NCERT Mathematics Class 6 Chapter 1: Numbers & ComparisonSection 1.1 & 1.2 YouTube Lecture Handouts
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What Will We Learn?

Numbers and Types

Place & Face Value

Positional Number System

Roman Numerals

Shifting Digits, Comparison and Ordering

Indian and International System

SI System

Large numbers

Estimation

Operations and BODMAS
Questions?

What is the place value of 0 in 209?

What is the largest roman numeral?

A notebook has 20 pages. 1 sheet of paper can make 10 pages. How many notebooks can be made from 40 sheets?

Find difference between greatest and smallest number formed using digits 6,2,4,3,1

Out of 40 m cloth, how many shirts can be made if each shirt requires 2 m 15 cm of cloth?
History of Numbers

Numbers developed as soon as man started counting
Used for trade, seed and food distribution, taxation etc.

Even mammals can count (upto 3)

Samarians, Egyptians, Greeks, and Indians contributed significantly to mathematics.

Romans developed roman numeral system which was widely used as they conquered the world.

Romans did not contribute to mathematics.
Nominal, Ordinal, and Cardinal

Use of numbers as counting is obvious cardinal numbers.

Numbers also denote order for example, 1^{st}, 2^{nd}, 3^{rd} etc. in a race.
Cant add, subtract ordinal numbers

Numbers are also used as just names. For example 18 for Kholi and for Dhoni is 7.

नाममात्र, क्रमवाचक, गणनसंख्या
Understanding Place Value and Face Value
2211

Zero is used as a placeholder to change the position of one.

In a decimal system each change in position makes the value of same digit 10 times bigger.

Thus the value of the first one is just one while the second 1 although it looks the same is “10”, that is it is 10 times greater.

Thus face value of both the ones is 1 while their place value is 1 and 10. Each zero placed to the right of the digit makes its value 10 times.

This is same as when you move from one class to next. Your face value remains the same while your value in school by virtue of your position (class) becomes larger.

अंकित मूल्य और स्थानीय मान
Non Positional Number System
4622

Ancient Egyptian numerals were base 10. They used hieroglyphs for the digits and were not positional.

Instead of using different positions for 1, 10, 100.

गैर स्थितीय संख्या प्रणाली
Positional Number System

As we said position of a number and also its symbol have significance.

After a certain count the same digit moves the next position you move to next class after 1 year.

As the number changes position its value changes.

In the decimal system the change in position happens after we count to 10. That is each position is 10 times the digit on right.

In other systems the change might happen after different counts. For example,

Sexagesimal: The Babylonian numeral system, base 60, was the first positional system developed, and its influence is present today in the way time and angles are counted in tallies related to 60, like 60 minutes in an hour, 360 degrees in a circle.

In binary system each digit is 2 times the previous one and so on.

The problem with positional systems is that they are susceptible to fraud.

स्थितीय संख्या प्रणाली
Two Meanings of Zero

In positional systems zero is used as a placeholder to change the position of the digits. We could have used –, space etc.

Zero also means count of zero.

Place value and face value of zero is always 0.

Zero was invented by Indians it is called śūnya in Sanskrit.

Aryabhata in 520 A.D., devised a positional decimal number system contained a word, “kha,” for the idea of a placeholder.

Brahmagupta, developed zero as an actual independent number, not just a placeholder, and wrote rules for adding and subtracting zero.

The Indian writings passed on to alKhwarizmi and then to Leonardo Fibonacci and others who continued to develop the concept and the number.

This Indian numberal system travelled all over the world and hence the current decimal numeral system is also called HinduArabic system
Rules for Roman Numerals
I V X L C D M
1 5 10 50 100 500 1000

Repetition: Repeated symbol => add value,

Max repeat = 3

No repeat for V, L, and D.


Addition: Smaller value to the right of greater value gets added.

Subtraction: Smaller value left of greater value is subtracted Symbols V, L and D are never subtracted

I => V & X only.

X => L, C, and M only.

Romans did not contribute to mathematicians but only to conquests.
For example

I=1, II = 2, XX=20 and XXX=30.

VI = 5 + 1 = 6, XII = 10 + 2 = 12, and LXV = 50 + 10 + 5 = 65.

IV = 5 – 1 = 4, IX = 10 – 1 = 9, XL= 50 – 10 = 40, XC = 100 – 10 = 90
रोमन संख्या
Roman Numerals
V(5) X(10) L(50) C(100) D(500) M(1000)

Repeated symbol (max 3) => add value. Except V, L, and D.

Smaller value to the right of greater value gets added.

Smaller value left of greater value is subtracted. Except V, L and D, I => V & X only, and X => L, M & C only.

The current year (2019) is MMXIX

1776 (M+DCC+LXX+VI) = MDCCLXXVI (the date written on the book held by the Statue of Liberty)

1910 is MCMX and not MDCCCCX

MMMCMXCIX which in our notation is 3,999

V (bar) = 5000 and X (bar) = 10,000
Convert from Roman Numerals
I (1) V(5) X(10) L(50) C(100) D(500) M(1000)

Repeated symbol (max 3) => add value. Except V, L, and D.

Smaller value to the right of greater value gets added.

Smaller value left of greater value is subtracted. Except V, L and D, I => V & X only, and X => L, M & C only.

