1. Understanding Large Numbers
We often come across numbers that are much larger than the numbers we normally use.Examples:
Population of a city, Number of people attending an event ,Distance between planets, etc...
One lakh
\(\text{1 lakh=1,00,000}\), It is \(1\) followed by \(5\) zeroes.Important relationships
\(\text{1 lakh = 100 thousand}\), \(\text{1 crore = 100 lakh}\)
Thus,
\(\text{1 crore = 1,00,00,000}\), A crore is \(1\) followed by \(7\) zeroes.
2. How Large is One Lakh?
A number becomes easier to understand when we compare it with something familiar.For example, about one lakh varieties of rice; one lakh people standing in a line;
Therefore, whether one lakh is “large” or “small” depends on the context.
Also, the size of a number is better understood when we compare it with familiar quantities.
Therefore, whether one lakh is “large” or “small” depends on the context.
Also, the size of a number is better understood when we compare it with familiar quantities.
3. Indian Place Value System
In the Indian system, commas are placed in groups of: \(\boxed{3,2,2,2,\ldots}\) starting from the right.Place-value pattern
| Period | Places |
| Ones | Ones, Tens, Hundreds |
| Thousands | Thousand, Ten Thousand |
| Lakhs | Lakh, Ten Lakh |
| Crores | Crore, Ten Crore |
| Arab | Arab |
4. International / American System and Important large - Number names
In the International system, commas are placed uniformly in groups of three digits from the right. \(\boxed{3,3,3,3,…}\)
| Indian Number | Indian name | International number | International name | Zeroes |
| \(1,000\) | One thousand | \(1,000\) | One thousand | \(3\) |
| \(10,000\) | Ten thousand | \(10,000\) | Ten thousand | \(4\) |
| \(1,00,000\) | One lakh | \(100,000\) | One Hundred thousand | \(5\) |
| \(10,00,000\) | Ten lakh | \(1,000,000\) | One million | \(6\) |
| \(1,00,00,000\) | One crore | \(10,000,000\) | Ten million | \(7\) |
| \(10,00,00,000\) | Ten crore | \(100,000,000\) | One Hundred million | \(8\) |
| \(1,00,00,00,000\) | One arab | \(1,000,000,000\) | One billion | \(9\) |
5. Exact and Approximate Values
Sometimes we do not need the exact value.For example, the exact population of Chintamani in the \(2011\) Census was given as: \(76,068\)
Instead of saying the exact number, we may say it is \(\boxed{\text{about }75,000}\). This is an approximate value.
Exact value is the actual or precise value.
Why do we approximate?
Approximation helps us: calculate quickly; understand the size of a quantity; make estimates; compare large quantities; make practical decisions, etc...
Approximation helps us: calculate quickly; understand the size of a quantity; make estimates; compare large quantities; make practical decisions, etc...
6. Rounding Up and Rounding Down
Rounding up: If the approximate value is greater than the actual value.Example: \(748\rightarrow750\)
Here, \(750>748\)
This is called rounding up.
Rounding down: If the approximate value is less than the actual value.
Example: \(472\rightarrow 470\)
Here, \(472<470\)
This is called rounding down.
This is called rounding up.
Rounding down: If the approximate value is less than the actual value.
Example: \(472\rightarrow 470\)
Here, \(472<470\)
This is called rounding down.
7. Nearest Neighbours
For a large number, we may want to know its nearest: thousand; ten thousand; lakh; ten lakh; or crore.Consider: \(6,72,85,183\)
Its nearest values are as a Place Approximation
| Thousand | \(6,72,85,000\) |
| Ten thousand | \(6,72,90,000\) |
| Lakh | \(6,73,00,000\) |
| Ten lakh | \(6,70,00,000\) |
| Crore | \(7,00,00,000\) |
General rule for rounding
Look at the digit immediately to the right of the place to which you are rounding.
- If it is 5 or more, increase the required digit by 1.
- If it is less than 5, keep the required digit unchanged.
- Replace all digits to the right by zeroes.
8. Estimation
An estimate should help us decide whether an answer is: reasonable; greater than a particular value; less than a particular value; close to a particular value.
9. Estimating Real-Life Quantities
Large numbers can be understood through familiar quantities.For example, if: one bus accommodates \(50\) people; there are \(1 \text{lakh}\) buses;
then total capacity is: \(1,00,000\times50 = 50,00,000 \)
Therefore, \(1 \text{lakh}\) buses can accommodate \(50 \text{lakh}\) people.
This type of calculation helps us decide whether a large quantity is sufficient for a particular purpose.
10.Patterns in Products
Multiplication Shortcuts
The chapter introduces ways to simplify multiplication by regrouping factors.Multiplication can be rearranged because \(a\times b=b\times a\); that is, the order of multiplication does not matter.
11. Useful Multiplication Conversions
\(25=\frac{100}{4}\); \(125=\frac{1000}{8}\); \(250=\frac{1000}{4}\); \(2500=\frac{10000}{4}\)
These forms can make calculations easier.
These forms can make calculations easier.
12. How long is the product?
The chapter investigates patterns such as:
\(11\times11 = 121\)
\(111\times111 = 12,321\)
\(1111\times1111 = 12,34,321\) and other similar products.
Observation:
The product forms symmetrical pattern. The digits start from \(1\), increase by \(1\) up to the number of digits in the original number, and then decrease by \(1\) back to \(1\).
So for \(4\) ones: \(1\rightarrow2\rightarrow3\rightarrow4\rightarrow3\rightarrow2\rightarrow1 = \boxed{1234321}\)
This is a nice number pattern for students to discover rather than simply memorise
The important things should be noted on patterns are:
- patterns in digits;
- number of digits;
- repeated digits; and
- relationships between factors and products.
13. Number of Digits in a Product
Suppose we multiply two numbers.There is a useful relationship between the number of digits in the factors and the possible number of digits in the product.
Examples:
1. \(1\text{-digit}\times1\text{-digit} = \boxed{1\text{-digit or 2-digit}}\)
2. \(2\text{-digit}\times2\text{-digit} = \boxed{3\text{-digit or 4-digit}}\)
3. \(3\text{-digit}\times3\text{-digit} = \boxed{5\text{-digit or 6-digit}}\)
2. \(2\text{-digit}\times2\text{-digit} = \boxed{3\text{-digit or 4-digit}}\)
3. \(3\text{-digit}\times3\text{-digit} = \boxed{5\text{-digit or 6-digit}}\)
In General idea, If two numbers have \(m\) and \(n\) digits respectively, their product can have either:
\(\boxed{m+n-1}\) or \(\boxed{m+n}\) digits.
\(\boxed{m+n-1}\) or \(\boxed{m+n}\) digits.