Monday, 21 July 2014
Sunday, 9 March 2014
Mathematics
1. Integers
Introduction:- To be able to subtract a larger number from a smaller number, and to be able to indicate the idea of oppositeness of things, man indicated each number with a positive and negative signs for them.
Definition :- Integers are, a bigger collection of numbers, formed by whole numbers and their negatives.
Properties of integers :- Closure property
(1) Integers are closed under addition.
In general, for any two integers a and b, a+b is an integer.
For example :- 5+4=9; -8+6=-2
(2) Integers are closed under subtraction.
In general, if a and b are two integers, a-c is also an integer.
For example:- 7-2=5; -22-(-9)=-13
(3) Integers are closed under multiplication.
In general, for any two integers a and b, ab is an integer.
For example:- 3×4=12; -7×-5=35
(4) Integers are NOT closed under division.
In general, if a and b are two integers, a/b may not be integer.
Commutative Property
(1) Addition is commutative for integers.
In general, for any two integers a and b, a+b=b+a
For example:- 2+(-3)=2-3=-1; but
(-3)+2=-3+2=-1
(2) Subtraction is NOT commutative for integers.
In general, for any two integers a and b, a-b≠b-a
For example:- 7-(-4)=7+4=11; but (-4)-7=-4-7=-11
5-(-2)=5+2=7; but (-2)-5=-2-5=-7
(3) Multiplication is commutative for integers.
In general, for any two integers a and b, ab=ba
For example:- 9×(-4)=-(9×4)=-36; but
(-4)×9=-4×9=-36
(4) Division is NOT commutative for integers.
In general, for any two integers a and b, a/b≠b/a
For example:- 2/8=1/4; but
8/2=4/1
Associative property
(1) Addition is associative for integers.
In general, for any three integers a, b and c, a+(b+c)=(a+b)+c
For understanding this statement, let us take an example:
-4, -3 and -6
(-6)+[(-4)+(-3)] [(-6)+(-4)]+(-3)
(-6)+(-7) (-10)+(-3)
= -13 =-13
(2) Subtraction is NOT associative for integers.
In general, for any three integers a, b and c, a-(b-c)≠(a-b)-c
For example:- 5-(6-4)= 5-2=3; but
(5-6)-4= 1-4=-5
(3) Multiplication is associative for integers.
In general, for any two integers a, b and c, a×(b×c)=(a×b)×c
For example:- [(-3)×(-2)×4]=6×4=24;
[-3×(-2×4)]=(-3)×(-8)=24
(4) Division is NOT associative for integers.
Distributive property
(1) Distributive property of multiplication over addition.
In general, for any three integers a, b and c, a×(b+c)=a×b+a×c
For example:- -2(4+3)=
-2(7)= (-2×4)+(-2×3)=
-14 (-8)+(-6)=
-14
(2) Distributive property of multiplication over subtraction.
In general, for any three integers a, b and c, a×(b-c)=a×b-a×c
For example:- -2(4-3)=
-2(1)= (-2×4)-(-2×3)=
-2 (-8)-(-6)=
-2
Identity under addition
'0' is the identity integer under addition. In general, for an integer a, a+0=a=0+a or in other words, a+0=0+a=a
For example:- 9+0=9=0+9
Operations on integers (formula only):- Addition and Subtraction:
Positive(+) {+/-} Positive(+) = Positive(+) sign
Negative(-) {+/-} Negative(-) = Negative(-) sign
Negative(-) {+/-} Positive(+) = The sign of the greater number(+/-)
Multiplication and division:
Positive(+) {×or/} Positive(+) = Positive(+) sign
Negative(-) {×or/} Negative(-) = Positive(+) sign
Negative(-) {×or/} Positive(+) = Negative(-) sign
2. Fractions
Introduction:- In our daily life, we use fractions in different situations. We use them when a whole thing is divided into parts.
Definition :- A fraction means a part from the whole thing, or more generally, equal parts from the whole thing.
Example :- The fraction 2/4 means: 2 out of 4 parts.
Components of fractions :- In a fraction, we call the top number, the Numerator and it indicates the part that we choose from the whole.
