NCERT Chapter Notes 1 - Knowing Our Numbers, Mathematics, Class 6 PDF Download

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Knowing Our Numbers

Comparing Numbers-

The arrangement of numbers from the smallest to the greatest is called ascending order.

• The arrangement of numbers from the smallest to the greatest is called ascending order. Ex: 2789, 3560, 4567, 7662, 7665

• The arrangement of numbers from the greatest to the smallest is called descending order. Ex: 7665, 7662, 4567, 3560, 2789

• If two numbers have an unequal number of digits, then the number with the greater number of digits is greater.

• If two numbers have an equal number of digits, then the number with the greater digit is greater.

• The greatest single-digit number is 9. When we add 1 to this single-digit number, we get 10, which is the smallest two-digit number.

Therefore, the greatest single-digit number +1=the smallest two-digit number.

• The greatest two digit-number is 99. When we add 1 to this two-digit number, we get 100, which is the smallest three-digit number. Therefore, the greatest two-digit number +1=the smallest three-digit number.

• The greatest three-digit number is 999. When we add 1 to this three-digit number, we get 1000, which is the smallest four-digit number. Therefore, the greatest three-digit number +1=the smallest four-digit number.

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• The greatest four-digit number is 9999. When we add 1 to this four-digit number, we get 10,000, which is the smallest five-digit number. Therefore, the greatest four-digit number +1=the smallest five-digit number.

• The greatest five-digit number is 99999. When we add 1 to this five-digit number, we get 1,00,000, which is the smallest six digit number. Therefore, the greatest five-digit number +1=the smallest six-digit number. The number, that is, one with five zeroes (100000), is called one lakh.

Commas in International system:

As per international numeration, the first comma is placed after the hundreds place. Commas are then placed after every three digits.

Ex:  (i) 8,876,547

The number can be read as eight million eight hundred seventy-six thousand five hundred and forty-seven.

(ii) 56,789, 056

The number can be read as fifty-six million seven hundred eighty-nine thousand and fifty-six.

• Use the following place value chart to identify the digit in any place in the international system.

Comparison of the Indian and the international numeration systems:

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Units of measurement:

• 1 metre= 100 centimetres
• 1 kilogram = 1,000 grams
• 1 kilometre = 1000 metres
• 1 litre=1,000 millilitres
• The greatest six-digit number is 999999. When we add 1 to this six-digit number, we get 10,00,000, which is the smallest seven-digit number.

Therefore, the greatest six-digit number +1=the smallest seven-digit number. The number, that is, one with six zeroes (1000000), is called ten lakh.

• The greatest seven-digit number is 9999999. When we add 1 to this seven-digit number, we get 10000000, which is the smallest eight-digit number. Therefore, the greatest seven-digit number +1=the smallest eight-digit number. The number, that is, one with seven zeroes (10000000), is called one crore.
 • Commas are placed to the numbers to help us read and write large numbers easily.

Commas in Indian numeration:

As per Indian numeration, the first comma is placed after the hundreds place. Commas are then placed after every two digits. Ex:

(i) 88,76,547

The number can be read as eighty-eight lakh seventy-six thousand five hundred and forty-seven.

(ii)5 , 67, 89, 056

The number can be read as five crore sixty-seven lakh eighty-nine thousand and fifty-six. Use the following place value chart to identify the digit in any place in the Indian system.

Commas in International system:

As per international numeration, the first comma is placed after the hundreds place. Commas are then placed after every three digits.

Ex: (i) 8,876,547

The number can be read as eight million eight hundred seventy-six thousand five hundred and forty-seven.

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(ii)56,789, 056

The number can be read as fifty-six million seven hundred eighty-nine thousand and fifty-six.

• Use the following place value chart to identify the digit in any place in the international system.

Comparison of the Indian and the international numeration systems:

Units of measurement:
• 1 metre = 100 centimetres
• 1 kilogram = 1,000 grams
• 1 kilometre = 1000 metres
• 1 litre = 1,000 millilitres
 • The greatest six-digit number is 999999. When we add 1 to this six-digit number, we get 10,00,000, which is the smallest seven-digit number. Therefore, the greatest six-digit number +1=the smallest seven-digit number. The number, that is, one with six zeroes (1000000), is called ten lakh.

• The greatest seven-digit number is 9999999. When we add 1 to this seven-digit number, we get 10000000, which is the smallest eight-digit number. Therefore, the greatest seven-digit number +1=the smallest eight-digit number. The number, that is, one with seven zeroes (10000000),is called one crore.
• Commas are placed to the numbers to help us read and write large numbers easily.

Commas in Indian numeration:

As per Indian numeration, the first comma is placed after the hundreds place. Commas are then placed after every two digits.

Ex: (i) 88,76,547
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The number can be read as eighty-eight lakh seventy-six thousand five hundred and forty-seven.  

(ii)5,67,89,056 The number can be read as five crore sixty-seven lakh eighty-nine thousand and fifty-six. Use the following place value chart to identify the digit in any place in the Indian system.

