Ascending

When we arrange the numbers in ascending order, the first number will be the smallest, and the last number will be the largest. In this order, the sequence of numbers gradually increases. The numbers in ascending order will increase in a pattern or an order.

A given set of numbers can be represented in ascending order using either commas (,) or the less than symbol (<). It indicates that the number on the right side of the symbol is greater than the number on the left side. We can arrange the numbers from 1 to 5 as:
1 < 2 < 3 < 4 < 5

While we arrange numbers in increasing order, we place them from the smallest to the largest value. To arrange the numbers in ascending order, we first need to identify the nature of the given numbers. Determine whether they are integers, negative numbers, fractions, or decimals. Then, compare the numbers and use the less-than symbol (<) to represent their order.

Imagine there is a horizontal line with a point at the origin (0). When we move towards the right of the line, the numbers increase in value. 

Look at the number line above. Here, -1 is smaller than 1. The smallest numbers are placed on the leftmost side, while the greatest numbers are positioned on the right side. The resulting arrangement on the number line is represented as:
-2 < -1 < 0 < 1 < 2 < 3 < 4

Whole numbers that can be zero, positive, or negative are called integers. While arranging integers in ascending order, certain rules must be followed. They are as follows:

• When arranging integers, keep in mind that positive numbers such as 1, 2, 3, etc. are greater than zero. Moreover, all the negative numbers like -1, -2, -3, and so on are less than zero. 

• Negative numbers are less than positive numbers. 

• All positive numbers are arranged in ascending order, meaning a smaller number comes before a larger number. 

• The absolute values of the negative numbers are arranged in descending order. For example, the absolute values of -2, -4, and -5 are 2, 4, and 5. Then, by using a negative sign, the sequence of negative numbers is written in ascending order. For instance, -7 is less than -4, which is less than -2. 

Let us take an example to understand the arrangement of integers in ascending order. 
-35 < - 24 < -5 < 0 < 2 < 22 < 34

In negative numbers, as the number increases, its value decreases. The larger the number, the smaller its value. The smallest negative number has the highest value. For example, the given negative numbers are -3, -12, and -67. Here the -3 is greater than -67. A number with a larger absolute value is smaller in numerical value. So, when we arrange negative numbers in ascending order, the absolute values of negative numbers are arranged in descending order.

For instance, 
-12 < -9 < -5 < -2 < -1 

The absolute value of the above negative numbers are: 
12, 9, 5, 2, 1

A fraction is a part of a whole number, and it has a numerator and a denominator. Arranging decimals in ascending order involves placing them from smallest to largest. There are two ways in which we can arrange them in ascending order.  The methods are as follows: 
 

Method 1: We can arrange the fractions in ascending order by dividing the given fraction's numerator by its denominator. Then, arrange the decimals in ascending order according to the place values of the whole number part and the decimal part. To understand, check this example: 
1/4, 1/2, 5/2, 2/7, 3/5. 

Arrange the fractions in ascending order. Here, we can divide the numerator by its denominator. 
1/4 = 0.25
1/2 = 0.5
5/2 = 2.5
2/7 ≈ 0.286
3/5 = 0.6

To arrange the decimals numbers in ascending order, from the smallest to largest:
0.25, 0.2857, 0.5, 0.6, 2.5

So, the fractions in ascending order will be as follows: 
1/4, 2/7, 1/2, 3/5, 5/2 

Method 2: By finding the LCM of the denominators, fractions can be arranged in ascending order using the second method. Here, we need to identify a common denominator for the fractions and then multiply the same number by the numerator and denominator. After that, compare the obtained numerators and arrange them in ascending order. To understand this, let us consider a simple example. Arrange the given fractions 1/2, 3/4, 5/8, 7/3. Here we need to find the least common multiple of the denominators. 

LCM of 2, 4, 8, 3 =  24 
Multiples of 2 include 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 24...
Multiples of 4 include 4, 8, 12, 16, 20, 24...
Multiples of 8 include 8, 16, 24...
Multiples of 3 include 3, 6, 9, 12, 15, 18, 21, 24...

Next, we multiply the numerator and denominator of each fraction by a factor to get the denominator as 24. 
1/2 = 1 × 12 / 2 × 12 = 12/24 
3/4 = 3 × 6 / 4 × 6 = 18/24
5/8 = 5 × 3 / 8 × 3 = 15/24
7/3 = 7 × 8 / 3 × 8 = 56/24

Arrange them in ascending order: 
12/24, 15/24, 18/24, 56/24

Therefore, the fractions in ascending order are: 
1/2 < 5/8 < 3/4 < 7/3 

Decimals are numbers in which the whole number part is separated from the fractional part by a decimal point (.). When we arrange decimals in ascending order, first observe the whole number part. If two numbers have the same whole number part, compare the decimal parts one by one. For example, the given decimals are 1.3, 4.11, 2.78, 1.8.
Here, 1.3 and 1.8 have the same whole number part (1). So compare the decimal part. 
1.3 < 1.8  

Therefore, we can arrange the given decimals in ascending order as follows: 
 1.3 < 1.8 < 2.78 < 4.11

In our daily lives, we arrange numbers, measurements, heights, or any values in ascending order to compare them easily and understand them better. Here are the real-world applications of ascending order are given below:

• Scholars and mathematicians often use ascending order to arrange various measurements or numbers in ascending order for better comparison. 

• Sports teams and coaches can classify and categorize their players according to their performance and competition results, in ascending order. 

• Travelers can figure out the altitude and height of a mountain by using ascending order. The height and altitude increase gradually, allowing them to predict their next steps. 

• In research and experiments, researchers can track growth in ascending order, which helps them understand the progress effectively.