Knowing Our Numbers

1.0Introduction to Numbers

Numbers play an important role in our life. We use numbers in our day to day life to count things.

While counting we use numbers to represent any quantity, to measure any distance or length.

The counting numbers starting from $1,2,3,4,5$ are termed as natural numbers.

The set of counting numbers and zero are known as whole numbers. Whole numbers are $0,1,2,3,4,5,6,7$, and so, on

The symbols used by different civilizations to represent numbers are as below:

Even Natural Numbers: Numbers which are divisible by 2 are called even numbers, e.g. $2,4,6,8,10$, ....

Odd Natural Numbers: Numbers which are not divisible by 2 are called odd numbers, e.g. $1,3,5,7,9,\dots$.

If at unit place, we have $0,2,4,6,8$ then number is even otherwise it is odd.

2.0Comparing and building numbers

To put large numbers in order, you must check the number of digits in them first. If the number of digits vary in each number, the smallest number is the one which is having the least number of digits and the greatest number is the one which is having the maximum number of digits.

Comparing numbers with the same number of digits

Comparison of the numbers with the same number of digits starts from the left-hand side. You must compare the face values of the digits having the same place value in the numbers until you come across unequal digits.

• 0 is the smallest whole number and largest whole number cannot be define because whole number goes to infinite. Building numbers Now, you will learn to build numbers, under different conditions.
• Greatest 4-digit number is 9999 and greatest 5-digit number is 99999 .

3.0Introducing 6-digit, 7-digit and 8-digit numbers

Till now you have learnt up to 5 digit numbers and you know that the greatest 5 digit number is 99,999 . On adding 1 to it, we get the smallest 6-digit number. $99999+1=1,00,000$, read as one lakh. The largest 6-digit number is 9,99,999. On adding 1 to it, we get the smallest 7 -digit number. $9,99,999+1=10,00,000$, read as ten lakh. The largest 7 -digit number is $99,99,999$. On adding 1 to it, we get the smallest 8 -digit number. $99,99,999+1=1,00,00,000$, read as one crore.

Ascending Order

When the numbers are arranged from the smallest to the largest number, those numbers are said to be in an ascending order. The numbers are arranged from left to right in increasing order.

Descending Order

When the numbers are arranged from the largest to the smallest number, those numbers are said to be in descending order. The numbers are arranged from left to right in decreasing order. Ascending order is represented by < (less than) symbol, whereas descending order is represented by > (greater than) symbol.

Shifting digits

Changing the position of digits in a number, changes magnitude of the number. Example: Take a number 257. The condition here is to exchange its hundreds and unit digit and form the new number. That is, exchange 2 to 7 and 7 to 2 .

Here comes a question. Which is greater and which is least among the numbers? To find that express the numbers formed in both ascending and descending order. The number before shifting is 257 . Exchanging the hundreds and the unit digits, the number after shifting is 752 . That is, if we exchange the hundreds and unit digit, the resultant number becomes greater.

4.0Place value and Face value

Every digit has two values the place value and the face value. The face value of a digit does not change while its place value changes according to its position and number. The face value of a digit in a numeral is its own value, at whatever place it may be. Place value or local value of a digit in a given number is the value of the digit because of the place or position of the digit in the number.

Expanded form of a Number

If we express a given number as the sum of its place values, it is called its expanded form.

5.0Indian and International system of numeration

Suppose a newspaper report state that Rs. 2500 crore has been allotted by the government for National Highway construction. The same amount of Rs. 2500 crore is sometimes expressed as 25 billion. In the Indian system, we express it as Rs. 2500 crore and in the International system, the same number is expressed as 25 billion. Hence, you need to understand both the systems and their relationship.

Indian system of numeration

The Indian system of numeration or Hindu-Arabic numeral system is a positional decimal numeral system developed between the 1st and 5th centuries by Indian mathematicians, adopted by Persian and Arabian mathematicians and spread to the western world by the High Middle Ages. It uses ten basic symbols $0,1,2,3,4,5$, $6,7,8,9$ (called digits) and the idea of place value. For a given numeral, we start from the extreme right as : Ones, Tens, Hundreds, Thousands, Ten Thousands, Lakhs, Ten Lakhs, etc. Each place represents ten times the one which is immediately to its right. Indian system of numbers

Indian place-value chart

$1\mathrm{C}=1$ crore $=1,00,00,000$

International system of numeration

International system of numeration is adopted by all the countries throughout the world. International system of numbers

International place-value chart

$1\mathrm{TM}=10$ million $=10,000,000$

6.0Use of commas

Commas help us in reading and writing large numbers. In our Indian system of numeration, commas are used to mark thousands, lakhs and crores. The first comma comes after hundreds place and marks thousands. The second comma comes after ten thousands place and marks lakh. The third comma comes after ten lakh place and marks crore.

