Knowing Our Numbers

 Mind Maps

Class VI - maths: Knowing Our Numbers
One Word Answer Questions:
Q) What is place value of 2 in 2341?

Q) Write the number 56798 in expanded form?

Q) Expand 78456 ?

Q) Write the numeral Sixty-two lakh five?

Q) Write the predecessor of the number 67,67,909?

Q) 1 million is equal to?

Q) 1 billion is equal to?

Q) LXI=?

Q) place value of 1 in 1024

Q) Identify the greatest and smallest numbers of 45,453,3546,78

Q) 14 lies between the numbers.

Q) 1 cm __________ mm

Q) Find the greatest and smallest numbers in 24,53,29,62,13,75,11 ?

Q) Round off 3934554 to the nearest ten in number?

Q) Compare 567654 and 567646 using >,< or= ?

Q) Write the hindu and arabic numerals for the following:
(a) CLIV?
(b) CX?

Q) Solve the following using the roman numerals
(a) XCXI - XV?
(b) LIX+CL+CV?

Q) Write the definition of shifting digits?

Q) What is sum?

Q) What is difference?

Q) What is product?

Q) What is quotient?

Q) A man saves \$ 653 every month. Estimate the amount of money saved by him in two year?

Q) A packet can hold 231 screws. Estimate the number of packet required to pack 34,46,876 screws?

Q) In a town, there are 6,200 men,3,876 women,and 2,908 children. find the estimate population of the town by rounding off the numbers to the nearest hundred?

Q) There are 2982 coins in a red bag and 2301 coins in a green bag. Estimate the the total number of coins to the nearest hundred in both the bag?

Q) A shopkeeper has 5653 kg of sugar. He sells 53 kg of sugar every day. Estimate how much sugar is left after a days sale to the nearest hundred ?

Q) What is rounding off the numbers?

Q) Round of number which is nearest to the 560000?

Q) Simplify (105 x 45)+9?

Q) Write the greatest 5-digit number ending in 1

Q) Write three successors of 865.

Q) 9999 + 999 = ?

Content
• Introduction
• Larger numbers
• Estimation by rounding off the numbers
• Rounding of Numbers
Summary
1. Given two numbers, one with more digits is the greater number. If the number of digits in two given numbers is the same, that number is larger, which has a greater leftmost digit. If this digit also happens to be the same, we look at the next digit and so on.
2. In forming numbers from given digits, we should be careful to see if the conditions under which the numbers are to be formed are satisfied. Thus, to form the greatest four digit number from 7, 8, 3, 5 without repeating a single digit, we need to use all four digits, the greatest number can have only 8 as the leftmost digit.
3. The smallest four digit number is 1000 (one thousand). It follows the largest three digit number 999. Similarly, the smallest five digit number is 10,000. It is ten thousand and follows the largest four digit number 9999. Further, the smallest six digit number is 100,000. It is one lakh and follows the largest five digit number 99,999. This carries on for higher digit numbers in a similar manner.
4. Use of commas helps in reading and writing large numbers. In the Indian system of numeration we have commas after 3 digits starting from the right and thereafter every 2 digits. The commas after 3, 5 and 7 digits separate thousand, lakh and crore respectively. In the International system of numeration commas are placed after every 3 digits starting from the right. The commas after 3 and 6 digits separate thousand and million respectively.
5. Large numbers are needed in many places in daily life. For example, for giving number of students in a school, number of people in a village or town, money paid or received in large transactions (paying and selling), in measuring large distances say between various cities in a country or in the world and so on.
6. Remember kilo shows 1000 times larger, Centi shows 100 times smaller and milli shows 1000 times smaller, thus, 1 kilometre = 1000 metres, 1 metre = 100 centimetres or 1000 millimetres etc.
7. There are a number of situations in which we 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, we state the approximate number, say 51,000, we do not need to state the exact number.
8. 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 our need.
9. In number of situations, we have to estimate the outcome of number operations. This is done by rounding off the numbers involved and getting a quick, rough answer.
10. Estimating the outcome of number operations is useful in checking answers.
11. Use of brackets allows us to avoid confusion in the problems where we need to carry out more than one number operation.
12. We use the Hindu-Arabic system of numerals. Another system of writing numerals is the Roman system.
Introduction
1. Ancient method of counting.
2. Why numbers are required.