And the number at the bottom, we call it the Denominator, and it indicates the part in which the whole thing is divided.
Example :- The ratio of 20 boys and 30 girls in a class can be represented by the fraction 2/3 (2 upon 3).
In this fraction-2/3, 2 is the numerator and 3 is the denominator.
What is Quarter?
Suppose you cut a cake in the middle, horizontally and vertically. We will get four equal parts. That's one-fourth of the whole cake. So the fraction is: 1/4 of the whole cake. It is also called quarter.
Fraction on the number line(only an idea) :- On a number line, we can divide the length between 0 and 1 into four equal parts. And if you take a fraction, suppose 1/4, it lies between 0 and 1 and one part can be one-fourth.
Classification of fractions:-
1. Decimal fractions:- Fractions in which the denominator is 10 or a higher power of 10 are called decimal fractions.
Examples :- 2/10; 3/100; 4/1000, etc.
2. Vulgar fractions:- Fractions in which the denominator is other than 10 or any powers of 10 are called vulgar fractions.
Examples :- 3/6; 8/16; 14/26; 17/36, etc.
3. Proper fractions:- Fractions in which the numerator is less than the denominator are called proper fractions.
English
1 : The Sentence
Introduction :- When we speak or write, we use words to express our thoughts and feelings. And we usually use these words in groups.
Definition:- A group of words which makes complete sense is called a sentence.
Examples:- 1. Humpty Dumpty sat on a wall.
2. Little Jack Horner sat in a corner.
Based on their difference, sentences are divided into four kinds.
(1) Assertive Sentences
Assertive sentences, also known as declarative sentences, says or states something.
Examples :- 1.Sherlock Holmes was a sharp-eyed detective.
2.We waited patiently at the station.
Note:- (a) An assertive sentence can be affirmative(positive) or negative.
For example:- She is tall.(positive)
She is not tall.(negative)
(b) They always end with a full stop(.)
(2) Interrogative Sentences
Interrogative sentences asks a question.
Examples :- 1.Where do you live?
2. Have you ever seen the Eiffel Tower?
Note:- (a) An interrogative sentence can be a yes or no-question or a wh-question.
(b)A yes or no-question can be answered in yes or no.
(c)A wh-question begins with a question word like when, who, why, what, whom, etc.
For example :- Were you absent yesterday?(yes or no-question)
Why were you absent yesterday?(wh-question)
(d) They always end with a question mark(?)
(3) Imperative Sentences
Imperative sentences expresses a command(order), a request, an advice or an entreaty.
Examples :- 1. Get that thing out my house at once.(command)
2. Please, tell me the story of the play in brief.(request)
3. You should learn to respect your elders.(advice)
4. Have mercy upon us.(entreaty)
Note:- They always end with a full stop(.)
(4) Exclamatory Sentences
Exclamatory sentences expresses a strong feeling.
Examples :- 1. What a great playwright Shakespeare was!
2. How cold and scary the night was!
Note:- They always end with an exclamation mark(!)
Subject and predicate
Every sentence contains two parts - a subject and a predicate.
The subject is what/whom the sentence is about.
The predicate says something about the subject.
To determine the subject of an interrogative sentence,
1) Find the verb
2) Ask the question 'what/who' to the verb.
*The answer will be the subject.
For example:- Where have you been?
*The verb is 'have...been'.
* When we ask the question 'who', we will get the answer as 'you', this is the subject.
Subjects in imperative sentences :- These type of sentences differ from normal sentences.
Read these sentences:
a. Stand up.
b. Come here.
c. Go away.
In these sentences, some may get confused and consider that there is no subject there and that there is only predicate. But it is wrong. We should always keep in mind that if there is a sentence, there must be a subject and a predicate.
So, what is the subject here?
Well, anyone can see that there is no subject typed, but it doesn't mean that its not there, and it is understood.
All these sentences mean that:
a. You stand up.
b. You come here.
c. You go away.
And so, as you can see, the subject is nothing other than 'you'.
Parts of speech
In order to study the English language, all words are placed in categories or classes called parts of speech. There are eight parts of speech: noun, pronoun, adjective, verb, adverb, preposition, conjunction and interjection.