MMXIV is 2014

MLXVI is 1066
Writing Large Numbers: Indian and International Numbering Systems

1,000; 10,000; 1,00,000 & Expansion of Numbers

999 + 1 = 1000

9999 + 1 = 10000

99999 + 1 = 1,00,000

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

653275829: Sixtyfive crore, thirtytwo lakh, seventyfive thousand, eight hundred and twenty nine

Short Scale

Million first used in 14^{th} Century

10 Lakhs make a million

10 millions make a crore
In international system after quintillion we have sextillion and then septillion. Thus, an nillion equals 10^{3n + 3}

Indian and International Numbering System, Expansion, Number Name
7 2 7 0 5 0 6 2
7 2 7 0 5 0 6 2

7,27,05,062
Seven crore, twentyseven lakh, fivethousand and sixtytwo

72,705,062
Seventytwo million, seven hundred and five thousand, sixtytwo
Indian and International Numbering System, Expansion, Number Name
6 5 3 2 7 5 8 2 9
6 5 3 2 7 5 8 2 9

65,32,75,829
Sixtyfive crore, thirtytwo lakh, seventyfive thousand, eight hundred and twentynine

653,275,829
Six hundred and fiftythree million, two hundred and seventyfive thousand, eight hundred and twentynine
Write Numeral
Two crore ninetylakh fiftyfive thousand eight hundred.
Comparing & Ordering

1473, 89423, 100, 5000, 310

15623, 15073, 15189, 15800

Find the largest possible 3 digit number using 9, 3, 4

Find the smallest possible 3 digit number using 0, 3, 9

Make the greatest and the smallest 4digit numbers using any four different digits with digit 9 in once place

Heat one end of rod. Particles get energy start vibrating similar to students sitting at their place they cant just move are stuck at one place.

Vibration travels to nearby particles and so on ultimately other end becomes hot.

Thus some material are good conductors and others are bad more on that later.

Students passing book
Shifting Digits
1473, 89423, 310
Sift the digits to make these numbers smallest and largest
SI Units: M, G, and L

How many centimeters make a kilometer?

Different prefixes are used depending on scale. For example, m is ok for measuring distances on track meet. But for measuring larger distances between cities need other unit (km). Similarly for smaller distances need another unit.

Same for grams (wheat/rice in kg vs. vegetables in g vs. medicine)

Same for liters (medicine bottles vs. bucker vs. overhead tank)


A box contains 2,00,000 medicine tablets each weighing 20 mg. What is the total weight of all the tablets in the box in grams and in kilograms?

4 kg

A bus has a speed of 60 km/hour. Find time taken by the bus to reach C from D (2,160,000 m)
36 hours

BODMAS (Using and Expanding Brackets)
Please Excuse My Dear Aunt Sally

(100 + 2) × 2 vs 100 + 2 × 2

48 * 2 + 2 / 12

48÷2∗(9+3) = 288

Please Excuse My Dear Aunt Sally

24 ÷ 48 + 54 ÷ 18 × 7 ÷ 14 + 2 × 3 ÷ 18 == (24 / 48) + (((54 / 18) * 7) / 14) + ((2 * 3) / 18)

Forty five divided by three times the sum of three and two== 45/(3 * (3 + 2))

Write five situations for the following where brackets would be necessary: 5(9 – 4): 5 times difference of 9 and 4. I have 5 crates of 9 apples each. From each crate 4 apples went bad. How many good apples are left.

Write five situations for the following where brackets would be necessary.(9 + 2) (6 – 3). Product of sum of 9 and 2 with difference of 6 and 3. I had 9 crates 6 of apples each. My friend gave me 2 more. From each of the crates 3 apples went bad. How many good apples are left?
Practicing Large Numbers: +, , *, /
A merchant had Rs 78,592 with her. She placed an order for purchasing 40 radio sets at 1200 each. How much money will remain with her after the purchase?
7859248000=30592
Practicing Large Numbers: +, , *, /
Instead of multiplying 7656 by 56 student multiplied it by 65? What is the difference between correct and incorrect answers?

7656 * 65  7656 * 56 = 497640428736=68904

7656 * (6556) = 7656 * (9) = 68904
Practicing Large Numbers: +, , *, /
A notebook has 20 pages. 1 sheet of paper can make 10 pages. How many notebooks can be made from 40 sheets?
40 * 10 = 400/20 = 20
Practicing Large Numbers: +, , *, /
Out of 40 m cloth, how many shirts can be made if each shirt requires 2 m 15 cm of cloth?

40 * 100 cm cloth = 4000 cm

215 cm cloth for each shirt

215 * 8 = 1720 and 215 * 9 = 1935

Thus number of shirts = 18 and130 cm cloth left
Practicing Large Numbers: +, , *, /
Find difference between greatest and smallest number formed using digits 6,2,4,3,1 (once and only once)

Largest number: 64321

Smallest number: 12346

6432112346 = 51975
Rounding
65437

What does rounding to nearest 10’s, 100’s, 1000’s etc. mean?

When we use rounding we are estimating.

7805, 65437
Estimating Sums
5,673 – 436

There is a compromise speed vs accuracy

Consider trader who receives 13,569, 26,785 but has to pay 37,000. Will he be able to pay?

Estimate: 5,673 – 436

Round to the greatest place for the smallest number
Estimating Products





Try multiplying 216 * 78 = 16848

220 * 80 = 17600

200 * 100 = 20,000

200 * 80 = 16,000

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

The Problem of 5’S: Breaking the Tie

Tie breaking!

Round half up

Convergent rounding, statistician’s rounding, or bankers’ rounding
Mayank