In International system of numeration, commas are used to mark thousands and millions. It comes after every three digits from the right.

Express 643871 in both the systems of numeration. Explanation Indian: 6,43,871 Six lakh forty three thousand eight hundred and seventy one

• Roman numbers were invented for the purpose of counting and performing other day-today transactions.

7.0Roman Numerals

The Roman numerals is the numeral system of ancient Rome. It uses combinations of letters from the Latin alphabet to signify values. The numbers 1 to 10 can be expressed in Roman numerals as follows:

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

This followed by XI for 11, XII for 12, ... till XX for 20. Some more roman numerals are :

The Roman numeral system is decimal but not directly positional and does not include a zero.

Rules to form Roman numerals

We can form different roman numerals using the symbols and the following rules.

Rule 1: If a symbol is repeated one after the other, its value is added as many times as it occurs. For example, $\mathrm{III}=1+1+1=3$ $XX=10+10=20$

Rule 2 : The symbols I, X, C and M can be repeated up to a maximum of three times. For example, $\mathrm{I}=1,\mathrm{II}=2,\mathrm{III}=3$ $X=10,XX=20,XXX=30$ $C=100,CC=200,CCC=300$ $M=1000,MM=2000,MMM=3000$ Rule 3: The symbols V, L and D (i.e., 5, 50, and 500, respectively) can never be repeated in a roman numeral. Rule 4: If a symbol with a smaller value is written on the right of a symbol with a greater value, then its value is added to the value of the greater symbol. For example, $\mathrm{XII}=10+2=12,\mathrm{LX}=50+10=60$, DCCCX $=500+300+10=810$ Rule 5 : If a symbol with a smaller value is written on the left of a symbol with a larger value, then its value is subtracted from the value of the greater symbol. For example, $\mathrm{IV}=5-1=4,\mathrm{IX}=10-1=9,\mathrm{CD}=500-100=400$ $\mathrm{VI}=5+1=6,\mathrm{XI}=10+1=11,\mathrm{DC}=500+100=600$ Note: 'I' can be subtracted from V and X once only. X can be subtracted from L and C once only. C can be subtracted from D and M once only. Thus, I or V is never written to the left of $L$ or $C.L$ is never written to the left of $C$.

• Zero is the only number that can't be represented in Roman numberals.
• VC is not possible because $V,L$ & $D$ are never subtracted.

8.0Use of brackets

Raju brought 6 pencils from the market, each at Rs. 2. His brother Ramu also bought 8 pencils of the same type. Raju and Ramu both calculated the total cost but in their own ways. Raju found that they both spent Rs. 28 and he used the following method: $(6\times 2)+(8\times 2)$ $=(12+16)$ $=28$ Here number of operations are two times multiplication and one time addition But Ramu found an easier way. He did $6+8=14$ and then $(2\times 14)=28$. The use of brackets makes this sum easy. It can be done as follows : Rs. $2\times (6+8)$ $=$ Rs. $(2\times 14)$ = Rs. 28 Here first solve the operation inside the bracket and then multiply it by the number outside.

Now number of operations are one addition and one multiplication. So, second method takes less time.

9.0BODMAS Explanation

$B\to$ Brackets first (parentheses) $0\to \mathbf{Of}$ DM $\to$ Division and Multiplication (start from left to right) AS $\to$ Addition and Subtraction (start from left to right)

Note:

(i) Start Divide/Multiply from left side to right side since they perform equally. (ii) Start Add/Subtract from left side to right side since they perform equally.

10.0Rounding Numbers

Rounding involves replacing one number with another number that's easier to work with. Rounded numbers can be easier to use. Suppose you want to find $18\times 43$, but had lost the calculator. You could find an answer close to $18\times 43$ by rounding to the nearest ten. "Rounding to the nearest ten" means replacing a number with the nearest multiple of 10 . Replacing a number with a higher number is called rounding up. Replacing a number with a lower number is called rounding down.

Procedure of round to different place values

You can round numbers to place values other than tens. Write the number. Underline the digit in the position you want to round to.

• If the digit to the right of the underlined digit is 5 or more, round up.
• If the digit to the right of the underlined digit is 4 or less, round down.

