To make it easier to read1 large numbers, we separate the figures of the numbers by commas into groups of three, counting from right to left. Each group is called a period and has its own name.
The system of numbers we use, called Arabic system, is a decimal system: that is, it is based on tens. In this system, the value a digit represents is determined by the place2 it in the number; if a digit is moved to the left one place, the value it represents becomes ten times as great3.
Zero in the decimal system is a "place-holder"; in the number 30, the zero shows that 3 has been moved to the left one place, thus counting tens instead of ones. The place value in numbers is shown below:
682,000,000,000 847,000,000 136,000 592
Billions Millions Thousands Ones
These numbers are read: six hundred eighty-two billion, eight hundred forty-seven million, one hundred thirty-six thousand, five hundred and ninety-two.
682,000,000,000 847,000,000 136,000 592
Billions Millions Thousands Ones or Units
4 periods 3 periods 2 periods 1 period
Rule to Remember. a) All periods of a number contain three digits, or places (the first period on the left may or may not). b) Zero is used as a place-holder.
Average. When we want to find a single number that will represent all the numbers in a group of unequal numbers or quantities we find the average (or arithmetic mean). To find the average of a group of unequal numbers, we add the numbers and then divide their sum by the number of addends.
Notes:
1 to make it easier to read - ,
2 is determined by the place
3 ten times as great
EXERCISES
I. Read the following words paying attention to the pronunciation:
to separate, period, system, zero, average, digit, unequal.
II. Form nouns of the following verbs:
to read, to count, to move, to place, to contain, to find, to determine, to represent.
III. Make up sentences of your own using the words and expressions given below:
quantity, unequal, sum, to make it easier to read, to separate the figures of the number, to be determined by, ten times as great, ten times as small.
IV. Answer the following questions:
1. Why do we separate the figures of the numbers by commas? 2. How is each group of three figures called? 3. How is the system of numbers we use called? 4. How many digits does a period of a number contain? 5. How do we find the average of unequal numbers?
V. Translate into Ukrainian:
Our present-day number-symbols are Hindu characters. It is important to notice that no symbols for zero occur in any of this early Hindu number system. They contain symbols for numbers like twenty, forty, and so on. A symbol for zero had been indented in India. The invention of this symbol for zero was very important, because its use enabled the nine Hindu symbols 1, 2, 3, 4, 5, 6, 7, 8 and 9 to suffice for the representation of any number, no matter how great. The work of a zero is to keep the other nine symbols in their proper place.
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VI. Translate into English:
䳿. ϳ , . , , .
, .
, (separate) . .
TEXT 3
ADDING, SUBTRACTING, MULTIPLYING AND DIVIDING
THE WHOLE NUMBERS
The result of additions of numbers is called the sum or total of the numbers. The numbers to be added1 are called the addends. In adding a series of numbers, begin with the column at the right. If the sum of a column of digits is ten or larger, carry the tens digits and add it to the sum of the digits in the next column to the left. Careless mistakes are sometimes made because the work was not checked. It is always wise therefore to check your answer.
In subtracting whole numbers, the number which is to be made smaller2, or diminished is called the minuend; the number "taken away" or subtracted is called subtrahend. The answer is the difference between the minuend and the subtrahend and it is called the remainder, or difference. In checking a subtraction example, add the remainder and the subtrahend. If your answer is correct, the result obtained by addition equals the minuend.
In multiplication, the number by which you multiply is called the multiplier, the number being multiplied is called multiplicand. The number resulting from the multiplication is called the product. Multiplication can be checked by interchanging the multiplier and multiplicand and multiplying again. Remember that the product of any number multiplied by zero is zero. The product of any number multiplied by one is the same number. The order in which numbers are multiplied does not change the product.
In division, the number that is to be divided is called the dividend. The number by which the dividend is to be divided is called the divisor. The answer is called the quotient. The remainder is what is left over after the dividend has been divided into equal parts. If there is a remainder, it may be written over the divisor and expressed as a fraction in the quotient.
Notes:
1 the numbers to be added - ,
2 is to be made smaller-
EXERCISES
I. Read the following words paying attention to the pronunciation:
to add, addends, adding, to subtract, subtrahend, minuend, remainder, to multiply, product, dividend, divisor, quotient.
II. Give all possible derivatives of the following verbs:
to differ, to check, to answer, to change, to obtain.
III. Make up sentences of your own using the words and expressions given below:
the numbers to be added, the exercise to be checked, the work to be done, the number to be divided, can be made, can be divided, can be checked.
IV. Answer the following questions:
1. How is the result of addition called? 2. What do we do while adding a series of numbers? 3. Why do we sometimes make mistakes in adding numbers? 4. What is the result of subtracting whole numbers called? 5. How do we check a subtraction example? 6. What is the result of multiplication called? 7. What is the result of division called?
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V. Make up 6 questions to the text and answer them.
VI. Translate into Ukrainian:
Signs of Operations Used in Arithmetic The signs most used in arithmetic to indicate operations with numbers are plus (+), minus (), multiplication (X), and division (:) signs. When either of these is placed between any two numbers it indicates respectively that the sum, difference, product, or quotient of the two numbers is to be found. The equality sign (=) shows that any indicated operation or combination of numbers written before it (on the left) produces the result or number written after it.
VII. Learn by heart:
Five times five are twenty five; five times six are thirty; five times seven are thirty five; five times eight are forty; five times nine are forty five; five times ten are fifty; five times eleven are fifty five; five times twelve are sixty; six times nine are fifty four; six times ten are sixty; seven times nine are sixty three, seven times ten are seventy; eight times nine are seventy two; eight times ten are eighty; nine times nine are eighty one; nine times ten are ninety.
VIII. Translate into English:
, , , , , , .
³ , (by means of which) .
, , ; , , . 䳿, , , .
, , ; , , ; , , .
TEXT 4
