ICSE Class 6 Maths Chapter 01 Number System

Unit 1 - Numbers

Chapter 1: Number System

Introducing Unit, Number, Numeral and Numeration

In mathematics unit (unity) means a single thing. For example, a pen, a boy, a metre, a day, etc.

The number written before the name of a unit indicates how many times that unit is taken.

For example:

(i) Four pens means a pen (unit/thing) is taken 4 times, i.e. 4 pens are taken.

(ii) Length = 3 m means unit of length is m (metre), and it is taken 3 times.

(iii) Weight = 63 kg means unit of weight used is kg (kilogramme), and it is taken 63 times.

Numeral and Numeration

A numeral is a symbol representing a given number and numeration represents that number in words.

Teacher's Note

Understanding units helps us measure and communicate quantities in everyday life, from buying groceries by weight to measuring distances and time.

Hindu-Arabic System of Numeration

The Hindu-Arabic system of Numeration is in fact the decimal system that is in use all over the world. This system was developed by the ancient Hindu-Mathematicians in India and was carried to the West by the Arabs. For this reason, it is called the Hindu-Arabic system of numeration.

In Hindu-Arabic number system (also known as denary system); the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are used to write a numeral (number). Each of these ten symbols is called a digit. Out of these digits:

(i) 0, 2, 4, 6 and 8 are even numerals.

(ii) 1, 3, 5, 7 and 9 are odd numerals.

Teacher's Note

The digits we use every day - 0 through 9 - originated in India and spread globally through Arab traders, making this system truly international.

Place Value (Local Value) and Face Value (True Value)

In a number, the place value of any digit, is the value of this digit according to its position in the number, whereas the face value of every digit used in the number is the digit itself.

Thus:

1. The place value of a digit depends upon the position it occupies in the number.

2. The face value of a digit is the digit itself.

3. Zero (0) is the only digit whose face value and place value are the same, irrespective of its position in the number. As a result, the face value of 0 is 0 and even the place value of 0 is 0.

Example 1

Write the place values of the two 6s (sixes) used in the number 36268 and find the sum of these two values.

Solution

In 36268, one 6 occurs at thousand's place, so its place value = 6000 (Ans.)

The other 6 occurs at ten's place, so its place value = 60 (Ans.)

The sum of these two place values of 6 = 6000 + 60 = 6060 (Ans.)

Example 2

Write the place values of the two 5s in 9,45,582 and find the difference of these place values.

Solution

In 9,45,582, the first 5 occurs at thousand's place

Its place value = 5 thousand = 5000 (Ans.)

The second 5 occurs at hundred's place

Its place value = 5 hundred = 500 (Ans.)

The difference of the two place values of 5 = 5000 - 500 = 4500 (Ans.)

A concrete number is a number which refers to a particular unit; such as: 8 metre, 12 kg, 18 km, 36 cm, etc.

An abstract number is a number which does not refer to any particular unit; such as: 8, 12, 18, 36, etc.

Teacher's Note

When you see a price tag showing 25.99 or a distance of 5 km, you're witnessing place value in action - each digit's position determines its actual worth.

Exercise 1(A)

1. Fill in the blanks:

(i) In 20 kg, the unit is -, which is taken - times.

(ii) In 80 m, the unit is -, which is taken - times.

(iii) If a unit cm (centimetre) is taken 5 times, the corresponding quantity is -.

(iv) If a unit km (kilometre) is taken 24 times, the corresponding quantity is -.

(v)

2. Fill in the blanks:

(i) In 24,673, the place value of 6 is - and the face value of 4 is -.

(ii) In 8,039, the face value of 8 is - and the place value of 9 is -.

(iii) In 3,25,648, the local value of 5 is - and the true value of 3 is -.

(iv) In 6,439, the true value of 6 is - and the face value of 6 is -.

3. Find the difference between the place value and the face value of 9 in the number 3945.

4. In the number 40562,

(i) the local value of 5 = -.

(ii) the true value of 6 = -.

(iii) the sum of the local value of 5 and the true value of 6 = -.

5. In the number 347825, write the difference between

(i) the place value and the face value of 2.

(ii) the place value and the face value of 5.

6. The number 978036 has six digits. Write the digit/digits used in this number that have the same place value and face value.

Forming Numbers Using Given Digits

(a) Use digits 4 and 7 to form all possible two-digit numbers such that no digit is repeated.

No digit is repeated. Neither 44 nor 77 is allowed.

Required numbers are 47 and 74.

(b) Use digits 2, 5 and 8 to form all possible three-digit numbers such that no digit is repeated.

Numbers of the type 225, 282, 222, 525, etc. are not allowed

Required numbers are 258, 285, 528, 582, 825 and 852.

To Form The Smallest and The Greatest Numbers Using Given Digits

(a) When the given digits include digit 0.

Example 3

Form the smallest and the greatest 6-digit number using the digits 2, 0, 7, 8, 9 and 5 without repetition.

Solution

To obtain the smallest number:

The smallest digit, other than zero, is put at the extreme left, then comes zero, and then the remaining digits in ascending (increasing) order of value.

Since, out of the given digits '2, 0, 7, 8, 9 and 5, the smallest digit other than 0 is 2, write 2 at the extreme left, then write 0 and then the remaining digits (7, 8, 9 and 5) in ascending order of value, i.e., 5, 7, 8 and 9.

Thus, the required smallest number is 205789. (Ans.)

The number 025789 is not a 6-digit number; it is a 5-digit number. A number cannot begin with the digit 0.

To obtain the greatest number:

Put the greatest digit at the extreme left, then put the remaining digits in descending order of their values, with 0 at the end.

Thus, the required greatest number is 987520. (Ans.)

(b) When the given digits do not include digit 0:

Example 4

Form the smallest and the greatest 4-digit number using the digits 3, 8, 5 and 2 without repetition.

Solution

To obtain the smallest number:

Write the digit with smallest value at the extreme left, and then the remaining digits in ascending order of their values.

The required smallest number = 2358 (Ans.)

To obtain the greatest number:

Write the digit with largest value at the extreme left, and then the remaining digits in descending order of value.

The required greatest number = 8532 (Ans.)

Example 5

(i) What is the smallest number of five digits?

(ii) What is the greatest number of five digits?

Solution

(i) To form the smallest number of five digits, place 1 (unity) at the extreme left, and then four zeros to the right of 1.

The required smallest number of five digits = 10000 (Ans.)

(ii) In forming the greatest number of five digits, we should have the greatest digit, i.e. 9, in all places.

The required greatest number of five digits = 99999 (Ans.)

Teacher's Note

When arranging digits to form numbers, remember that the leftmost position has the greatest influence on the final value - this is why 0 can never start a number.

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