Adverbs in -ly form their comparative and superlative with more and most. (But not early)

1. Adverbs in -ly form their comparative and superlative with more and most. (But not early)

Could you say that more slowly, please?
Tom can shoot the most accurately.
You will just have to get up earlier.

2. Adverbs with the same form as adjectives form their comparative and superlative with -er and -est.

Sarah run the fastest.

Some adverbs form their comparative and superlative irregularly.

Positive	Comparative	Superlative
badly	worse	worst
well	better	best
little	less	least
far	farther, further	farthest, furthest
much	more	most

Comparison as...as, less etc.

We use as...as to compare 2 things that are the same in the same way.
I cannot do crosswords as quickly as you.

Less and least are the opposites of more and most.
The old man's son visits him less often nowadays.

We can repeat a comparative after and to talk about a change in something else.
They went faster and faster down the hill.

We use the+comparative to talk about a change in one thing which causes a change in something else.
The more you practise, the better you will play.

Ten sentences:

I am not the tallest student in our group.

James Hetfield plays electric guitar much better than I.

I think that tomorrow it will be colder than today.

On my girlfriend’s birthday I bought the most expensive present I could afford.

Sevastopol is the most picturesque city in Crimea.

T-1 line Internet connection is much faster than dial-up one, however it is more expensive.

The more I study, the better will be results.

On average, people eat about twice as much proteins as they need.

Calculus is simpler than statistics.

10) Mice are smaller than elephants.

Text(~1500 symbols):

The first computers with their electronics filled more room than up-to-date computers do. Photographs of early computers show men and women in business suits and laboratory coats standing in the middle of a room surrounded by a U-shape machine. In reality, people operating and developing the first computers did not wear suits. Air-conditioning was poorer than people needed and they dressed in T-shirts and tennis shoes.

The development of the transistor in 1948 made it possible to build smaller electronic devices. Computers became smaller and smaller and in our days personal computer can easily be fitted on the desk. Notebooks have less size than personal computers and they widely used by businessmen. For the most pretentious people engineers created a Pocket personal computer that can find room in pocket. Pocket PC is the smallest PC, on the other hand, it is the slowest and the least powerful than other personal computers. For people who do not need features like Infrared or Bluetooth connection or color display that offers Pocket PC, Palm designed small electronic organizer. It has fewer features and it is less powerful than Pocket PC, but it is also less expensive.

Computer designers are trying to create more friendly interface computer, because it is sold better. The one of the most important factors is a noise uttered by computer. Nobody likes noise and people are ready to pay more to buy quieter PC.

The progress never stops and smaller, more powerful and quieter computers will appear soon.

Can, Could, Be able to

Can and could are modal auxiliary verbs. Be able to uses the verb "to be" as a main verb. It is not an auxiliary verb, but we look at it here for convenience.

Can

Can is an auxiliary verb, a modal auxiliary verb. We use "can" to:

talk about possibility and ability

make requests

ask for or give permission

Structure of Can

subject + can + main verb

The main verb is always the bare infinitive (infinitive without "to").

	subject	auxiliary verb	main verb
+	I	can	play	tennis.
-	He	cannot	play	tennis.
		can't
?	Can	you	play	tennis?

Notice that:

Can is invariable. There is only one form of can.

The main verb is always the bare infinitive.

Use of Can can: Possibility and Ability

We use can to talk about what is possible, what we are able or free to do:

She can drive a car.

John can speak Spanish.

I cannot hear you. (I can't hear you.)

Can you hear me?

Normally, we use can for the present. But it is possible to use can when we make present decisions about future ability.

Can you help me with my homework? (present)

Sorry. I'm busy today. But I can help you tomorrow. (future)

can: Requests and Orders

We often use can in a question to ask somebody to do something. This is not a real question - we do not really want to know if the person is able to do something, we want them to do it! The use of can in this way is informal (mainly between friends and family):

Can you make a cup of coffee, please.

Can you put the TV on.

Can you come here a minute.

Can you be quiet!

can: Permission

We sometimes use can to ask or give permission for something:

Can I smoke in this room?

You can't smoke here, but you can smoke in the garden.

(Note that we also use could, may, might for permission. The use of can for permission is informal.)

Could

Could is an auxiliary verb, a modal auxiliary verb. We use "could" to:

talk about past possibility or ability

make requests

Structure of Could

subject + could + main verb

The main verb is always the bare infinitive (infinitive without "to").

	subject	auxiliary verb	main verb
+	My grandmother	could	speak	Japanese.
-	She	could not	speak	Chinese.
		couldn't
?	Could	your grandmother	speak	Japanese?

Notice that:

Could is invariable. There is only one form of could.

