Read the text and summarise the main ways of expressing numbers in English.


If a number is in the range 21 to 99, and the second digit is not zero, we should write the number as two words separated by a hyphen:

25 twenty-five; 57 fifty-seven; 89 eighty-nine.

Numbers over 100 are generally written in figures. However if you want to say them aloud or want to write them in words rather than figures you put 'and' in front of the number expressed by the last two figures. For example:

203 two hundred and three (AmE: two hundred three)

622 six hundred and twenty-two (AmE: six hundred twenty-two)

Numbers between 1000 and 1,000,000 is usually said or written in words as:

1,803 one thousand, eight hundred and three
(AmE: one thousand, eight hundred three)

1,963 one thousand, nine hundred and sixty-three
(AmE: one thousand, nine hundred sixty-three)

2,840 two thousand, eight hundred and forty
(AmE: two thousand, eight hundred forty)

Four-figure numbers ending in 00 can also be said or written as a number of hundreds. For example, 1800 can be said or written as "eighteen hundred".

If the number 1963 is being used to identify something, it is said as "one nine six three". We always say each figure separately like this with telephone numbers. If a telephone number contains a double number, we use the word "double":

561 6603 five six one [pause] double six 'oh' three (AmE: five six one [pause] six six 'oh' three)

Saying years. We normally say a year in two parts. In the case of years ending in "00", we say the second part in "hundred":

1058 ten fifty-eight

1706 seventeen hundred and six (or 'seventeen oh six')

1865 eighteen sixty-five

1900 nineteen hundred

There are two ways of saying years ending in "01" to "09" before 2000. For example: "1901" can be said as "nineteen oh one" or "nineteen hundred and one". The year 2000 is read "two thousand", 2006 "two thousand and six" (AmE: two thousand six). Post-2010 dates are often said as normal (2010 would be "twenty ten").

Flight numbers. We pronounce a flight number in two parts or digit-by-digit. For example:

110 one ten (or 'one one oh')

1248 twelve forty-eight

2503 twenty-five oh three

3050 three oh five oh (or 'three zero five zero', 'thirty fifty')


Expressing millions


1,412,605 one million four hundred (and) twelve thousand six hundred (and) five

2,760,300 two million seven hundred (and) sixty thousand three hundred

Remember: The British use 'and' before tens and ones but the Americans usually leave the 'and' out.

Ways of expressing the number 0

  0 = nil in football and other sports, for scores of 0 (AmE: zero or nothing) 0 = love in tennis 0 = zero in temperatures to refer to freezing point (0 Celsius, 0 Fahrenheit)   0 = nought in mathematics (AmE: zero)   0 = oh for telephone numbers   0 = oh (or zero) for flight numbers



1. We use zero to express some numerical values such as temperatures, taxes, and interest rates.

2. We can pronounce "0" like the letter "o", when we are reading out numbers figure by figure (e. g. telephone number, flight number, credit card number, etc.)


Fractional numbers

½ a half 0.5 British English nought point five American English zero point five
two and a half 2.5 two point five
¼ a quarter 0.25 British English nought point two five American English zero point two five
¾ three quarters American English also three fourths 0.75 British English nought point seven five American English zero point seven five


Writing full stops and commas in numbers

Use a full stop (.) to separate the main part of a number from the decimal part. 3.062 means 'three point nought six two'.

Say point to refer to the full stop. You can use a comma (,) in large numbers to separate the hundreds, thousands, and millions.3,062 means 'three thousand and sixty-two'. In British English, spaces are sometimes used instead of commas (3 062). Remember: Speakers of some other languages use (,) and (.) in the opposite way the commas for the decimals and the points for thousands, millions, etc.


Read the text and be ready to prove every of the following laws:

The Closure Law of Addition.

The Commutative Law of Addition.

The Associative Law of Addition

The Identity Law for Addition.

The Inverse Law of Addition.

The Closure Law of Multiplication.

The Associative Law of Multiplication.

The Identity Law of Multiplication.

The Inverse Law of Multiplication.

The Distributive Law of Multiplication


Many who have been out of school for a number of years find, if they want to refresh their knowledge of mathematics, that there has been a great change, a sort of mathematical revolution while they were away from school. The old, classical math has had its face lifted and has taken on a new look which modern instructors claim is a great improvement.

In the classical math often taught in high-school courses, many simple truths were taken for granted and there was a failure to analyze these truths to find out why they are true and under what particular conditions they might not be true.

During the past centuries, great, world-shaking theories were born, notably the Maxwell electromagnetic theory, the theory of relativity, and the concept of differential and integral calculus. And all these extremely important doctrines came about as a result of questioning and continually asking WHY?

The results obtained using the New Math agree, of course, with those obtained using the old, classical math, but the method of the former is much more thorough and therefore more satisfactory to the student who has never before studied math. The New Math teaches a student to think a problem through rather than try to recall tricks of manipulation.