Ancient method of counting :

• Facts clearly state that counting sense in humans emerged long before the names of the numbers 1,2,3,4......
• In ancient times,the method of counting was based on distinct,uniform objects like fingers,stones,knots and lines.

For Example : Kharosthi Numerals : Indian writing found in third century B.C.

Here, 1 is represented by I line,2 is represented by II,3 is represented by III and so on.... this is more of a symbolic representation to counting rather than a numerical representation.

Why numbers are required :

• Numbers are required to count concrete objects.
• Numbers help to create sequence by arranging smaller to bigger and bigger to smaller.
• Helps in creating order.

Shifting digits

Definition : Shifting given set of numbers from one place to another.

Let us consider set of certain numbers :

Examples : Exchange digit at hundreds place to the digit at ones place 1.2.3.Introducing 10000

As we all know that 99 is the largest two digit number,1 added to 99 gives us the smallest 3 digit number.

99+1 =100 ( smallest three digit number )

Similarly,

999+1 =1000 (smallest four digit number )
Largest 3 digit number + 1 = smallest 4 digit number.

9999+1 =10000 (smallest five digit number )
Largest 4 digit number + 1 = smallest 5 digit number.

 10 → 1×10 100 → 10×10 1000 → 10×100 10000 → 10×1000 10000 is actually 10 times 1000.so it is 10000

So,we concluded that

 Greatest single digit number+1=Smallest 2 digit number Greatest 2 digit number  +1=Smallest 3 digit number Greatest 3 digit number  +1=Smallest 4 digit number Greatest 4 digit number  +1=Smallest 5 digit number

1.2.4 :Revisiting place value

Let us consider few examples.

• 28 = 20 + 8 = 2 × 10 + 8
• 528 = 500 + 20 + 8 = 5 × 100 + 2 × 10 + 8 × 1
• 4528 = 4000 + 500 + 20 + 8 = 4 × 1000 + 5 × 100 + 2 × 10 + 8 × 1
• 64528 = 60000 + 4000 + 500 + 20 + 8 = 6 × 10000 + 4 × 1000 + 5 × 100 +2 × 10 + 8 × 1.

Let us illustrate above example using a table :

Number Ten Thousand Thousand Hundreds Tens Ones
28 2 8
528 5 2 8
4528 4 5 2 8
64528 6 4 5 2 8

Expansion of numbers:

Example :

 Number Number Name Expansion 50000 Fifty Thousand 5 × 10000 28000 Twenty Eight Thousand 2 × 10000 + 8 × 1000 68250 Sixty Eight Thousand Two Fifty 6 × 10000 + 8 × 1000 + 2 × 100 + 5 × 10 89264 Eighty Nine Thousand Two Hundred and Sixty Four 8 × 10000 + 9 × 1000 + 2 × 100 + 6 × 10 + 4 × 1

1.2.5 Introducing 100000:

Greatest 5 digit number.
Adding 1 to the greatest 5 digit number gives the smallest 6 digit number.

Example :

99,999 + 1 =100000 ( This number is one lakh )

10 × 10000 = 100000

Let us consider 6 digit number in the expanded form for example.

8,56,243 = 8 × 100000 + 5 × 10000 + 6 × 1000 + 2 × 100 + 4 × 10 + 3 × 1.

This number has 3 in one's place,4 in ten's place,2 in hundred's place,6 in thousand's place,5 in ten thousands place and 8 in lakhs place and the number is "eight lakhs fifty six thousand two hundred and forty three".

Few more examples of expansion :

 Number Number Name Expansion 500000 Five Lakh 5 × 100000 450000 Four Lakh Fifty Thousand 4 × 100000 + 5 × 10000 398029 Three Lakh Ninety EightThousand and Twenty Nine 3 × 100000 + 9 × 10000 + 8 × 1000 + 2 × 10 + 9 × 1
Larger numbers
• Introduction to larger numbers
• How to write large numbers - Introduction to comma system.