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Estimation of the Numbers

The estimation of a number is a reasonable guess of the actual value. Estimation means approximating a quantity to the accuracy required. This is done by rounding off the numbers involved and getting a quick, rough answer.

The numbers 1, 2, 3 and 4 are nearer to 0. So, these numbers are rounded off to the lower ten. The numbers 6, 7, 8 and 9 are nearer to 10. So, these numbers are rounded off to the higher ten. The number 5 is equidistant from both 0 and 10, so it is rounded off to the higher ten.

Eg:

(i) We round off 31 to the nearest ten as 30.

(ii) We round off 57 to the nearest ten as 60.

(iii) We round off 45 to the nearest ten as 50. The numbers 1 to 49 are closer to 0. So, these numbers are rounded off to the nearest hundred.

The numbers 51 to 99 are closer to the lower hundred. So, these numbers are rounded off to the higher hundred. The number 50 is rounded off to the higher hundred.

Eg: (i) We round off 578 to the nearest 100 as 600.

(ii) We round off 310 to the nearest 100 as 300.

Similarly, 1 to 499 are rounded off to the lower thousand, and 501 to 999 to the higher thousand. The number 500 is equidistant from both 0 and 1000, and so it is rounded off to the higher thousand.

Eg: (i)We round off 2574 to the nearest thousand as 3000.

(ii)We round off 7105 to the nearest thousand as 7000.

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Estimation of sum or difference:

• When we estimate sum or difference, we should have an idea of the place to which the rounding is needed.

Eg: (i) Estimate 4689 + 19316. We can say that 19316 > 4689

We shall round off the numbers to the nearest thousands. 19316 is rounded off to 19000 and 4689 is rounded off to 5000.

Estimated sum: 19000 + 5000=24000

(ii) Estimate 1398-526. We shall round off these numbers to the nearest hundreds.

1398 is rounded off to 1400. 526 is rounded off to 500 

Estimated difference: 1400-500=900

Estimation of the product:

• To estimate the product, round off each factor to its greatest place, then multiply the rounded off factors.

Eg:

Estimate 92 x 578 The first number, 92, can be rounded off to the nearest ten as 90. The second number, 578, can be rounded off to the nearest hundred as 600.

Hence, the estimated product =90 x 600 = 54,000

• Estimating the outcome of number operations is useful in checking the answer.
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Roman Numerals:

Many years ago, Hindus and Arabs developed a number system called the Hindu-Arabic number system.

Hindu-Arabic number system:

Many years ago, Hindus and Arabs developed a number system called the Hindu-Arabic number system. It is the name given to the number system that we use today.

Roman numerals:

It is the numeral system that originated in ancient Rome. This numeral system is based on certain letters, which are given values and are used as numerals. The following are the seven number symbols used in the Roman numeral system, and their values:

Seven letters of English alphabet, i.e. I, V, X, L, C, D and M, are used to represent Roman numerals. Roman numerals do not have a symbol for zero. Roman numerals are read from left to right, and are arranged from the largest to the smallest.

Multiplication, division and other complex operations were difficult to perform on Roman numerals. So Arabic numerals were used. The Roman numerals for the numbers 1 - 15 are shown below:

We can find these roman numerals in some clocks.

Rules for Roman numerals:

1. In Roman numerals, a symbol is not repeated more than thrice. If a symbol is repeated, its value is added as many times as it occurs.

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For example, if the letter I is repeated thrice, then its value is three.

1. The symbols V, L and D are never repeated.

2. If a symbol of smaller value is written to the right of a symbol of greater value, then its value gets added to the symbol of greater value.

3. For example, in case of VI, I is written to the right of V. It means that 1 should be added to 5. Hence, its value is 6. If a symbol of smaller value is written to the left of a symbol of greater value, then its value is subtracted from the symbol of greater value.

For example, in case of IV, I is written to the left of V. It means that 1 should be subtracted from 5. Hence, its value is 4.

4. The symbols V, L and D are never written to the left of a symbol of greater value, so V, L and D are never subtracted.

For example, we write 15 as XV and not VX.

• The symbol I can be subtracted from V and X only. For example, the value of IV is four and the value of VI is six.

• The symbol X can be subtracted from L, M and C only. For example, X is subtracted from L to arrive at 40, which is represented by XL.

Using brackets:

Brackets help in simplifying an expression that has more than one mathematical operation. If an expression that includes brackets is given, then turn everything inside the bracket into a single
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number, and then carry out the operation that lies outside. Eg: 

1. (6 + 8) x 10 = 14 x 10 = 140 

2. (8 + 3) (9 - 4) = 11 x 5 = 55

Expanding brackets:

The use of brackets allows us to follow a certain procedure to expand the brackets systematically.

For example:
1. 8 x 109 = 8 x (100 + 9) = 8 x 100 + 8 x 9 = 800 + 72 = 872
2. 105 x 108 = (100 + 5) x (100 + 8)
= (100+5) x 100+(100+5) x 8 =100 x 100 + 5 x 100 + 100 x 8 + 5 x 8 =10000 + 500+ 800 + 40 =1134