How to Write Numbers in General Form: Step-by-Step Guide
We see numbers everywhere around us. We have studied different types of numbers, such as natural numbers, whole numbers, integers, and rational numbers. You have also studied how to find factors and multiples and the relationships among them.In this article, we will explore the general numbers in more detail.
A number is said to be in a generalized form if it is expressed as the sum of the product of its digits with their respective place values. Thus, a two-digit number having a and b as its digits at the tens and the one’s places respectively can be written in a general form as 10a + b, where ‘a’ and ‘b’ can be any of the digits from 1 to 9. Similarly, a three-digit number can be written in the generalized form as 100a + 10b + c, where ‘a’, ‘b’, and ‘c’ can be any one of the digits from 1 to 9.
For Any Two-Digit Number:
We know that the face value and place value of the unit's place digit remains the same, but the place value of the tens place digit is ten times the face value of the digit. The value of the number is the sum of the individual place values of each digit.
Let us consider an example
For Example a Two-Digit Number 85 –
85 = 10 × 8 + 5
8 is at ten places so we multiply it by 10 and 5 is at one place so it remains the same.
63 = 10 × 6 + 3 → 6 is at tens place so we multiply it by 10 and 3 is at one place so it remains the same.
Thus, any two-digit number can be written in a general form as 10 × x + y i.e 10x + y, where x is the digit in tens place which can be any digit from 1 to 9 and y is the digit in tens place that can be any digit from 1 to 9.
Similarly, for the Three-Digit Number
General form is,
The three-digit number xyz can be written as 100 × x + 10 × y + z i.e 100x + 10y + z,
where x is the digit in the hundreds place, y is the digit in the tens place, and z is the digit in one place.
Example:
329 = 100 × 3 + 2 × 10 + 9;
266 = 100 × 2 + 10 × 6 + 6;
Games With Numbers
When we reverse the two digits, the numbers become 10y + x
Sum of the original number and the reversed number = (10x + y ) + ( 10y + x)
= 10x + y + 10y + x
= 11x + 11y
= 11( x+ y)
Hence the sum of the original number and the reversed number will always be a multiple of 11.
If the answer is divided by 11 the remainder is 0.
Also, the difference between an original 2-digit number and the number obtained by reversing its digits is always divisible by 9.
When we reverse the three-digit numbers it will be written as 100z + 10y + x
The difference of the two numbers = (100x + 10y + z) - (100z + 10y + x)
= 100x + 10y + z - 100z -10y - x
= 99x + 0 - 99z
= 99(x - z)
Hence, the difference will always be a multiple of 99.
Tests of Divisibility
Learning about the general number form now lets us study how it works with rules of divisibility.
Test of Divisibility by 2:
The rule of divisibility states that a number is divisible by 2, if its units digit is even, i.e., if its units digit is any of the digits 0, 2, 4, 6, or 8.
For a number in the generalized form:
A general two-digit number 10a + b is divisible by 2 if ’b’ is any of the digits 0, 2, 4, 6 or 8.
A general three-digit number 100a + 10b + c is divisible by 2 if ’c’ is any of the digits 0, 2, 4, 6 or 8.
For example, each of the numbers 74, 22, 12, 344, 406, 864, 130, etc., is divisible by 2.
Test of Divisibility by 3:
The rule of divisibility by 3 states that a number is divisible by 3 if the sum of its digits is divisible by 3.
For a number in the generalized form:
A general two-digit number 10x + y is divisible by 3 if (x + y) is divisible by 3.
A general three-digit number 100a + 10b + c is divisible by 3 if (x + y + z) is divisible by 3.
For example, each of the numbers 72, 12, 93, 342, etc., is divisible by 3. Also, each of the numbers 71, 53, 42, 82, etc., is not divisible by 3.
Test of Divisibility by 5:
The rule of divisibility by 5 states that a number is divisible by 5 if its unit’s digit has either 0 or 5.
For a number in the generalized form:
A general two-digit number 10x + y is divisible by 5 if ‘y’ is either 0 or 5.
A general three-digit number 100x + 10y + z is divisible by 5 if ‘z’ is either 0 or 5.
For example, each of the numbers 25, 30, 75, 115, 190, 540, 275, 825, etc., is divisible by 5.
Test of Divisibility by 9:
The rule of divisibility by 9 states that a number is divisible by 9 if the sum of its digits is divisible by 9.
For a number in the generalized form:
A general two-digit number 10x + y is divisible by 9 if (x + y) is divisible by 9.
A general three-digit number 100x + 10y + z is divisible by 9 if (x + y + z) is divisible by 9.
For example, each of the numbers 18, 81, 72, 324, 459, 792, etc., is divisible by 9.