Example :

9,99,999 + 1 = 1000000 ( It is ten lakh )

Similarly, the greatest 7 digit number gives us the smallest 8 digit number.
99,99,999 + 1 = 10000000 ( It is one crore )

Pattern or sequence of numbers :

 9+1 = 10 99+1 = 100 999+1 = 1000 9,999+1 = 10000 99,999+1 = 100000 9,99,999+1 = 1000000 99,99,999+1 = 10000000

Indian system to numeration :

Values of the place in the Indian system of numeration are one's,ten's,hundred's, thousand's,ten thousand's,lakh,ten lakh,crore and so on.....

Following place value chart can be used to identify the digit in any place in the Indian system.

 Periods Crores Lakhs Thousands Ones Places Tens Ones Tens Ones Tens Ones Hundred Tens Ones

International system of numeration:

Values of the places in international system of numeration are one's,ten's, hundred's,thousand's,ten thousand's,hundred thousand's,millions,ten million and so on....

1 million = 1000 thousands
1 billion = 1000 millions

Following place value chart can be used to identify the digit in any place in the international system.

 Periods Billions Million Thousand Ones Places Hundred Tens Ones Hundred Tens Ones Hundred Tens Ones Hundred Tens Ones

Comparison of the Indian and international system of numeration:

Indian
numeration
Crore Ten Lakhs Lakh Ten thousand Thousand Hundred Tens Ones
Numbers 10000000 1000000 100000 10000 1000 100 10 0
International
numeration
Ten
millions
Million Hundred
Thousands
Ten
Thousands
Thousand Hundred Tens Ones
Estimation

Finding a value that is close enough to the right answer, usually with some thought or calculation involved.

Example: Rahul's estimation of 150 bricks to build a wall was very good as they were only 5 bricks left.

Estimation only finds the approximate value which is a value close to the actual value.

Estimation by rounding off the numbers

This method is commonly used to round off the numbers nearest to tens, hundreds, thousands and so on..... Estimating sum, difference, product and quotient

Estimating sum, difference, product and quotient Rounding of Numbers
Rounding of a number to the nearest ten:

Example: let us consider numbers 14 and 18 on the number line.

14 lies between 10 and 20.we observe that the gap between 10 and 14 is less than the gap between 14 and 20 i.e., 14 is nearer to 10 than 20, so we round off 14 to 10,that is to the nearest ten.
Similarly, if we consider the number 18 which is more nearer to 20 than 10, we will round off 18 to 20 which is the nearest ten
First example shows rounding of 14 to the nearest ten.
Second example shows rounding of 18 to the nearest ten. But if a number is equidistant from 10 and 20 both it is rounded off from 10 and 20.

Rounding of a number to the nearest ten thousand

Example: let us consider numbers 18652 and 53468.

a) Consider the digits in the thousands place of the given number.
b) If it is less than 5, replace the digits in thousands, hundreds, tens and ones by 0, keeping the digits in other places as they are.
c) If it is 5 or more than 5,replace the digits in thousands,hundreds,tens,and ones by 0 and increase the digit in ten thousands by 1.

i. 18652

Since, the digit in the thousands place is 8, and 8 > 5, so when rounded off the number nearest to ten thousand is 20000.

ii. 53468

In the number 53468, the digit in the thousands place is 3 and 3 < 5, so when rounded off the number nearest to ten thousand is 50000

Rounding of a number to the nearest lakh

Example :let us consider numbers 470518 and 621059

a) Consider the digits in the ten thousands place of the given number.
b) If it is less than 5, replace the digits in the ten thousands, thousands, hundreds, tens and ones by 0, keeping the digits in other places as they are.
c) If it is 5 or more than 5, replace the digits in ten thousands, thousands, hundreds, tens, and ones by 0 and increase the digit in lakhs by 1.

I. 470518

Since, the digit in the ten thousands place is 7 and 7 > 5, so the rounded off number is 500000.

II. 621059

In the number 621059, the digit in the ten thousands place is 2 and 2 < 5, so the rounded off number is 600000.

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