Note: When we round a number to nearest place, all other digits to the right of the place becomes zero. Ex round 24912 to nearest hundred, we will get 24900 . Digits to the right of 9 become zero.

• While converting 27381 into nearest thousands, we will focus on the digit which is on hundred place.

Using rounded numbers

Now, you will learn more about using rounded numbers. You'll think about how much certain numbers should be rounded. You'll also see how rounded numbers are useful for checking your work. People round numbers to different place values depending on what the numbers are being used for.

The amount of rounding affects the accuracy

If you use rounding to estimate a sum, be careful how much you round. Rounding to higher place values usually gives an estimate farther from the actual answer than rounding to lower place values.

Rounded numbers can be used to check work

Many times you'll want to check your work without doing the calculation all over again. Rounding is a way to see if your answer is reasonable. Note: Using rounded numbers to check your answer won't ever tell you that your answer is definitely right, only whether it is reasonable. Your answer might be close to the real answer but could still be wrong.

11.0Estimation

Estimation means "making a good guess." We can use it if we don't need to know an exact answer, or if a question has no exact right answer. You can estimate when there's no exact answer Sometimes in math there is no exact right answer. You can use the information you do have to make an estimate.

Using estimation

Estimation is really useful in a lot of real-life situations, where you might not be able, or don't need, to do an exact calculation. There are other times when it's better to figure out the exact answer.

Estimates aren't always a good idea There are some situations where you definitely shouldn't use an estimate.

12.0Use of numbers in everyday life

Numbers are used immensely in our everyday life, such as measuring the length of a small object as pencil, the distance between two given places, the weight of an orange, the weight of a ship, the amount of juice in a glass and the amount of water in a lake. Small lengths are measured in millimeter ( mm ) and centimeter ( cm ) while bigger lengths are measured in meter ( m ) and kilometer ( km ). Meter ( m ) is the standard unit of length and we define it as : $1\mathrm{m}=100\mathrm{cm}=1000\mathrm{mm}$ $\therefore 1\mathrm{cm}=10\mathrm{mm}$ $\therefore 100\mathrm{cm}=100\times 10=1000\mathrm{mm}$ $1\mathrm{km}=1000\mathrm{m}$ Also, $1\mathrm{km}=(1000\times 1000)\mathrm{mm}=1000000\mathrm{mm}$ Similarly, the units of weight are as under: $1\mathrm{g}=1000\mathrm{mg}$ $1\mathrm{kg}=1000\mathrm{gm}$ $1\mathrm{kg}=(1000\times 1000)\mathrm{mg}=1000000\mathrm{mg}$ For capacity or volume, $1\ell =1000\mathrm{mL}$ and $1\mathrm{k}\ell =1000\ell$ $1\mathrm{k}\ell =1000\times 1000\mathrm{m}\ell =1000000\mathrm{m}\ell$

13.0Numerical Ability

How many odd numbers are there between 151 and 168 ?

• Explanation The odd numbers between 151 and 168 are - $153,155,157,159,161,163,165,167$ The total number is 8 .

(i) Find the smallest natural number. (ii) Find the number of four-digit natural numbers.

• Solution (i) The smallest natural number is 1 . (ii) The number of four-digit natural numbers is 9000 .

Compare 45967 and 45861. Explanation As number of digits are same so starting from the left hand side, we notice that 2 digits are the same. $\underline{45}967$ and $\underline{45861}$ On comparing the digits at the hundred places in both the numbers we find that 9 in 45967 is greater than 8 in 45861. $\therefore 45967>45861$

Make the greatest and the smallest four-digit numbers by using different digits such that digit 6 is always in the tens place.

• Explanation We know that the digits written in the descending order are $9,8,7,6,5,4,3,2,1,0$. Keeping 6 in the tens place, we have Greatest number $=98\underline{6}7$ Smallest number $=10\underline{6}2$

Make the smallest and the greatest 5 -digit numbers using the digits 4, 6, 3, 1 and 0 only once.

• Solution: Smallest number: 10,346 Greatest number: 64,310

Arrange the numbers in ascending order.

• Explanation
  

Arrange the numbers in descending order.

• Explanation
  

Express the following in expanded form. (i) $3,54,039$ (ii) 3,85,00,386

• Explanation (i) Place value of $3=3\times 100000$ Place value of $4=4\times 1000$ Place value of $3=3\times 10$ Place value of $5=5\times 10000$ Place value of $0=0\times 100$ Place value of $9=9\times 1$ $\therefore$ The expanded form of $3,54,029$ is $3\times 100000+5\times 10000+4\times 1000+0\times 100+3\times 10+9\times 1$. (ii) Likewise, the expanded form of $3,85,00,386$ is $3\times 10000000+8\times 1000000+5\times 100000+0\times 10000+0\times 1000+3\times 100+8\times 10+6\times 1$.