The main verb is always the bare infinitive.

Use of Could could: Past Possibility or Ability

We use could to talk about what was possible in the past, what we were able or free to do:

I could swim when I was 5 years old.

My grandmother could speak seven languages.

When we arrived home, we could not open the door. (...couldn't open the door.)

Could you understand what he was saying?

We use could (positive) and couldn't (negative) for general ability in the past. But when we talk about one special occasion in the past, we use be able (positive) and couldn't (negative). Look at these examples:

	Past
	General	Specific Occasion
+	My grandmother could speak Spanish.	A man fell into the river yesterday. The police were able to save him.
-	My grandmother couldn't speak Spanish.	A man fell into the river yesterday. The police couldn't save him.

could: Requests

We often use could in a question to ask somebody to do something. The use of could in this way is fairly polite (formal):

Could you tell me where the bank is, please?

Could you send me a catalogue, please?

Be able to

Although we look at be able to here, it is not a modal verb. It is simply the verb "to be" plus an adjective (able) followed by the infinitive. We look at "be able to" here because we sometimes use it instead of "can" and "could". We use "be able to":

to talk about ability

Structure of Be able to

The structure of be able to is:

subject + be + able + infinitive

	subject	be main verb	able adjective	infinitive
+	I	am	able	to drive.
-	She	is not	able	to drive.
		isn't
?	Are	you	able	to drive?

Notice that be able to is possible in all tenses, for example:

I was able to drive...

I will be able to drive...

I have been able to drive...

Notice too that be able to has an infinitive form:

I would like to be able to speak Chinese.

Use of Be able to be able to: ability

We use be able to to express ability. "Able" is an adjective meaning: having the power, skill or means to do something. If we say "I am able to swim", it is like saying "I can swim". We sometimes use "be able to" instead of "can" or "could" for ability. "Be able to" is possible in all tenses—but "can" is possible only in the present and "could" is possible only in the past for ability. In addition, "can" and "could" have no infinitive form. So we use "be able to" when we want to use other tenses or the infinitive. Look at these examples:

I have been able to swim since I was five. (present perfect)

You will be able to speak perfect English very soon. (future simple)

I would like to be able to fly an airplane. (infinitive)

Have To (objective obligation)

We often use have to to say that something is obligatory, for example:

Children have to go to school.

Structure of Have To

"Have to" is often grouped with modal auxiliary verbs for convenience, but in fact it is not a modal verb. It is not even an auxiliary verb. In the "have to" structure, "have" is a main verb. The structure is:

subject + auxiliary verb + have + infinitive (with "to")

Look at these examples in the simple tense:

	subject	auxiliary verb	main verb "have"	infinitive (with "to")
+	She		has	to work.
-	I	do not	have	to see	the doctor.
?	Did	you	have	to go	to school?

Use of Have To

In general, "have to" expresses impersonal obligation. The subject of "have to" is obliged or forced to act by a separate, external power (for example, the Law or school rules). "Have to" is objective. Look at these examples:

In France, you have to drive on the right.

In England, most schoolchildren have to wear a uniform.

John has to wear a tie at work.

In each of the above cases, the obligation is not the subject's opinion or idea. The obligation is imposed from outside.

We can use "have to" in all tenses, and also with modal auxiliaries. We conjugate it just like any other main verb. Here are some examples:

	subject	auxiliary verb	main verb "have"	infinitive
past simple	I		had	to work	yesterday.
present simple	I		have	to work	today.
future simple	I	will	have	to work	tomorrow.
present continuous	She	is	having	to wait.
present perfect	We	have	had	to change	the time.
modal (may)	They	may	have	to do	it again.

Must (subjective obligation)

We often use must to say that something is essential or necessary, for example:

I must go.

Structure of Must

"Must" is a modal auxiliary verb. It is followed by a main verb. The structure is:

subject + must + main verb

The main verb is the base verb (infinitive without "to").

Look at these examples:

subject	auxiliary verb "must"	main verb
I	must	go	home.
You	must	visit	us.
We	must	stop	now.

Use of Must

In general, "must" expresses personal obligation. "Must" expresses what the speaker thinks is necessary. "Must" is subjective. Look at these examples:

I must stop smoking.

You must visit us soon.

He must work harder.

In each of the above cases, the "obligation" is the opinion or idea of the person speaking. In fact, it is not a real obligation. It is not imposed from outside.

We can use "must" to talk about the present or the future. Look at these examples:

I must go now. (present)

I must call my mother tomorrow. (future)

There is no past tense for "must". We use "have to" to talk about the past.

Must Not (prohibition)

We use must not to say that something is not permitted or allowed, for example:

Passengers must not talk to the driver.

