of 2 integers are known as irrational numbers.

These include all rational, irrational, fractions and integers.

These are decimal numbers which stop after a certain number of decimal places.

E.g. 7/8 = 0.875 it stops after three decimal places.

These are decimal numbers which keep repeating a digit or a group of digits.

E.g. 137/259 = 0.528 957 528 957 528 957… the six digits 528 957 repeat in this order.

Numbers which are divisible by 2

E.g. 2, 4, 6, 8…

Numbers which are not divisible by 2

E.g. 1, 3, 5, 7

It is the result of multiplying a number by itself.

E.g. 1^{2}, 2^{2}, 3^{2} ….. 1, 4, 9….

It is the result of multiplying a number by itself 3 times.

E.g. 1^{3}, 2^{3}, 3^{3} …. 1, 8, 27….

The following are few examples showing clearly what is meant by significant figures.

8064 = 8000 (correct to 1 significant figures)

8064 = 8100 (correct to 2 significant figures)

8064 = 8060 (correct to 3 significant figures)

0.00208 = 0.005 (correct to 1 significant figures)

0.00208 = 0.0021 (correct to 2 significant figures)

3.00508 = 3.01 (correct to 3 significant figures)

The following are few examples showing clearly what is meant by decimal places.

0.0647 = 0.1 (correct to 1 dp)

0.0647 = 0.06 (correct to 2dp)

0.0647 = 0.065 (correct to 3dp)

2.0647 = 2.065 (correct to 3dp)

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]]>The post Set Language And Notation appeared first on TeachifyMe.

]]>A set is a collection of objects, things or symbols which are clearly identified.

The individual objects in the set are called the elements or members of the set.

Elements may be specified in two ways:

- By listing the elements
- By description

E.g.

**Listing Description**

{2, 4, 6, 8, 10} The set of even numbers between 1 and 11

{a, e, i, o,u} The set of vowels in the alphabet

{} These braces stand for the word “the set of”

e.g. {Even numbers between 1 and 11}

Usually, we use Capital letters to denote a set and small letters to denote members of the set.

**n ( )** indicates the total number of members in a set.

E.g.

A= {1, 3, 5, 7, 9, 11} n (A) = 6

B= {2, 4, 6, 8} n (B) = 4

Є is an element of (is a member of ) (belongs to)

is not an element of (is not a member of) ( does not belongs to)

E.g.

A = {1, 3, 5, 7, 9, 11}

Set in which all the elements can be listed.

A= {1,3,5,7,9} n(A) =5

B= {days of the week beginning with T} n(B) = 2

Sets in which it is possible to list all the members of a set.

C= {2, 4, 6, 8,10….}

E={x:x is a natural number}

The set which contains all the elements.

All proper subsets formed within the universal set draw their elements from the available elements of the universal sets.

(Ƹ element – A element} = A**`**

If Ƹ = {2, 3, 5, 7, 11, 13} and A = {2, 3, 7, 13}

A**` **= {5, 11}

Two sets A and B are said to be equal if and only they have exactly the same elements.

Two equal sets are also subsets (denoted by __C__) of each other.

A={2,4,6,8} B={8,6,2,4}

Then

A=B B=A A __C__ B B __C__ A

When each member of a set A is also a member of a set B, then A is a subset of B.

C** is a subset** **of :**When two sets have exactly same elements or elements in the first set are also elements in the second set.

**is not a subset** **of:** There is at least one element in the first set that does not belong to the second set.

When each member of a set A is also a member of a set B, but set B has MORE elements than set A, then set A is a proper subset of B, denoted by “A C B”. Therefore, in this case set B is not a proper subset of A (B A)

**C is a proper subset of:** When each element in the first set also belongs to the second set, but the second set has more elements than the first set.

**C is not a proper subset**: When there is at least, one element in the first set that does not belong to the second set.

E.g. A = {1, 5, 9} B = {1, 3, 5, 9}

A set which contains **NO** elements

An empty set in a subset of any set.

Common elements in different sets.

A= {1,2,3,4,5,6}

B= {2,4,8,10}

**Union Of Sets: U**

The Union of set A and set B is the set of all elements which are in A, or in B, or in both A and B. It is denoted by ‘A U B’ and is read as “the union of A and B”.

A= {1,3,4} B={6,7,8}

A U B = {1,3,4,6,7,8}

If the two sets have No element in common then the two sets are called disjoint.

The intersection of two disjoint sets is null or empty.

e.g. A = {1,3,5,7} and B = {2,4,6,8,9}

A**B = thus A and B are disjoint sets.**

**De Morgans Law:**

In a venn diagram, we use a rectangle to denote a universal set Ƹ and a loop such as circle or an oval to represent any set in Ƹ .

**Examples of Venn Diagrams:**

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]]>The post Mensuration appeared first on TeachifyMe.

]]>**Area:** 2 × Length

**Perimeter**: 4(Length) or ( L+L+L+L )

**Area:** L×B

**Perimete**r: 2(L+B) or (L+L+B+B)

**Area:** × Base × Height

**Perimeter: **Sum of all three sides.

**Area:**

**Circumferenc**e:

**Area:**

**Perimeter**:

**Area:** base × height

**Perimeter:** 2 (a+b) or Sum of all sides.

Area: (a+b) h

Perimeter: Sum of all sides

Surface Area: 6L^{2}

Volume: L^{3} or L × L × L

** Surface Area**: 2 (BL+BH+LH)

**Volume:** L × B × H

**Surface Area:** (Perimeter of the base area (p) + 2 (base area))

**Volume:** Base area × height

For **Hemisphere:**

**Surface area:**

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]]>The post Matrices appeared first on TeachifyMe.

]]>Null Matrix is that matrix, that only contains number 0 in it.

