Basic Knowledge on Whole Numbers

Numeral System

       We use numbers every day. The number system that we use is the Hindu-Arabic system. It is based on ten symbols called digits. They are

0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

       They are also the ten smallest whole numbers. Any other whole number can be written using the ten digits and the idea of place values.

       Using two or three digits, we can write the next nine hundre and ninety whole numbers:

10, 11, ..., 20, 21, ..., 98, 99, 100, 101, ..., 200, 201, ..., 998, 999

Very large whole numbers can be written with more digits. In June 2010, the estimated poulation of the Philippines was 94,013,200 people or ninety-four, thirteen thousand, two hundred people, in words. Here are the place values for the 8 digits of the number that represents the Philippine population.

9 - ten millions (10 000 000)
4 - millions (1 000 000)
0 - hundred thousands (100 000)
1 - ten thousands (10 000)
3 - thousands (1 000)
2 - hundreds (100)
0 - tens (10)
0 - ones (1)

Hence, 94,013,200

=(9 x 10,000,000) + (4 x 1,000,000) + (0 x 100,000) + (1 x 10,000) +

(3 x 1,000) + (2 x 100)

       In the above illustration, we could see that the Hindu-Arabic numeration system is built on groups of 10, 10=10 x 1, 100=10 x 10, 1000=10 x 100, and so on.

       Thus, the system is known as the base ten system or the decimal system.

Uses of Numbers

       It is hard to live even for one day without using or thinking about numbers. Numbers are used on clocks, calendars, license plates, rulers, scales, and so on. The major uses of numbers are listed here.

Kinds of Numbers

Four Basic Arithmetic Operations (+, -, x, /)

       An operation in mathematics is something we do to numbers, such as adding, subtracting, multiplying, or dividing. For instance, in

13+41,

the operation of addition is performed on the numbers 13 and 41.

       The following table shows the arithmetic operations.

Operation

Addition

Subtraction

Multiplication

Division

Symbol

+

-

x

 /
Result Sum Difference Product Quotient (and
remainder)

       The following tables show some examples of the elementary mathematics operations, and terms that we use to describe them.

Expression 25+38=63 387-259=128
Statement
  1. The sum of 25 and
    38 is 63.

  2. 25 plus 38 is 63.
  1. The difference
    between 387 and
    259 is 128.

  2. 387 minus 259 is 128


Expression 48x13=624 461/55=8 R21
Statement
  1. The product of 48
    and 13 is 624.
  2. 48 times 13 is 624
 Divide 461 by 55, the
quotient is 8 and the
remainder is 21.

The Number Line

       In mathematics, it is often useful to present whole numbers by points on a line called the number line

       To illustrate a number line, draw a line. Choose any point on the line and label it 0. Starting with 0, mark off equal intervals of any suitable length. Label the points marked 1, 2, 3, 4, ..., then draw arrows on the extreme right and left. The arrows on the extreme right and left indicate that the list of numbers continues in the same way indefinitely.

       A number on the number line is always grater than any number on to its left and smaller than any number to its right, that is, 4 > 3 and 4 < 5.

Properties of the Four Operations

       There are four mathematical properties which involve addition and multiplication



Commutative Property

2+7 is the same as 7+2,

2x7 is the same as 7x2.

       That is, the order in which the two numbers appear does not affect the answer.

MATH FOCUS
For any numbers a and b
  • a + b=b + a      (Commutative Property of Addition)

  • a x b=b x a      (Commutative Property of Multiplication)



Associative Property

       In addition and multiplication, we can see that

(2+3)+4 = 2+(3+4),

(2x3)x4 = 2x(3x4).

That is, the way in which the three numbers are grouped does not affect the answer

MATH FOCUS
For any numbers a, b, and c,
  • (a + b) + c=a + (b + c)      (Associative Property of Addition)

  • (a x b) x c=a x (b x c)      (Associative Property of Multiplication)



Distributive Property

       Mark and Joseph counted the books in the bookshelves in two different ways.

Mark: "There are altogether 3 rows of (5 + 3) books."

Joseph: "There are 3 rows of 5 books in the left shelf and 3 rows of 3 books in a right shelf."

Who do you think was correct?

Consider the following:



3 x (5 + 3)=3 x 8

=24
and 3 x 5 + 3 x 3=15 + 9

=24
Thus, we can see that

3 x (5 + 3)=3 x 5 + 3 x 3.

MATH FOCUS
For any numbers a, b, and c,
  • a x b (b + c)=a x b + a x c      (Distributive Property of Multiplication over addition)

  • a x (b - c)=a x b - a x c      (Distributive Property of Multiplication over Subtraction)



Flexible Use of Distributive Property

       The distributive property of multiplication over addition can also be extended to each of the following operations:



Multiplication by Zero

       It is useful to know that any number multiplied by 0 gives an answer of 0.

For example, 519 x 0=0.

       That is, for any number a, a x 0=0.

Multiplication by powers of 10

       A power of any number is either 1 or that number is multiplied by itself one or more times. We will discuss this idea in details in another section. Some of the powers of ten are 1, 10, 100, 1000, 10 000, and so on.

1
10
10 x 10=100
10 x 10 x 10=1,000
10 x 10 x 10 x 10=10,000

       Multiplication by powers of 10 can be done mentally. The following examples illustrate an important pattern:

6 x 1=6
6 x 10=60
6 x 100=600
6 x 1,000=6,000
6 x 10,000=60,000

       If one of two whole number factors is 1,000, the product will be the other factor with three zeros (000) written to its right. Two zeros (0) are written to the right of the factor when multiplying by 100, and one zero (0) is written when multiplying 10. Will multiplication by one million (1,000,000) result in writing six zeros to the right of the other factor? The answer is yes.

MATH FOCUS

Multiplication by powers of 10

The product of a number and a power of 10 is equal to the number and as many zeros to its right as power of 10.
Hence the product when a number is multiplied by

10 is the number and 0 to its right;
100 is equal to the number and 00 to its right;
1,000 is equal to the number and 000 to its right;
10,000 is equal to the number and 000 to its right;
and so on.



Shortcut Computation

       Computation of large numbers with the use of pencil and paper can sometimes be tedious and difficult. In some cases, however by applying the properties of operations, computations could be simpler and shorter.