KNOWING OUR NUMBERS�CLASS – VI�MATHEMATICS

By

D.L.N.Achary, TGT(Maths),

JNV, Nayagarh, Odisha

Knowing Our Numbers

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Chapter – 1

Natural Numbers

• These are the numbers that we all know, like One, Two and Three ,…. etc!

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• They are represented by symbols 1,2,3 ,… etc.

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• But more importantly they can be ordered .

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• The numbers which are used for counting purpose are called Natural Numbers.

Disciplining Numbers�See this messy picture of undisciplined numbers.

How do we bring order to the numbers ?

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Solution :

One way to do that is to arrange them in a number line .

What is a number line ?

• Number Line is nothing but the collection of ‘Positive’ and ‘Negative’ numbers arranged serially according to their sizes with zero as center .

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Comparing Numbers ( Positive )

• Notice that any two natural numbers can be compared, i.e. given two natural numbers that are not equal, one is larger than the other.

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• For example, Take 11 and 5. We can say that 11 is greater than 5 and 5 is less than 11 .

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• The symbol used to represent greater than is ‘>’ and the symbol used for less than is ‘<’.

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• The above example can be stated as ‘11>5’ or ‘5<11’ in terms of symbolic notation .

Arranging Positive Numbers based upon their size ( Serially )

The magnitude of the numbers increase as one goes to the right of the number line !

Negative Numbers

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• Negative numbers are numbers marked with ‘-’ sign . They are -1,-2,-3 … etc.

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• They play an important role in representing loss or often , they act as an opposite of positive numbers.

Comparing negative numbers

• In the same way we compared two positive numbers using “<” and “>” we can compare the negative numbers using the same signs .

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• But there is a principle to be followed while

comparing them. “The larger the negative number the smaller , is its size” .

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• For example, -11 is less than -5 and -100 less than -10 .

Splitting the number line !

• We can see the number line described above is formed by joining positive number line and negative number line with zero in the middle and can be decomposed into negative and positive numbers , as follows :

Comparing numbers�

Comparing numbers when the total number of digits is different

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• The number with most number of digits is the largest number by magnitude and the number with least number of digits is the smallest number.

�Example: Consider numbers: 22, 123, 9, 345, 3005. The largest number is 3005 (4 digits) and the smallest number is 9 (only 1 digit)

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Comparing numbers…�

• Comparing numbers when the total number of digits is same

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The number with highest leftmost digit is the largest number. If this digit also happens to be the same, we look at the next leftmost digit and so on.

�Example: 340, 347, 560, 280, 265. The largest number is 560 (leftmost digit is 5) and the smallest number is 265 (on comparing 265 and 280, 6 is less than 8).

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�Ascending and Descending Order �

• Ascending order: Arranging numbers from the smallest to the greatest.

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• Descending order: Arranging numbers from the greatest to the smallest number.

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Example: Consider a group of numbers:

32, 12, 90, 433, 9999 and 109020.

�They can be arranged in descending order as

109020, 9999, 433, 90, 32 and 12,

They can be arranged in descending order as

12, 32, 90, 433, 9999 and 109020.

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�How many numbers can be formed using a certain number of digits?�

• If a certain number of digits are given, we can make different numbers having the same number of digits by interchanging positions of digits.

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Example: Consider 4 digits: 3, 0, 9, 6. 

Using these four digits,�(i) Largest number possible = 9630�(ii) Smallest number possible = 3069

(Since 4 digit number cannot have 0 as the leftmost number, as the number then will become a 3 digit number)

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�Shifting digits�

• Changing the position of digits in a number, changes magnitude of the number.

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Example: Consider a number 789. If we swap the hundredths place digit with the digit at units place, we will get 987 which is greater than 789.

�Similarly, if we exchange the tenths place with the units place, we get 798, which is greater than 789.

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�Place value�

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• Each place in a number, has a value of 10 times the place to its right.

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Example: Consider number 789.

� (i) Place value of 7 = 700� (ii) Place value of 8 = 80� (iii) Place value of 9 = 9

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�Introducing large numbers�

• Large numbers can be easily represented using the place value. It goes in the ascending order as shown below

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For example : 9951024 can be placed in place value chart

Place Value�( Indian and International )

Indian & International System

USE OF COMMAS - Rules

• 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.

Example : The number 5,08,01,592 is read as five crore

eight lakh one thousand five hundred ninety two.

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• In the International System of Numeration, as it is being used we have ones,tens, hundreds, thousands and then millions. One million is a thousand thousands.

Example : The number 50,801,592 is read as fifty million

eight hundred one thousand five hundred ninety two.

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�Estimation�

• When there is a very large figure, we approximate that number to the nearest plausible value. This is called estimation.
• Estimating depends on the degree of accuracy required and how quickly the estimate is needed.

Example:

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Estimating sum or difference

• Estimations are used in adding and subtracting numbers.

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• Example of estimation in addition: Estimate 7890 + 437.�Here 7890 > 437.

�Therefore, round off to hundreds.�7890 is rounded off to       7900�437 is rounded off to      +   400�Estimated Sum =              8300�Actual Sum        =              8327

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• Example of estimation in subtraction: Estimate 5678 – 1090. �Here 5678 > 1090.

�Therefore, round off to thousands.�5678 is rounded off to       6000�1090 is rounded off to    – 1000�Estimated Difference =     5000�Actual Difference        =     4588

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�Estimating products of numbers�

• Round off each factor to its greatest place, then multiply the rounded off factors.

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• Estimating the product of 199 and 31:�199 is rounded off to 200�31 is rounded off to 30�Estimated Product = 200 × 30 = 6000�Actual Result = 199 × 31 = 6169

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�BODMAS - Rule�

• We follow an order to carry out mathematical operations. It is called as BODMAS rule.

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• While solving mathematical expressions, parts inside a bracket are always done first, followed by of, then division, and so on.

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BODMAS Rule- Example

• [(5 + 1) × 2] ÷ (2 × 3) + 2 – 2 = ?

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Ans : [(5 + 1) × 2] ÷ (2 × 3) + 2 – 2….

{Solve everything which is inside the brackets}

= [6 × 2] ÷ 6 + 2 – 2…..

{Multiplication inside brackets}

= 12 ÷ 6 + 2 – 2…… {Division}

= 2 + 2 – 2…… {Addition}

= 4 – 2……. {Subtraction}

= 2

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�Roman Numerals�

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• Digits in Roman are represented as :

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

• Some other Roman numbers are :

I = 1, V = 5 , X = 10 ,

L = 50 , C = 100 , D = 500 ,

M = 1000

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�Rules for writing Roman numerals�

• If a symbol is repeated, its value is added as many times as it occurs.�Example: XX = 10 + 10 = 20
• A symbol is not repeated more than three times. But the symbols X, L and D are never repeated.
• 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.�Example: VII = 5 + 2 = 7
• 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 greater symbol.�Example: IX = 10 – 1 = 9.
• Some examples : 105 = CV , 73 = LXXIII and 192 = 100 + 90 + 2 = C  XC  II = CXCII

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Thanking You ….