Numbers in General Form

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