FOUR BASIC OPERATIONS OF ARITHMETIC

MY STUDY

Now I’m a first-year student at the Cherkasy Bohdan Khmelnytsky National University. You know, since I’m not a citizen of Cherkasy, I live in the students’ hostel. To tell the truth, I am happy about it, since I like to be independent. In the hostel I share a room with two other girls.

One of them is from Horodyshche. Her name is Helen. The other girl is from Novomyrhorod, a town in the Kirovohradska oblast. She is Ukrainian. Her name is Irene. As a matter of fact they are not only my room-mates but also my good friends. So you see, students from different oblasts take their course at the Cherkasy University.

We have a very nice and comfortable room. It’s also very light, since it faces the West. All of us are early risers. We usually get up at 7 o’clock. After the usual morning exercises and a shower we have breakfast. Since we don’t like to be late for our classes we try to come to the University a little before 9 o’clock. We usually have lectures and seminars in the morning and sometimes in the afternoon. After classes we have dinner at the students’ dining-room. It does not take us long. Then I usually take a walk round the University building. I like to do it by myself. That’s my idea of a good rest. After that I go to the reading-room and look through jurnals and periodicals. On Tuesday and Friday I also do my English homework in the reading-room. I get back to my room in the hostel rather late in the evening.

On my day off, that is on Sunday I usually go to the centre of Cherkasy. There is so much to see and there are so many places to go to. There are old and modern houses, a concert hall, various museums, beautiful ancient buildings among them. There are also lovely green parks and stadiums.

Today is Sunday. I should like to go to some theatre with my boy-friend. His name is Ihor. He’s a lovely boy, very nice and bright. He is also a student. He does physics. As a matter of fact, his parents are also physicists. His father does research in the field of atomic physics.

I hope we will get tickets for the theater tonight. If we do not get the tickets we can go to the cinema. There are some new films on.

I must take my terminal exams in December. Our first term lasts from the first of September, through October and November. In November we have our credit-tests and if we take them successfully we can take our exams. The exams are usually over by the 24^th of December. Then we’re free.

FOUR BASIC OPERATIONS OF ARITHMETIC

We cannot live a day without numerals. Numbers and numerals are everywhere. On this page you see number names. They are zero, one, two, three, four, and so on. And here are the numerals: 0, 1, 2, 3, 4 and so on. In a numeration system, numerals are used to repre­sent numbers, and the numerals are grouped in a special way. The numbers used in our numeration system are called digits.

In our Hindu-Arabic system we use only ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 to represent any number. We use the same ten digits over and over again in a place-value system whose base is ten. These digits may be used in various combinations. Thus 1, 2, and 3 are used to write, 123, 132, 213, and so on. One and the same number could be represented in various ways. For example, take number 3. It can be represented as 2 + 1, 4 - 1 and so on.

A very simple way to say that each of the numerals names the same number is to write an equation - a mathematical sentence that has an equal sign (=) between them. For example, 3 + 4 = 5 + 2, or 3 – 1 = 6 - 4. The + is a plus sign. The - is a minus sign. We say three plus six equals five plus four, or three minus 1 is equal to six minus four. Another example of an equation is 3 + 5 = 8. In this equation three is an addend. Five is also an addend. Eight is the sum. We add three and five and we get eight.

There are four basic operations of arithmetic that you all know of. They are addition, subtraction, multiplication and division. In arith­metic an operation is a way of thinking about two numbers and getting one number. As you remember from the above in the operation of addition the two numbers with which you work are called addends or summands and the number that you get as a result of this operation is the sum. In subtraction again you use two numbers. In the equati­on 7 - 2 = 5 seven is the minuend and two is the substrahend. As a re­sult of this operation you get the difference. We may say that subtra­ction is the inverse operation of addition since 5 + 2 = 7 and 7 - 2 = 5.

The same might be said about multiplication and division, which are also inverse operations. In multiplication there is a number that must be multiplied. It is the multiplicand. There is also a multiplier. If we multiply the multi­plicand by the multiplier we shall get the product as a result. When two or more numbers are multiplied, each of them is called a factor. For example, in the expression 5 x 2 (five multiplied by two), the 5 and the 2 will be factors. The multiplicand and the multiplier are na­mes for factors.

In the operation of division there is a number that is divided and it is called the dividend; the number by which we divide is called the divisor. As a result of the operation of division we shall get the quoti­ent. In some cases the divisor is not contained a whole number of ti­mes in the dividend. For example, if you divide 10 by 3 you will get a part of the dividend left over. This part is called the remainder. In our case it will be 1.

Since multiplication is the inverse operation of division you may check division by using multiplication.

There are two very important facts that must be remembered abo­ut division.

a) The quotient is 0 whenever the dividend is 0 and the divisor is not 0. That is, 0 : n for all values of n except n = 0.

b) Division by 0 is meaningless. If you say that you cannot divide by 0 it really means that division by 0 is meaningless. That is, n :0 is meaningless for all values of n.

BASE TWO NUMERALS

During the latter part of the seventeenth century a great German philosopher and mathematician Gottfried Wilhelm von Leibnitz (1646 - 1716) was doing a research on the simplest numeration system. He developed a numeration system using only the symbols 1 and 0. This system is called a base two or binary numeration system.

Leibnitz actually built a mechanical calculating machine which until recently was standing useless in a museum in Germany. Actually he made his calculating machine some three centuries before they were made by the modern machine makers.

The binary numeration system introduced by Leibnitz is used only in some of the most complicated electronic computers. The numeral 0 corresponds to “off” and the numeral 1 corresponds to “on” for the electrical circuit of the computer.

