Counting things is easy now. You can count objects in large numbers and represent them through numerals. It is not as if we always knew how to convey large quantities in conversation or through symbols. Many thousands years ago, people knew only small numbers.

Gradually, they learnt how to handle larger numbers. They also learnt how to express large numbers in symbols. As human beings progressed, there was greater need for development of Mathematics and as a result Mathematics grew further and faster. 

We use numbers and know many things about them. Numbers help count concrete objects. They help to say which collection of objects is bigger and arrange them in order e.g., first, second, etc. Numbers are used in many different contexts and in many ways.

Ordering of Numbers

Ascending order

Ascending order means arrangement from the smallest to the greatest.

Descending order

Descending order means arrangement from the greatest to the smallest.

Place Value of Numbers

For example, 

45278 = 4 × 10000 + 5 × 1000 + 2 × 100 + 7 × 10 + 8

Here 8 is at ones place, 7 at tens place, 2 at hundreds place, 5 at thousands place and 4 at ten thousands place. The number is read as forty five thousand, two hundred seventy eight.

Large Numbers

1 hundred = 10 tens

1 thousand = 10 hundreds = 100 tens

1 lakh = 100 thousands = 1000 hundreds

1 crore = 100 lakhs = 10,000 thousands

Use of commas

In writing large numbers, you often use commas. Commas help in reading and writing large numbers. In Indian System of Numeration we use ones, tens, hundreds, thousands and then lakhs and crores. Commas are used to mark thousands, lakhs and crores.

The first comma comes after hundreds place (three digits from the right) and marks thousands. The second comma comes two digits later (five digits from the right). It comes after ten thousands place and marks lakh. The third comma comes after another two digits (seven digits from the right). It comes after ten lakh place and marks crore.

For example: 5,08,01,592; 3,32,40,781; 7,27,05,062

International System of Numeration

In the International System of Numeration, we have ones, tens, hundreds, thousands and then millions. One million is a thousand thousands. Commas are used to mark thousands and millions. It comes after every three digits from the right. The first comma marks thousands and the next comma marks millions. For example, the number 50,801,592 is read in the International System as fifty million eight hundred one thousand five hundred ninety two.

In the Indian System, it is five crore eight lakh one thousand five hundred ninety two.

To express numbers larger than a million, a billion is used in the International System of Numeration. 1 billion = 1000 million.

Estimation

There are a number of situations in which you do not need the exact quantity but need only a reasonable guess or an estimate. For example, while stating how many spectators watched a particular international hockey match, you can state the approximate number, say 51,000, you do not need to state the exact number.

Estimation involves approximating a quantity to an accuracy required. Thus, 4117 may be approximated to 4100 or to 4000, i.e. to the nearest hundred or to the nearest thousand depending on the need.

In number of situations, you have to estimate the outcome of number operations. This is done by rounding off the numbers involved and getting a quick, rough answer.  

Roman Numerals

One of the early systems of writing numerals is the system of Roman numerals. This system is still used in many places. The Roman numerals are from 1 to 10:

I, II, III, IV, V, VI, VII, VIII, IX, X

This is followed by XI for 11, XII for 12, ... till XX for 20. Some more Roman numerals are:

  • I: 1
  • V: 5
  • X: 10
  • L: 50
  • C: 100
  • D: 500
  • M: 1000

The rules for the system are:

  1. If a symbol is repeated, its value is added as many times as it occurs. II is equal 2, XX is 20 and XXX is 30.

  2. A symbol is not repeated more than three times. But the symbols V, L and D are never repeated.

  3. If a symbol of smaller value is written to the right of a symbol of greater value, its value gets added to the value of greater symbol. VI = 5 + 1 = 6, XII = 10 + 2 = 12 and LXV = 50 + 10 + 5 = 65

  4. If a symbol of smaller value is written to the left of a symbol of greater value, its value is subtracted from the value of the greater symbol. IV = 5 – 1 = 4, IX = 10 – 1 = 9, XL= 50 – 10 = 40, XC = 100 – 10 = 90

  5. The symbols V, L and D are never written to the left of a symbol of greater value, i.e. V, L and D are never subtracted. The symbol I can be subtracted from V and X only. The symbol X can be subtracted from L, M and C only.