Note:- A word may not always belong to the same category. We should never say that a word is a noun, adjective or verb. It may be used as a noun in one sentence and as a an adjective in another. So it is important to know that the category to which a word belongs depends upon the work it does in a particular sentence.
Look at these examples:- My mother patted me on the back.(noun)
I must have left the back door unlock.(adjective)
The contractor backed out of the agreement.(verb)
My dad came back immediately.(adverb)
Nouns
Definition :- A noun is the name of a person, animal, place or thing. It may also name a feeling or idea. In short, we can say that a noun is a naming word.
Examples:- hospital, girl, Olivia, stone, beauty, army, etc.
Based on their characteristics, nouns are divided into four kinds.
(1) Proper nouns
A noun which names a particular person,place or thing is called a proper noun.
Look at these examples:- Sachin Tendulkar is an Indian cricket player.
Jaipur is famous for its pink buildings.
The Hindu is a famous newspaper in India.
In these sentences, Sachin is the name of a person. Jaipur names a particular person. The Hindu names a particular thing.
Note:- Proper nouns include the names of persons, countries, cities, towns, villages, historical monuments and places, rivers, ships, streets, mountains, months of the year, days of the week, festivals, books, newspapers, etc.
(2) Common nouns
A noun that gives a common name to persons, places, animals or things of the same kind is called a common noun.
Examples :- book, pen, river, mountain, train, city, dog, animal, etc
Note:- India, Russia, China, America, etc are all names of countries. The word country is a common noun. It is a name shared by all the four. But India is the name of a particular river. So the word India is a proper noun. Similarly, the words Russia, China and America are also proper nouns.
(3) Abstract nouns
A noun that states the name of some quality, feeling or idea that we can only think of or feel but cannot touch or see.
Examples:- happiness, freedom, morality, health, friendship, wisdom
More detailed:- We can feel anger but cannot see or touch it. We can see or touch an angry person but not anger.
(4) Collective nouns
A noun that states the name of a collection of persons or things taken together and spoken as of one whole.
Examples:- bunch, pack, fleet, herd, troupe, etc
Now, let us learn more collective nouns:-
a fleet or an armada of ships
a suite of rooms
a clan of hyenas
a heap of stones
a clutch of eggs
a pair of shoes
an army or a battalion, or regiment of soldiers
a host of sparrows
a shoal or school of fish
a chain of islands
a cluster or galaxy of stars
a pack of wolves or dogs or hounds or cards
a clump or forest of trees
a swarm of flies or mosquitoes
a hive or cluster or swarm of bees
a bundle of sticks
a troupe of dancers
a library of books
a flock of sheep or birds
a pride of lions
a fleet of airplanes, cars
a crowd or mob of people
an orchard of fruit trees
a team of players or oxen or horses
a litter of puppies or kittens
a brood of hens or chickens
a crew of sailors
a murder of crows
a bouquet or bunch of flowers
a caravan of camels
a quiver of arrows
a flight of stairs or birds
a clutter of cats
a gaggle of geese
a herd of cows, cattle, deer or elephants
an army of ants
a colony of ants or frogs
a bunch of keys, grapes, bananas
Abstract and concrete nouns:-
Abstract nouns name feelings(like anger), qualities(like bravery), states of mind(like hope), or ideas(like knowledge). These words suggest no physical form that we may see or touch.
Concrete nouns stand for something we can see or touch, something that has a physical form, something that has a position in space. Words indicating people(like aunt), animals(like rabbit), places(like hospital) and things(like watch) are concrete nouns.
Nouns: Number
Singular and Plural nouns
Introduction:- Look at these words: doll - dolls; baby - babies; tree - trees
The first word of each pair denotes one thing, the second word of each pair denotes more than one.
- A noun that refers to one person or thing, is said to be singular (or in the singular number).
- A noun that refers to more than one person or thing, is said to be plural (or in the plural number).
- Only countable nouns can have a plural form.
Formation of plurals:-
Certain changes are to be made while changing singular nouns to plural.
(1) Most plurals of nouns are generally formed by adding -s to the singular noun;
apple-apples; cow-cows; flower-flowers
kite-kites; window-windows; dog-dogs
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