Whole numbers

DEVELOPMENT OF NUMBER SYSTEM

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. Table 1.1 shows how to use the letters

Table 1.1

EXAMPLE 1: Write these numbers in Roman numerals.

1. a) 25 b) 105 c) 49 d) 2011

Solution

1. a) 25 = XXV
2. b) 105 = CV
3. c) 49 = XLIX
4. 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

CLASS ACTIVITY

1. Write these numbers in Roman Numerals
2. a) 352 b) 1 257 c) 2456
3. Add the following Roman numerals and give your answers in figures
4. a) XXV and CV b) XXIV and MDCIX.

What are whole numbers?

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

25. a) 25.436? Answer: three hundredths
26. b) 5.368? Answer: three tenths
27. c) 346.12? Answer: three hundreds.

CLASS ACTIVITY

1. What is the value of 5 in: a) 3572? b) 5372? c) 25347869?
2. State the place value of the quoted digit in each of the following
3. i) 43’7’68
4. ii) 56.73’2’4

iii) 4.’5’21096

9431. iv) ‘7’09431.58

Counting in tens, hundreds, thousands, ten thousands, hundred thousands, millions and billions

1. Numbers written in tens contains 2 digits. Examples: 12, 78, 73 etc.
2. 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.

1. 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.

CLASS ACTIVITY:   State what the following numbers stands for

1. i) 178 000 000 (ii) 234 000 000 000 (iii) 23 000 (iv)500 000 000 000 (v) 67 000 000 000 000

TRANSLATION OF NUMBERS WRITTEN IN FIGURES TO WORDS AND VICE-VERSA

Example 1

Write the following numbers in words:

• 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

• 6 006 006 006

Solution:

• 6006006006 = 6 006 006 006 stands for six billion, six million, six thousand and six

Example 2

Write the following words in numerals

• Three hundred and fifty four thousand, seven hundred and twenty
• Seven billion, two hundred and sixty four million, one hundred and one thousand, two hundred and two

Solution:

• Using expanded form,

300 000 + 50 000 + 4000 + 700 + 20 = 354 720

• 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.

CLASS ACTIVITY:

Question 1.   Write the following figures in words:

(i).   15284037     (ii).  789030861     (iii).   512278374415

(iv).  734015090700018     (v).  89780260044784 .

Question 2.   Express the following in figures:

                 (i). Seven hundred and ninety-eight million, one hundred and thirty- two thousand five

Hundred and forty- five.

                (ii). Twenty-four billion, seventy-eight million, four hundred and thirty-six thousand, one

Hundred and forty -eight.

(iii). Thirteen trillion, nine hundred and forty-one billion, three hundred and twenty-four million, forty-seven thousand, one hundred and ninety-eight.

   (iv). Four hundred and seventeen trillion, two hundred and eighty billion, five hundred and        six thousand, eight hundred and eighteen.

                 (v). Eighteen million, twenty-five thousand, six hundred and one.

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)

• (7:49) (10:100)           (11:?)

Here first numbers in a bracket are squared to have the second number.

(11:?) = (11: 121)

• (490000: 700) (1210000:?)

Here the square root of the first numbers in a bracket gives second number.

(1210000: ?) = (1210000: 1100)

Evaluation:

Now do the following:

1. (12300 : 151290000) (45600 : ?)
2. (10000 : 100000000) (200000 : ?)
3. (9000 : 30) (?:250)

Class activity

Now do the following using the above samples

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, 18) mean MY FATHER.

Class activity

Given that the English alphabets is represented with figures 1 to 26, translate

1. a) (20, 8, 5) (2, 15, 25) b) (14, 9, 7, 5, 18) (1, 18, 5, 1)
2. Change the following messages to codes:
3. a) A funny boy
4. b) I love figures
5. c) Maths is fun.

PRACTICE QUESTIONS

1. 1. Write the year you were admitted to DLHS in Roman numerals.
2. What is the place value of 7 in: a) 76500000000; b) 4.752 c) 347108.345?
3. Write 90457623145 in words
4. Write Three trillion, four hundred and fifty-eight billion, five hundred and four in figures.
5. The place value of 5 in 43.576 is ……

ASSIGNMENT

1. Find the sum of MCDV and XCIV leave your answer in Roman numeral.
2. Write 1999 in Roman numeral
3. Express in words only: a) 2.5 billion b) 1.3 trillion
4. Express a) 560 000 b) 3 500 000 in a mixture of digits and words.
5. Write in figures only, three billion, four hundred and fifty thousand, and seven.

KEYWORDS

1. i) Numbers ii) Place value iii) Numeral.