Test of Divisibility by 10:
The rule of divisibility by 10 states that a number is divisible by 10 if its unit’s digit is 0.
For a number in the generalized form:
A general two-digit number 10x + y is divisible by 10 if ‘y’ is equal to 0.
A general three-digit number 100x + 10y + z is divisible by 10 if ‘z’ is equal to 0.
For example, each of the numbers 10, 140, 100, 340, 530 180, etc., is divisible by 10.
Conclusion
Numbers are seen everywhere. We know of the different types of numbers, such as natural numbers, whole numbers, integers, and rational numbers. A two-digit number that has a and b as its digits at the tens and the one’s places respectively is written as 10a + b, where ‘a’ and ‘b’ is any digit from 1 to 9. Likewise, a three-digit number can also be written as 100a + 10b + c, where ‘a’, ‘b’, and ‘c’ is any digit from 1 to 9.
Solved Examples
Write the Generalized Number Form
1. 9500
Solution:
9500 = (9 x 1000) + (5 x 100) + (0 x 10) + (0 x 1)
= (9 x 103) + (5 x 102) + (0 x 101) + (0 x 100)
2. 379
Solution:
379 = (3 x 100) + (7 x 10) + (9 x 1)
= (3 x 102) + (7 x 101) + (9 x 100)
Write in General Number Form
1. (8 x 100) + (5 x 10) + (8 x 1)
Solution:
= 800 + 50 + 8
= 858
2. (6 x 1000) + (4 x 10) + (8 x 1)
Solution:
= 6000 + 40 + 8
= 6048
FAQs on Numbers in General Form: Definition, Rules & Uses
1. What does it mean to write a number in its general form?
Writing a number in its general form means expressing it as the sum of the products of its digits with their respective place values. This method breaks a number down into its constituent parts based on the base-10 system, making it easier to analyse its properties and prove mathematical rules. For example, the number 85 in its usual form is written as 8 × 10 + 5 in its general form.
2. How do you write a two-digit and a three-digit number in general form? Provide examples.
To express numbers in general form, you multiply each digit by its place value (ones, tens, hundreds, etc.) and then sum up the results. The rules are as follows:
- A two-digit number represented as 'ab', where 'a' is the tens digit and 'b' is the units digit, is written as 10a + b. For example, the number 62 = 10 × 6 + 2.
- A three-digit number represented as 'abc' is written as 100a + 10b + c. For example, the number 349 = 100 × 3 + 10 × 4 + 9.
3. Why is using the general form of numbers, such as 10a + b, so important in mathematics?
The general form of numbers is very important because it allows us to prove properties and rules that apply to all numbers of a certain type, rather than just observing patterns in specific examples. Its main uses include:
- Proving Divisibility Tests: It helps in algebraically proving the tests for divisibility by numbers like 2, 3, 5, 9, and 10.
- Solving Number Puzzles: It is the key to solving cryptarithmetic problems, where letters stand for unknown digits.
- Understanding Number Patterns: It explains the logic behind number games, such as tricks involving reversing digits.
4. How does the general form of a number help explain the divisibility test for 3?
The general form provides a logical proof for the divisibility test of 3. Let's take a three-digit number 'abc', which in general form is 100a + 10b + c. We can rewrite this expression as:
- (99a + a) + (9b + b) + c
- By rearranging the terms, we get: (99a + 9b) + (a + b + c)
- This can be factored as: 9(11a + b) + (a + b + c)
5. What is the difference between a digit's face value and its place value in the context of its general form?
The general form of a number clearly illustrates the difference between face value and place value:
- Face Value: This is the inherent value of a digit, regardless of its position. In the number 257, the face value of '5' is simply 5.
- Place Value: This is the value a digit contributes to the total number due to its position. In 257, the '5' is in the tens place, so its place value is 50 (or 5 × 10).
6. How does the general form explain the outcome of reversing a two-digit number and subtracting it from the original?
The general form clearly explains this number trick. Let the original two-digit number be 'ab' (where a > b), which in general form is 10a + b. The reversed number 'ba' is 10b + a. When you subtract the smaller number from the larger one:
- (10a + b) - (10b + a) = 10a + b - 10b - a
- This simplifies to 9a - 9b, which can be written as 9(a - b).
7. What are the main rules to follow when solving letter-for-digit puzzles (cryptarithmetic) using the general form?
When using the general form to create equations for cryptarithmetic puzzles, you must follow two fundamental rules:
- Unique Digits: Every letter represents a single, unique digit from 0 to 9. If 'A' is 3, no other letter in the puzzle can be 3.
- No Leading Zeros: The first digit of any multi-digit number cannot be zero. For example, in the number 'SEND', 'S' cannot be 0.
