Whole Numbers: Counting

 CONTENT

• Introduction
• System of Counting
• Counting in Millions
• Counting in Billions and Trillions

INTRODUCTION

1. Counting

It is likely that mathematics began when people started to count and measure. Counting and measuring are part of everyday life.

Ancient people used fingers and toes to help them count or group numbers in different number bases. This led them to collect numbers in groups: sometimes 5s (fingers of one hand), sometimes 10s (both hands) and even in 20s (hands and feet). When people group numbers in 5s, we say they use a base five method. The most common bases used were five, ten and twenty. For example, a person with thirty two cows would say ‘I have six fives and two cows’ when counting in base ten. The most widely used base is base ten also called the denary system.

Other bases of counting: seven and sixty

7 days = 1 week

60 seconds = 1 minute

60 minutes = 1 hour

In English, ‘dozen’ means 12, ‘score’ means 20 and ‘gross’ means 144

System of Counting

1. Tally System

Tally marks were probably the first numerals.

The ancient people employed tally marks to count large numbers. The tally marks were scratched on stones or sometimes cut on sticks but today we use tally marks to count or record large data, especially in statistics.

A tally mark of 5 is written by putting a line across a tally count of 4.

2. Roman numerals

The Romans used capital letters of the alphabets to represent numbers. Many people believe that the Romans used the fingers to represent numbers as follows:

I for one finger, II for two fingers, III for three fingers, V for five fingers and X for the combination of two hands ( or two V’s) .

The Roman also used L for fifty, C for hundred, D for five hundred and M for one thousand as shown below.

Hindu-Arabic	Roman Numeral	Hindu-Arabic	Roman Numeral
1	I	20	XX
2	II	40	XL
3	III	50	L
4	IV	60	LX
5	V	90	XC
6	VI	100	C
7	VII	400	CD
8	VIII	500	D
9	IX	900	CM
10	X	1000	M

The Roman used the subtraction and addition method to obtain other numerals. For example

• IV means V- I i.e.  5- 4 = 4
• VI means V+ I, i.e. 5 + 1 = 6
• IX means X- I, i.e. 10 – 1 = 9
• XXIV means XX + IV = 20 + 4 = 24
• CD means D- C = 500 – 100 = 400
• MC means M + C = 1000 + 100 = 1100

3. The Counting board

A counting board is a block of stone or wood ruled in columns. Loose counters, pebbles, stones or seeds in the columns show the value of the numbers in the columns.

Counters in the right-hand column (U) represent units, counters in the next column (T) represent tens, and so on.

An Abacus showing 2703

5. Place Value of Numbers

Numbers of units, tens, hundreds,…….., are each represented by a single numeral.

(a).For a whole number:

– the units place is at the right-hand end of the number.

– the tens place is next to the units place on the left, and so on

For example: 5834 means ↓

5  thousands,  8 hundreds, 3 tens, and 4 units.

See the illustration below:

6    →   units

.     →   decimal

7    →    tenths

9    →    hundredths

8    →    thousandths

Example 1:

What is the place value of each of the following?

• the 9 in 10269
• the 2 in 2984

Solution:

• the 9 in 10269 is = 9 units or nine units
• the 2 in 2984 is = 2 thousands or two thousands

Example 2

What is the value of each of the following?

• the 8 in  1.85
• the 0 in 16.08

Solution:

• the 8 in 1.85 is = 8 tenths or eight tenths
• the 0 in 16.08 is =0 in tenths or zero tenths