Playing With Numbers

General Form of a Number

Can you count the stars of the universe without numbers or can you tell your age to someone without numbers? No! Because everything is just a game of numbers. Let us now see the general form of a number. Let us do it right away.

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General Form of a Number

General Form of a Number

Two-Digit Numbers

Let us first start with two digit numbers. In general, any two digit number ab can be written as,

ab = 10 × a + 1 × b = 10a + b

where a  is a ten’s place digit and b  is a unit’s place digit. Let us consider the example of a two digit number- 67. Now when we start from the right i.e from 7 to 6, the place of seven is known as ones and that for 6 is known as tens.

Three-Digit Numbers

In general, any three-digit number xyz can be written as,

xyz = 100 × x + 10 × y + 1 × z = 10x + 10y + z

where, x is a hundredth place digit, y is a ten’s place digit and z is a unit’s place digit. Hence, if it’s a three digit number, the places will be ones, tens and hundreds from right to left.

Similarly, for a 4 digit number, it’ll be ones, tens, hundreds and thousands. For a five digit number, the leftmost place will be called ten thousand. Let’s understand this concept.

Games with Numbers

Reverse Three Digit Numbers

The difference between a 3-digit number and a number obtained by reversing its digits is always divisible by 11. Let 3-digit numbers be:
xyz = 100x + 10y + z

Reversing the order of digits we get,
zyx = 100z + 10y + x

On substraction, if x > z  then difference between the numbers is
(100x + 10y + z) – (100z + 10y + x)
= 100x + 10y + z – 100z + 10y + x
= 99x – 99z → 99 ( x-z)

In case of z > x
(100z + 10y + x) – (100x + 10y + z)
= 100z + 10y + x – 100x + 10y + z
= 99z – 99x
= 99 (z-x)

In case of x =z the diffrence is zero.

Forming Three-Digit Numbers from Given Three Digits

Example: Find the smallest three-digit number from the given digits 0, 7,8

Solution: Arrange the given digits in ascending order (as the smallest three-digit number) we have 0 < 7 < 8. Now as the number is a three digit number hence we cannot choose the first digit of the resultant to be 0 as it will make the  number a two digit (say 078), hence the required number is 708

Forming a Four Digit Number from Given Four Digits

Example: Find the largest four-digit number from the given digits 0, 7,8,6

Solution: Arranging the digits in descending order we get 8> 7 >6 > 0. Now the resultant is a four-digit number hence we cannot choose the first digit of the resultant to be 0. Hence the answer is 8760 choosing the digits of the resultant number in descending order.

 Letters for Digits

Here we have puzzles in which letters take the place of a digit in arithmetic sum and the problem is the to find out which letter represents which digit. There are two rules we follow while doing such puzzles.

  • Each letter in the puzzle must stand for just one digit. Each digit must be represented by just one letter.
  • The first digit of a number cannot be zero.
  • Thus, we write the number sixty-three as 63 and not as 063 or 0063.

For example, R and Y can be represented by 18 and 25 respectively, as per their alphabetical sequence.

Solve Examples for You

Question 1: On multiplying 121  and its reverse, we get

  1. 14641
  2. 14541
  3. 14441
  4. None of the above.

Answer: A is the correct option. The reverse of 121 is 121, hence 121 × 121 = 14641.

Question 2: Which of the numbers are in general form?

  1. 2 × 100 + 3 × 10 + 7
  2. 2 × 10 + 3× 10 + 7
  3. 2 × 100 + 2 × 100 + 7
  4. 2 × 100 + 3 × 10 + 7

Answer: A is the correct option. The general form of any three digits is
abc = a× 100 + b × 10+ c
So, 237 = 2 × 100 + 3 × 10 + 7
So only one expression is in general form.

Question 3: Can we say that 70 is a number?

Answer: One can regard the number 70 as a natural number. Furthermore, 70 is a whole number that is without a fraction. Moreover, one can regard 70 is a “counting number”. Also, 70 is an even and composite number. Another categorization of 70 can be as a real number. 70 also happens to be a rational number.

Question 4: What are the possible natural numbers that exist from 1 top 100?

Answer: The natural numbers from 1 to 100 are all the numbers from 1 to 100, including 1 and 100 themselves. Furthermore, when we count from 1 to 100, all the numbers that one counts, without any exception, are natural numbers.

Question 5: What is the number 1 and its fundamental property?

Answer: 1 is a number or numerical digit that is representative of a single entity. For example, if there is a line segment of unit length then it means that it means that it has a length of 1. Furthermore, multiplicative identity is a fundamental property of 1.

Question 6: Explain why number 1 is unique?

Answer: 1 is unique because it is neither a composite number nor a prime number.

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