Whole numbers
There were many ancient ways of writing numbers part of which are the Hindu Arabic
system, tally system, Roman system, etc. While so many have gone into extinction, the
Roman system is still in use up to date.
ROMAN NUMBER SYSTEM
The Roman number system was developed about 300BC. The Romans used capital letters
of the alphabet for numerals. The table below shows how to use the letters
EXAMPLE 1: Write these numbers in Roman numerals.
1. a) 25 b) 105 c) 49 d) 2011
Solution
(a) 25 = XXV
(b) 105 = CV
(c) 49 = XLIX
(d) 2011 = MMXI
EXAMPLE 2: What numbers do these Roman numerals represent?
1. XLVI 2. XCIX 3. MMCMLIV 4. MMMDCI
Solution:
1. XLIV = 46 2. XCIX = 99 3. MMCMLIV = 2954 4. MMMDCI = 3601
What are whole numbers?
The whole numbers are the part of the number system which includes all the positive integers from
0 to infinity. These numbers exist in the number line. Hence, they are all real numbers. We can say,
all the whole numbers are real numbers, but not all the real numbers are whole numbers
The whole numbers are the numbers without fractions and it is a collection of positive integers and
zero. It is represented by the symbol “W” and the set of numbers are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9,}.
Zero as a whole represents nothing or a null value.
Whole Numbers: W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Natural Numbers: N = {1, 2, 3, 4, 5, 6, 7, 8, 9,}
Integers: Z = {….-9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,…}
Counting Numbers: {1, 2, 3, 4, 5, 6, 7.}
Whole Numbers are also called Integers. There are positive Integers and negative
Integers. Examples of positive integers are 1, 2, 3, 4, 5, etc., while examples of negative
integers are – 1, – 2, – 3, – 4, – 5, etc.
The figure 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are called digits or units which form counting
numbers.
PLACE VALUES
The value of the position of a digit within a number is called the place value. When any
whole number is written, the value of each digit depends on its position in the number. In
the common decimal system that we use, the value of a digit increases each time it moves
from left to right by ten times,
E.g. 4 = 4 units
40 = 4 tens
400 = 4 hundreds
4 000 = 4 thousands
The number 7483 is represented as
THOUSANDS HUNDREDS TENS UNITS
7 4 8 3
EXAMPLE1: What is the place value of 6 in 8643?
Solution: The place value of 6 in 8643 is six hundreds.
EXAMPLE2: What is the place value of 3 in
(a) 25.436?
Answer: three hundredths
(b) 5.368?
Answer: three tenths
(c) 346.12?
Answer: three hundreds.
Numbers written in tens contains 2 digits. Examples: 12, 78, 73 etc.
Numbers written in hundreds are always in 3 digits. Examples: 185, 359,
675, etc.
Numbers written in thousand contains 4 digits. Examples: 1254, 7566,
9081, etc.
Numbers written in ten thousands contains 5 digits.
Examples: 12,000 stands for 12 thousand
78,000 stands for 78 thousand
1. Numbers written in hundred thousand contains 6 digits.
Examples: 460 000 stands for 460 thousands
300,000 stands for 300 thousands
1. Numbers written in millions must contain at least 7 digits. The seven digits
must have two spaces separating them in “threes” from the right hand
side.
Examples: 12,000,000 stands for 12 million.
2,000,000 stands for 2 million.
1,000,000 stands for 1 million.
238,000,000 stands for 238 million.
Counting in billions:
Numbers written in billions must contain at least ten digits with three spaces separating
them in “threes” from the right hand side.
Examples: 12,000,000,000 stands for 12 billion.
4,000,000,000 stands for 4 billion.
7,456,201,456 stands for 7 billion, four hundred and fifty six million, two hundred and
One thousand, four hundred and fifty six.
835,000,000,000 stands for 835 billion.
Counting in trillions
Numbers written in trillions must contain at least thirteen digits with four spaces separating
them in “threes” from the right hand side.
Examples: 7 000 000 000 000 stands for 7 trillion.
25 000 000 000 000 stands for 25 trillion.
714 000 000 000 000 stands for 714 trillion.
1 000 million is called a trillion.
TRANSLATION OF NUMBERS WRITTEN IN FIGURES TO WORDS AND VICE-VERSA
Example 1
Write the following numbers in words:
1. 51 807 508 051 754
Solution:
51 807 508 051 754 = 51 807 508 051 754 stands for fifty one trillion, eight hundred and
seven billion, five hundred and eight million, fifty one thousand, seven hundred and fifty four
1. 6 006 006 006
Solution:
1. 6006006006 = 6 006 006 006 stands for six billion, six million, six
thousand and six
Example 2
Write the following words in numerals
1. Three hundred and fifty four thousand, seven hundred and twenty
2. Seven billion, two hundred and sixty four million, one hundred and one
thousand, two hundred and two
Solution:
1. Using expanded form,
300 000 + 50 000 + 4000 + 700 + 20 = 354 720
1. 7 000 000 000 + 200 000 000 + 60 000 000 + 4 000 000 + 100 000 + 1000
+ 200 + 2
= 7 264 101 202
NOTE: We no longer use commas between the groups of digits. Many countries use
a comma as a decimal point; thus, to avoid confusion do not use commas for
grouping the digits.
QUANTITATIVE APTITUDE REASONING
Problem solving in quantitative aptitude reasoning using large numbers
Sample1:
Study these examples and use them to answer the given questions.
(81:9) (100:10) (144:12)
(5:25) (8:64) (13:169)
1. (7:49) (10:100) (11:?)
Here first numbers in a bracket are squared to have the second number.
(11:?) = (11: 121)
1. (490000: 700) (1210000:?)
Here the square root of the first numbers in a bracket gives second number.
(1210000: ?) = (1210000: 1100)
Simple codes
A way of sending messages is by using numbers to represent letters of the alphabet. The
method is called coding.
Example: What does (13, 25) (6, 1, 20, 8, 5, 18) mean if 1 – 26 is represented by the letters
of the English alphabet A – Z.
Solution: 13 = M; 26 = Y; 6 = F; A = 1; T = 20; H = 8; E = 5; 18 = R.
Thus, (13, 25) (6, 1, 20, 8, 5, and 18) means MY FATHER.