Let's take a simple example of the two methods:

We all learned that if x24=0, x must equal either 2 or 2. Either of these numerical replacements for the letter x makes the statement meaningful. This is so elementary it hardly needs comment. But just how did we arrive at this 2? Did we actually "transpose" the 4 to the other side of the equal sign where it became +4" the equation becoming x2=4 and x becoming 2? Any child might well ask, "Why do we change signs when we "transpose" from one side to the other in an equation". This, of course, is a sensible question. In the New Math this is dealt with before the child asks the question. We say:

If x24=0, then by adding +4 to both sides of the equation we get x24+4=0+4.

Next we show that 4 and +4 cancel each other and that 0+4=4.

Then x2+0=4 or x2=+4. Thus x=2.

As a matter of fact, it is not at all difficult to demonstrate that we solved our little problem by making use of some of the eleven laws that form the foundation of arithmetic. Yes, that is a truly startling fact and a truly startling discovery. Numbers are one of the most basic of the great ideas of mathematics. And believe it or not" eleven laws not an infinity of manipulative devices are the tools available to us when we want to solve problems. These are the eleven laws of real numbers:

1. The Closure Law of Addition. The sum of any two real numbers is a unique real number. For example, the sum of 10 and 117 is 127.

2. The Commutative Law of Addition. The order in which we add is trivial. For example, the sum of 3 and 4 is 7; the sum of 4 and 3 is also 7.

3. The Associative Law of Addition. Since addition is defined for pairs of numbers, the addition of three numbers depends on our first adding any two of the numbers and then adding their sum to the third number; the order in which we do this is trivial. For example, when 3, 4 and 5 are added in three different orders, the same sum is obtained:


3+4=7, 7+5=12

4+5=9, 9+3=12

3+5=8, 8+4=12


4.The Identity Law of Addition. The number zero is the additive identity, for the addition it to any other number leaves the second number unchanged. For example, the sum of 0 and 9 is 9.

5.The Inverse Law of Addition. The sum of any number and its negative is zero. For example, the sum of 5 and -5 is 0.

6.The Closure Law of Multiplication. The product of any two real numbers is a unique real number. For example, the product of 177 and 10 is 1,170.

7.The Associative Law of Multiplication. Since multiplication is defined for pairs of numbers, the multiplication of three numbers depends on our first multiplying two of the numbers and then multiplying their product by the third number; the order in which we do this is trivial. For example:

3 4 = 12, 12 5 = 60

3 5 = 15, 15 4 = 60

4 5 = 20, 20 3 = 60


8. The Identity Law of Multiplication. The number one is the multiplicative identity, for the product of it and any other number leaves the second number unchanged. For example, the product of 1 and 8 is 8.

9. The Inverse Law of Multiplication. The product of any number (except zero) and its reciprocal is one. For example, the product of 3 and 1/3 is 1; the product of 5 and 1/5 is 1; the product of 3/10 and 10/3 is 1. Division of a number by zero is meaningless.

10.The Distributive Law. Multiplication "distributes> across addition. For example:

6 (4+5) =6 9=54

6 (4+5) = (6 4) + (6 5) =24+30=54



Read the text and be ready to tell about the origin of the term "algorithm" and its main properties. Make use of the vocabulary list:

sequence of operations

distinguishing feature ,

to carry out

to take into account

to have an influence

due to ,

both ... and ..., ...

to come into usage


The term "algorithm" has come into usage quite recently. Its appearance in our life is due to the rapid rise of computer science which has the study of algorithm as its focal point.

The word "algorithm" originated in the Middle East. It comes from the Latin version of the last name of the Persian scholar Abu Jafar Mohammed ibn Musa al-Khowaresmi (Algorithmi), whose textbook on arithmetic, written in 825 A. D., gave birth to algebra as an independent branch of mathematics. In the 12th century this textbook was translated into Latin and it had a great influence for many centuries on the development of computing procedures. The name of the textbooks author became associated with computation in general and used as a term "algorithm".

The concept of an algorithm is now one of the most fundamental notions, both in mathematics and engineering. An algorithm is defined as an exact and intelligible order for a certain executor to carry out a sequence of operations, aiming at getting certain results or solving a given problem.

An algorithm has 5 properties of its own. The first of them is called discreteness. This property means that the process under description is to be separated into certain steps (instructions).

The second property of an algorithm may be called its intelligibility. It means that an algorithm should take into account, what orders an executor can understand and carry out and what orders he or it cannot.

The next distinguishing feature of an algorithm is that all vagueness must be eliminated each instruction must have one single meaning. This property of an algorithm is called the property of determinacy.

Another property of an algorithm is its mass character, which means that a given algorithm may be used for solving a certain class of problems.

The last property of an algorithm is its effectiveness. It means that the exact carrying out of all orders of the algorithm should lead to termination of the process after a finite number of steps.


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