Also known as square matrix, in which all element zero except the diagonal upper left to lower left.

The elements in the diagonal are one’s only.

Order = Number of Rows * Number of Columns

**Example:**

Order = 3*2

Order = 2 * 3

Matrices of the same order are added (or subtracted) by adding (or subtracting) the corresponding elements in each matrix.

Adding A + B: Subtracting A – B:

*Rules:*

**A+B = B+A :** A and B can change position when adding.

**(A+B) + C = A + (B+C)**: order of operation bracket first.

**A – B ≠ B –A:** A and B should not change positions when subtracting.

Example:

Example:

If two matrices A and B are of the same order and their corresponding elements are equal, then A = B.

Matrices can only be multiplied only if they are compatible. They are compatible when the number of rows of the second matrix is the same as the number of coloumns of the first matrix.

*Rules:*

**AB ≠ **BA : A and B should not change positions

**(AB)C = A(BC):** If 3 or more matrices you can choose whichever 2 to multiply first.

Example:

Determinant A = ad – bc

A^{-1 = }

*Remember:*

- If Determinant = 0 then the matrix has no inverse.
- Multipying by the inverse of the inverse of a matrix gives the same result as dividing by the matrix.

E.g.

If AB = C

A^{-1}AB = A^{-1}C

B= A^{–}^{1}C

**Example of Inverse of Matrix:**

- If Determinant = 0 then matrix has no inverse.
- Multipying by the inverse of the inverse of a matrix gives the same result as dividing by the matrix.

E.g.

Example:

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]]>The post Properties Of A Circle appeared first on TeachifyMe.

]]>If 2 chords in a circle area congruent, then the 2 angles at the centre of the circle are identical.

AB = CD (Equal Chords)

∠ AOB = ∠ COD

The perpendicular bisector of a chord passes through the centre of a circle

E.g.

AD = DB

∠CDA = ∠CDB

=90°

2 congurent chords in a circle are of the same distance from the centre of the circle.

E.g.

AB = CD

OX = OY

Angle in a semi circle is a right angle.

∠AXB = ∠AYB = 90°

Angles in the same segment are equal

∠p = ∠p = ∠p

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]]>The post Trigonometry appeared first on TeachifyMe.

]]>For all the right angled triangles “the square on the hypotenuse (c^{2}) is equal to the sum of all the squares on the other two sides (a^{2} + b^{2 }).

c^{2 }= a^{2} + b^{2}

All angles in a right angle triangle are acute angles. (between 0 to 90 degrees).

*Tip to remember this SOH CAH TOA.*

**Example:**

Q: Find the unknown values of :

Sin 30 = Opposite / Hypotenuse

Sin 30 = x / 13

X = 6.5 Ans

Y^{2} + X^{2} = 10^{2}

Y^{2 }= 10^{2 }– 6.5^{2}

Y = 7.60 (Ans)

The trigonometric ratios of an obtuse angle (between 90 and 180 degree) can be expressed in terms of the adjacent acute angle that lies in the same straight line.

For a known right angled triangle with any two sides given and the area.

Area = ½ ab sin C

Area = ½ bc sin A

Area = ½ ac sin B

It can be used when:

- Two angles and one side given.
- Two sides and one non-included angle given.

a^{2=} b^{2 }= c^{2} – (2bc cos A)

b^{2=} a^{2 }= c^{2} – (2ac cos B)

c^{2=} a^{2 }= b^{2} – (2ab cos C)

It can be used when:

- Three sides are given.
- Two sides and one angle is given.

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]]>The post Bearings appeared first on TeachifyMe.

]]>- An angle measured north to at A.
- In a clockwise direction.
- Written as three- figured number (i.e from 000 to 300 degrees).

The bearing of B from A is 050 degree.

When a person looks at something above his or her location e.g (The top of a building). The angle formed between the horizontal ground and the line of sight is called the angle of elevation.

When a person looks at something below his or her location, the angle between the imaginative horizontal and vertical and the line of sight is the angle of depression.

Angle of depression is always of equal value as angle of depression because they are alternative angles.

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]]>The post Congurence And Similarity appeared first on TeachifyMe.

]]>Two geometric figures are congurent, if they are of the same shape and size.

AB =XY ∠a = ∠x

BC = YZ ∠b = ∠y

AC = XZ ∠c = ∠z

- Side Side Side (SSS) – all sides are equal
- Angle Angle Angle (AAA) – all angles are same
- Side Angle Side (SAS) – two sides and one angle are equal
- Right Angled – Hypotenuse – Side – one right angle and hyp & side equal

≡ – To denote “ is congruent to”

≢ – To denote “ is not congruent to”

** **

When two figures are similar, they are exactl the same in shape but not in size.

- Equal corresponding Angles:

The angles of one triangle are equal to corresponding angles of the other triangle.

- Proportional Corresponding Angles.
- Two Pairs of Proportional Corresponding Sides with equal angles.

We right similarity as** ( **~ ) i.e ΔABC ~ ΔDEF

** **

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]]>The post Vectors (In Two Dimensions) appeared first on TeachifyMe.

]]>Resultant Vector:

Single Letter in bold. i.e **a.**

Position vectors are also called as end points.

**To Find the Modulus / Magnitude of a position vector:**

(Parallelogram Law of Addition)

(Triangular Law of Addition) nose to tail

- The negative sign reverses the directionof a vector.
- The result of a-b is a + – b e. subtracting b is equal to adding the negative of b.

If two line segmants AB and CD are parallel, in same direction and equal in length, then AB and CD are equal vectors.

Vectors are parallel, if they have the same direction.

Both components of one vector must be in the same ratio to the corresponding components of the parallel vector.

Vector b kb (k>0) kb (k<0)

If a = kb, the vectors a and b are parallel and are in same direction.

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