Base two numerals indicate groups of ones, twos, fours, eights and so on. The place value of each digit in 1101 in base TWO as shown by the above words (on or off) and also by powers of 2 in base TEN notation as shown below.

The numeral 1101 in base TWO means “one multiplied by two in the cube” plus “one multiplied by two in the square” plus “zero multiplied by two” plus “one multiplied by one” equals (1X8) + (1X4) + (0X2) + (1X1) = 8 + 4 + 0 + 1 = 13. Therefore 1101 in base TWO = 13.

A base ten numeral can be changed to a base two numeral by dividing by powers of two. From the above you know that the binary numeration system is used extensively in high-speed electronic computers. The correspondence between the two digits used in the binary system and the two positions (on and off) of a mechanical switch used in an electronic circuit accounts for this extensive use.

The binary system is the simplest place-value, power-position system of numeration. In every such numeration system there must be symbols for the numbers 0 and 1. We’re using 0 and 1 because we’re well familiar with them.

CLOSURE PROPERTY

If we add two natural numbers, the sum will also be a natural number. For example, 5 is a natural number and 3 is a natural num­ber. 5 + 3 = 8 and only 8.

The sum, 8, is also a natural number. Following are other exam­ples in which two natural numbers are being added and the sum is another natural number. 19 + 4 = 23 and only 23; 6 + 6 = 12 and on­ly 12; 1429 + 357 = 1786 and only 1786. Actually if any two natural numbers are being added, the sum again is a natural number. Sin­ce this is true we say that the set of natural numbers is closed un­der addition. This is a statement of closure, one of special properti­es of addition.

Notice that we can name the sum in each of the above equati­ons. That is, the sum of 5 and 3 exists, or for example, there is a number which is the sum of 19 and 4. In fact the sum of any two numbers exists. This is called the existence property. Notice also, that when 5 and 3 are being added the sum is 8 and only 8 and not some other number. Since there is one and only one sum for 19 + 4, we say that the sum is unique. This is called the uniqueness property. Both existence and uniqueness are implied in the definition of closure.

If a and b are numbers of a given set, then a + b is also a num­ber of that same set. For example, if a and b are any two natural numbers, then a + b exists, it is unique, and it is again a natural number.

If we use the operation of subtraction instead of the operation of addition, we cannot make the statement we made above. If one na­tural number is being subtracted from another natural number the result produced is sometimes a natural number, and sometimes not. 11 - 6 = 5 and 5 is a natural number. 9—9 = 0 and 0 is not a natural number. Consider the equation 4 - 7 = n. It cannot be solved if we must have a natural number as an answer. Therefore, the set of natural numbers is not closed under subtraction.

When two natural numbers are being multiplied there is always a natural number which is the product of the two numbers. Every pair of natural numbers has a unique product which is again a na­tural number. Thus the set of natural numbers is closed under mul­tiplication.

In general, the closure property may be defined as follows: if x and у are any elements, not necessarily the same, of set A and * (asterisk) denotes an operation *, then set A is closed under the operation * if (x*y) is an element of set A.

It must be pointed out that it is impossible, to find the sum or the product of every possible pair of natural numbers. Hence, we accept the closure property without proof, that is as an axiom.

WHOLE NUMBERS

Many statements in mathematics are concerned not with a single number but with a set of numbers that have some common property. For example, such a set of numbers is the set of odd numbers 0, 2, 4, 6… or the set of even numbers 1, 3, 5, 7 … What property is common to all even numbers? What property is common to all odd numbers?

You ought to know that the result of multiplication is called a product, and the numbers to be multiplied are called factors. When you write 6X3 = 18 it means that you write number 18 as a product of two whole number factors.

Another pair of whole number factors will be 9 and 2, since 9X2 = 18. Will you be able to name other factors of 18? Because 6X3 = 3X6 let us agree to call 6 and 3 just one pair of factors of 18.

When you use 0 as one of the factors, what should the product be? That is, 0 times 5 equals what number? Or 7 times 0 equals what number? The answers to these questions are summarized in the following statement: For any statement a, ax0 = 0 = xa. In some cases when we have to name a whole number in a factorial name more than two factors can be used. We can, for example name 60 as a product of 3 factors.

Since multiplication is associative, we know that (3X4)X5 = 3X4X5 = 3X(4X5). We may also write 60 = 3X4X5; 60 = 3X5X4, and so on.

Since aX1 = a for any number a, we know that 1 is a factor of every whole number. Let us agree to omit 1 as a factor when naming a number in factored form.

In each of the above equations the same set of factors is used, namely, 3, 4 and 5. Regardless of the order in which they’re written, 3, 4 and 5 should be considered just as one set of three factors of 60. Also 60 can be written as the product of four factors as shown in the equation 60 = 3X2X2X5. In previous exercises you probably noticed that some of the factors you used could be factured further and others could not.

In the equation 18 = 6X3, the factor 6 can in turn be written as 3X2. If you do this, you will get 18 = 2X3X3. None of these three factors can be written in factured form if you do not use 1 as a factor. Hence 2X2X3 is the form containing the smallest factors of 18.

You will be able to do the same with an odd number, say 105, where 105 = 3X35 = 3X5X7. You already know that every whole number has 1 and itself as a factor. That is 9X1 = 9 and 11X1 = 11. Some such numbers have only 1 and themselves as a factor. Since its only factors are 1 and 5, 5 is such a number.

A whole number is called a prime number, or just a prime if:

a) It is greater than 1.

b) Its only factors are 1 and itself.

Any whole number other than 0 and 1 which is not a prime number is called a composite number, or just a composite.