The population of Rajasthan is $5,64,73,122$, and of Goa is $13,43,998$ and of Karnataka is $5,27,33,958$. What is the combined population of the three states?

• Solution Population of Rajasthan is $5,64,73,122$ Population of Goa is $13,43,998$ Population of Karnataka is 5,27,33,958 $\therefore$ Total population of three states $=5,64,73,122+5,27,33,958+13,43,998=11,05,51,078$ i.e., Eleven crore five lakh fifty-one thousand seventy-eight.

What must be added to $34,52,629$ to make it equal to 6 crores?

• Solution 6 crores =6,00,00,000 $\therefore$ Required number $=6,00,00,000-34,52,629$

$=5,65,47,371$

There are 785 students on roll in a residential public school. If the annual fee per student is Rs. 62,606. What is the total fee collected annually by the school.

• Solution Annual fee of one student $=$ Rs. 62,606 62606 Number of students $=785$ Total Annual collection of fee $=$ Rs. $62,606\times 785$ $=$ Rs. $4,91,45,710$

Find the number of pages in a book which has on an average 207 words on a page and contains 201411 words altogether.

• Solution Number of pages $=201411÷207$ Thus, the number of pages in the book $=973$.

Write the numeral for each of the following numbers: (i) Ninety-eight crore two lakh seventy five. (ii) Six million, four hundred and twelve thousand, two hundred and twenty.

• Solution: (i) Ninety-eight crore two lakh seventy-five is $98,02,00,075$. (ii) Six million, four hundred and twelve thousand, two hundred and twenty is $6,412,220$.

Write the following in Roman numerals: (i) 52 (ii) 44 (iii) 85 (iv) 49 (v) 99

• Explanation (i) $52=50+2=\mathrm{L}+\mathrm{II}=\mathrm{LII}$ (ii) $44=40+4=\mathrm{XL}+\mathrm{IV}=\mathrm{XLIV}$ (iii) $85=80+5=$ LXXX $+V=$ LXXXV (iv) $49=40+9=$ XL + IX $=$ XLIX (v) $99=90+9=\mathrm{XC}+\mathrm{IX}=\mathrm{XCIX}$

Write the following in Hindu-Arabic numerals: (i) XLV (ii) LXIII (iii) LXXVI (iv) XCII (v) XXXVIII

• Solution (i) $\mathrm{XLV}=\mathrm{XL}+\mathrm{V}=(50-10)+5=40+5=45$ (ii) LXIII $=\mathrm{L}+\mathrm{X}+\mathrm{III}=50+10+3=63$ (iii) LXXVI $=\mathrm{L}+\mathrm{XX}+\mathrm{VI}=50+(2\times 10)+6=76$ (iv) XCII $=$ XC + II $=(100-10)+2=90+2=92$ (v) XXXVIII $=$ XXX + VIII $=(3\times 10)+8=30+8=38$

$78-$ [5 + 3 of (25-2×10)]

• Explanation: $78-[5+3$ of $(25-2\times 10)]$ $=78-[5+3$ of $(25-20)]$ (Simplifying 'multiplication' $2\times 10=20$ ) $=78-[5+3$ of 5] (Simplifying 'subtraction' 25-20=5) $=78-[5+3\times 5]$ (Simplifying 'of) $=78-[5+15]$ (Simplifying 'multiplication' $3\times 5=15$ ) $=78-20$ (Simplifying 'addition' $5+15=20$ ) $=58$ (Simplifying 'subtraction' 78-20 $=58$ )

Simplify: 25-[ 22 - {17-(5-2) $\}$ ]

• Solution $25-[22-\{17-(5-2)\}]$ $=25-[22-\{17-3\}]$ $=25-[22-14]$ $=25-8=17$

Round $18\times 43$ to the nearest ten.

• Explanation You need to decide whether to round up or down. Look at the digit in the ones place: If the ones digit is 5 or more, round up. If the ones digit is 4 or less, round down. Start with 18 : The digit in the ones place is 8 and 8 is more than 5 , so round up. 18 rounded up to the nearest ten is 20 . Next, 43: The digit in the ones place is 3 and 3 is less than 5 , so round down. 43 rounded down to the nearest ten is 40 . By rounding, you can replace $18\times 43$ with $20\times 40$. This is much easier to solve: $20\times 40=800$ 800 is fairly close to the real answer: $18\times 43=774$

Round 25,281 to the nearest hundred.