Structure of Must Not

"Must" is an auxiliary verb. It is followed by a main verb. The structure for "Must Not" is:

Subject + "Must Not" + Main Verb

The Main Verb is the base verb (infinitive without "to").

"Must Not" is often contracted to "mustn't".

Look at these examples:

subject	auxiliary "Must" + "Not"	main verb
I	mustn't	forget	my keys.
You	mustn't	disturb	him.
Students	must not	be	late.

NB: like all auxiliary verbs, "must" cannot be followed by an infinitive. So, we say:

You mustn't arrive late. (not You mustn't to arrive late.)

Use of Must Not

"Must Not" expresses prohibition - something that is not permitted, not allowed. The prohibition can be subjective (the speaker's opinion) or objective (a real law or rule). Look at these examples:

I mustn't eat so much sugar. (subjective)

You mustn't watch so much television. (subjective)

Students must not leave bicycles here. (objective)

Policemen must not drink on duty. (objective)

We use "Must Not" to talk about the present or the future:

Visitors must not smoke. (present)

I mustn't forget Tara's birthday. (future)

We cannot use "Must Not" for the past. We use another structure to talk about the past, for example:

We were not allowed to enter.

I couldn't park outside the shop.

Shall and Will

People may sometimes tell you that there is no difference between shall and will, or even that today nobody uses shall (except in offers such as "Shall I call a taxi?"). This is not really true. The difference between shall and will is often hidden by the fact that we usually contract them in speaking with 'll. But the difference does exist.

The truth is that there are two conjugations for the verb will:

1st Conjugation (objective, simple statement of fact)
	Person	Verb	Example	Contraction
Singular	I	shall	I shall be in London tomorrow.	I'll
	you	will	You will see a large building on the left.	You'll
	he, she, it	will	He will be wearing blue.	He'll
Plural	we	shall	We shall not be there when you arrive.	We shan't
	you	will	You will find his office on the 7th floor.	You'll
	they	will	They will arrive late.	They'll

2nd Conjugation (subjective, strong assertion, promise or command)
	Person	Verb	Example	Contraction
Singular	I	will	I will do everything possible to help.	I'll
	you	shall	You shall be sorry for this.	You'll
	he, she, it	shall	It shall be done.	It'll
Plural	we	will	We will not interfere.	We won't
	you	shall	You shall do as you're told.	You'll
	they	shall	They shall give one month's notice.	They'll

It is true that this difference is not universally recognized. However, let those who make assertions such as "Americans never use 'shall'" peruse a good American English dictionary, or many American legal documents, which often contain phrases such as:

Each party shall give one month's notice in writing in the event of termination.

Note that exactly the same rule applies in the case of should and would. It is perfectly normal, and somewhat more elegant, to write, for example:

I should be grateful if you would kindly send me your latest catalogue.

Ten sentences:

Children have to go to school.

I must go to the university.

People mustn’t drive a car when they drink alcohol.

I needn’t do math today, I can do it later.

I should study harder before exams.

Elephants and mice can’t fly.

I could play snooker much better two years ago than I can now.

I can’t have made a mistake in my calculations because I used a calculator.

Can you run 100 meters in 5.5 seconds? 10)

Students mustn’t eat or drink during the lection.

Texts:

Combinatorial mathematics.

Specialists in a broad range of fields have to deal with problems that involve combinations made up of letters, numbers or any other objects.

The field of mathematics that studies problems of how many different combinations can be built out of a specific number of objects is called combinatorial mathematics (combinatorics).

This branch of mathematics has its origin in the 16^th century, in the gambling games that played such a large part in high society in those times. These games gave the initial impetus to develop combinatorial mathematics and the theory of probability.

Italian and French mathematicians were the first to enumerate the various combinations achieved in games of dice. Further advances in the theory of combinations were connected with the names of German scientists.

In recent years combinatorial mathematics has seen extensive developments associated with grater interest in problems of discrete mathematics. Combinatorial methods can be employed in solving transport problems, in particular scheduling; the scheduling of production facilities and of the sale of goods. Links have been established between combinatorics and problems of linear programming, statistics, etc. Combinatorial methods are used in coding and decoding and in the solution of other problems of information theory.

The combinatorial approach also plays a significant role in purely mathematical problems such as the theory of groups and their representations, in the study of the main principles of geometry, some branches of algebra, etc.

Probability.

Probability is a mathematical expression of the likelihood of an event. Every probability is a fraction. The largest probability can be 1. The smallest probability can be is 0, meaning that it’s something that cannot happen. You can find the probability that something will not happen by subtracting the probability that it will happen from 1. For example, if the weatherman tells you that there is a 0.3 probability of rain today, then there must be a 0.7 probability that it won’t r