Finding greatest and smallest numbers
When two numbers are given,
- The number with more digits is greatest.
- If number of digits is same, then the number with greater leftmost digit is greatest
- If number of digits is also same, then examine the next digit from left and so on.
Making numbers from digits
- To form different numbers from given digits,
For example, take four digits 7, 8, 3, 5
- No digit must be repeated
- All the given digits must be used
To make greatest number, arrange digits in descending order.
To make smallest number, arrange digits in ascending order except zero. Zero can occupy any place except leftmost place.
When we shift the digits from one place to other, the number can become large or sometimes even small.
- The greatest number we get is 8753
- The smallest number we can get is 3578
Ordering of numbers
Ascending order – The arrangement from the smallest to the greatest
Descending order – The arrangement from the greatest to the smallest
The smallest and the largest numbers
- The smallest two-digit number is 10 (ten) and it follows the largest one digit number 9
- The smallest three-digit number is 100 (one hundred) and it follows the largest two digit number 99
- The smallest four-digit number is 1,000 (one thousand) and it follows the largest three digit number 999
- The smallest five digit number is 10,000 (ten thousand) and it follows the largest four digit number 9,999
- The smallest six digit number is 100,000 (one lakh) and it follows the largest five-digit number 99,999 and so on.
Look at the pattern: 9 + 1 = 10 = 10 × 1
99 + 1 = 100 = 10 × 10
999 + 1 = 1000 = 10 × 100
9999 + 1 = 10000 = 10 x 1000
We observe 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 and so on.
Use of Commas
Use of commas helps in reading and writing large numbers. In India, Hindu-Arabic numeral system and also Roman numeral system are used.
In the Indian system of numbers,
- We use ones, tens, hundreds, thousands, lakhs and crores
- Commas are placed after 3 digits starting from the right and then every 2 digits thereafter. Commas are used to mark hundreds, thousands, lakhs and crores. For example: 12, 34, 56, 789
- The commas after 3, 5 and 7 digits separate thousand, lakh and crores respectively
In the International system of numbers,
- We use ones, tens, hundreds, thousands and millions
- Commas are placed after every 3 digits starting from the right to mark hundreds, thousands and millions, billions. For example: 123, 456, 789
- The commas after 3 and 6 digits separate thousand and million respectively.
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- The estimation refers to the approximation. The estimate gives rough idea before the exact value.
- In a number of situations we do not need the exact quantity but only a reasonable guess or an estimate would be enough.
- For example, while stating how many kilometers is the school from the house, we state the approximate number; may be 3 or 5 km we do not need to state the exact number.
- Estimation is useful in a number of situations, when we need a quick and rough answer. It is useful in checking answers.
- In estimating the products, it is a general rule to, Round off each factor to its greatest place, then multiply the rounded off factors.
- Estimating to the nearest tens by rounding off
Numbers 1 to 4 are rounded off to 0 and numbers 5 to 9 are rounded off to 10.
- Estimating to the nearest hundreds by rounding off
Numbers 1 to 49 are rounded off to 0 and numbers 50 to 99 are rounded off to 100.
- Estimating to the nearest thousands by rounding off
Numbers 1 to 499 are rounded off to 0 and numbers 500 to 999 are rounded off to 1000.
- We perform various functions on the numbers like addition, subtraction, multiplication, division etc.
- Sometimes we may have to execute many functions together. To avoid confusion while doing so, brackets are used.
- The numbers on which we perform the same function are packed into a bracket and are treated as a single number.
- While solving the problems with brackets, first turn everything inside the brackets into a single number and then do the operations outside the brackets
- In India we use Hindu-Arabic numeral system. Other than this, there are many other numeral systems
- Roman numeral system is one of the oldest systems of writing numerals.
- This system uses the following special symbols to represent numbers.
- Other roman numerals are formed by the combination of these special symbols
- The rules for the system are:
- If a symbol is repeated, its value is added as many times as it occurs:
Ex: II is equal 2, XX is 20 and XXX is 30.
- A symbol is not repeated more than three times. And the symbols V, L and D are never repeated.
Ex: We do not write VV to resemble 10, Instead we have a symbol for 10 that is X; Similarly we do not write LL to resemble 100, instead we have a symbol for 100 that is C and so on
- If a symbol of smaller value is written to the right of a symbol of greater value, its value gets added to the value of greater symbol.
Ex: VI = 5 + 1 = 6, XII = 10 + 2 = 12 and LXV = 50 + 10 + 5 = 65
- If a symbol of smaller value is written to the left of a symbol of greater value, its value is subtracted from the value of the greater symbol.
Ex: IV = 5 – 1 = 4, IX = 10 – 1 = 9
XL= 50 – 10 = 40, XC = 100 – 10 = 90
- The symbols V, L and D are never written to the left of a symbol of greater value, i.e. V, L and D are never subtracted.
Ex: The symbol I can be subtracted from V and X only.
The symbol X can be subtracted from L, M and C only.
Basic combinations of roman numbers
Other roman numerals are formed by the combination of these special symbols,
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