• Explanation Write the number, and underline the hundreds digit: 25, $\underline{2}81$ You're rounding to the nearest hundred, so that's going to be either 25,200 or 25,300. The digit to the right of the underline is 8 . That's greater than 5 , so round up. So, 25,281 rounds up to 25,300 , to the nearest hundred. Rounding a number to the nearest 1000 To round off a number to the nearest thousand, we get the nearest multiples of 1000 for that number. Rule : Look at the digit in the hundreds place. If it is 5 or more, then round up, i.e., replace the digits at ones, tens and hundreds place by 0 and add 1 to the digit at thousands place. If it is less than 5 , then round down, i.e., replace the digits at ones, tens and hundreds place by zeros and leave the digit at thousands place unchanged.

Round 8691 to the nearest thousand.

• Explanation
  

Round 4392 to the nearest thousand.

• Explanation
  

Round off the following numbers to the nearest tens, hundreds, thousands. (i) 7848 (ii) 5164

• Explanation (i) $7848\underset{\text{ to nearest }10}{\overset{\text{ Rounded off }}{\to }}78507848\underset{\text{ to nearest }100}{\overset{\text{ Rounded off }}{\to }}7800$ $7848\underset{\text{ to nearest }1000}{\overset{\text{ Rounded off }}{\to }}8000$ (ii) $5164\underset{\text{ to nearest }10}{\overset{\text{ Rounded off }}{\to }}51605164\underset{\text{ to nearest }100}{\overset{\text{ Rounded off }}{\to }}5200$ $5164\underset{\text{ to nearest }1000}{\overset{\text{ Rounded off }}{\to }}5000$

Lucas wants to add 3439 and 5482. He doesn't need an exact answer, so he decides to use rounding. Look at Lucas's work below. How could he have found a more accurate answer?

Actual:

Rounded to the nearest thousand:

Rounded to the nearest hundred:

• Explanation Lucas rounded to the nearest thousand, so he got an estimate of 8000 . If he had rounded to the nearest hundred, he would have got 8900 , which is much closer to the actual 8921.

Calculate 2343 +5077. Then check your work by rounding to the nearest hundred.

• Solution Actual sum: 2343 Rounded sum: 2300

The answer to the rounded sum is close to the answer to the actualsum, so the answer to the rounded sum is reasonable.

Martin is trying to solve $29.6\times 9.8$. He gets the answer 192.08. Check Martin's answer by rounding to the nearest ten.

• Solution Rounded product : $30\times 10=300$ Martin's answer is a long way from the rounded answer, so it looks like his answer of 192.08 might be wrong. In fact, $29.6\times 9.8=290.08$ This is much closer to the rounded estimate.

Carla has a tall bookshelf and a short bookshelf. When full, the tall bookshelf can hold about 60 books. Estimate from the picture how many books the small bookshelf will hold.

• Explanation There is no exact number of books you can fit on a bookshelf, because not all books are the same size. To estimate the answer, compare the bookshelves. The tall one has 3 shelves, and the small one only 2 . All the shelves has the same size, so the small bookshelf will hold around two-thirds the number of books. So, you can estimate that the small bookshelf will hold about 40 books. You can estimate if you don't need an exact answer You don't always need to use an exact figure. Sometimes an estimate is enough.

A auto started its journey and reached different places with a speed of $20\mathrm{km}/\mathrm{hour}$. The journey is shown below.

• Explanation (i) Total distance covered by the auto from A to D $=4180+3650+2360$ $=10,190\mathrm{km}$ (ii) Total distance covered by the auto from D to G $=8250+4930+2610$ $=15790\mathrm{km}$ (iii) Total distance $=4180+3650+2360+8250+4930+2610+1120$ $=27100\mathrm{km}$ (iv) Difference of distances from C to D and D to E $=8250-2360=5890\mathrm{km}$ (v) (a) $\frac{4180}{20}=209\mathrm{hrs}$ (b) $\frac{2360}{20}=118\mathrm{hrs}$ (c) $\frac{4930+2610}{20}=377\mathrm{hrs}$ (d) $\frac{\text{ Totaldistance }}{\text{ speed }}=\frac{27100}{20}=1355\mathrm{hrs}$

14.0Memory map