Ascending Order

Ascending order is a method of arranging numbers from smallest to largest. We can also say arranging the numbers in increasing order. For example, a set of natural numbers are in ascending order, such as 1, 2, 3, 4, 5, 6, 7, 8… and so on. The inverse of increasing order is decreasing order or also called descending order. Various Mathematical symbols are used to precisely represent the concept statement.  Ascending order is represented by the less than symbol ‘<‘. Ascending order is represented by the symbol ‘<‘. For example, 1<2<3<4<5<6<7<8<9. The symbol represents that the succeeding number is greater than the preceding number in the arrangement. In this article, let us discuss in detail about the definition of ascending order, its symbol, ascending order for integers, fractions and decimals with many solved examples.

Definition

Ascending order is a sorting method in which the sort starts from the lowest to the highest or the smallest value to the largest value. It means the smallest or earliest value in the order that will come at the top of the list.

Examples of Ascending Order

• For amount and numbers, the sort is from smallest value to largest value. Lower amounts or numbers will be at the top of the list. Ex: 16, 23, 45, 76, 80.
• For letters and words, the sort is from alphabetical order like A to Z. Example: A, B, C, D, E……Y, Z.
• For information with numbers, letters and words, like address lines, the sort is most likely to be alphanumeric values (0-9) then followed by A-Z. Example: 119, D – Building, XYZ.
• For dates, the sort will be the oldest dates that will appear at the top of the list.

Ascending order symbol

To represent this order of numbers, we need to use the ascending order symbol “< “.

It represents numbers from lowest to highest, which is opposite to the concept of Descending Order, where numbers are arranged from largest to smallest. For example:- If you want to arrange numbers 1 to 20 in increasing order, you will make use of this symbol “<“.

1 < 2 < 3< 4 < 5 < 6 < 7 < 8 <  9 < 10 < 11 < 12 < 13 < 14 < 15 < 16 < 17 < 18 < 19 < 20 

Ascending Order of Negative Numbers

It is easy to arrange the positive integers in ascending order, but for negative integers, we have to be careful. The highest number with minus sign (-), is the smallest value. 

For example, if we take -1 to -10 numbers, then on arranging them in increasing order, we get;

-10 < -9 < -8 < -7 < -6 < -5 < -4 < -3 < -2 < – 1

The above arrangement shows, -10 is the smallest value and -1 is the largest one.

Fact: The integers on the number line, when seen on the right side of the 0 are arranged in ascending order whereas, on the left side of the 0, they are arranged in descending order.

How to Arrange Numbers in Ascending Order?

To arrange the numbers in the ascending order, first, we need to compare the values and then order it in the ascending order. The procedure to arrange the numbers in the smallest order to the largest order is as follows:

Step 1: The first and foremost step is to count the number of digits in the number. The number with the least number of digits is the smallest number. If the list has more numbers with the least digit, then compare the values and write down the smallest number. 

Step 2: Write down the smallest number first, and then compare with all the remaining numbers with the same number of digits.

Step 3: Continue the comparison of numbers with the next level of digits, and write down the numbers in the ascending order till all the numbers are arranged in the form of ascending order.

For example, we have to arrange the given numbers in the ascending order. 

2346, 34, 7, 123, 30, 4 

In this given list of numbers, there are numbers with one digit, two digits, three digits and four digits. So, first we have to categorize the numbers from single digit, double digit and so on.

Numbers with one digit: 7 and 4

Numbers with two digits: 34 and 30

Numbers with three digits: 123

Numbers with four digits: 2346.

So, first compare the numbers 7 and 4. Here 4 is smaller than 7. Next, compare the two digit numbers 34 and 30. Here 30 is smaller than 34. Also, we have only one three digit number and four digit numbers.

So, the ascending order of the given number 2346, 34, 7, 123, 30, 4 is 4, 7, 30, 34, 123, 2346. This can be represented as 4 < 7< 30< 34< 123< 2346. 

What does Ascending order mean?

In mathematics, you must have learned many fundamentals and concepts. Each of these has its own importance in the mathematical calculations. In the same way, ascending order also has its own significance in Math.

Ascending order means increasing the order of a series, sequence or a pattern. In terms of numbers, the increasing order is written from the least value to the highest value. Series and sequences follow a pattern where the numbers are written in an increasing or decreasing order, based on the common difference between the terms. Suppose, 2, 4, 6, 8, 10 is a series which is written in ascending order with a common difference of 2. We can represent it as 2 < 4 < 6 < 8 < 10.

Ascending Order in Math

In Mathematics, the numbers are said to be in ascending order when they are arranged from the smallest number to the largest number. This order in math helps the children at primary school to learn about the arrangement of numbers in increasing order to solve the problem. The arrangement of numbers can be done in any of the real number systems.

Example: Consider the numbers 34, 10, 6, 4, 45, 25 are arranged randomly.

The increasing order of the numbers are 4, 6, 10, 25, 34 and 45

Since the numbers are arranged in increasing order, the increasing order symbol is denoted by “ < “ symbol. The above example is also represented as 4 < 6 < 10 < 25 < 34 < 45

Solved Problem

Question: Arrange the following numbers in increasing order :

6³ , 7² , 4² , 2⁶ ,3⁴ and 10²

Solution: Given , 6³ , 7² , 4² , 2⁶ ,3⁴ and 10²

6³ = 6 x 6 x 6 = 216

7²= 7 x 7 =49

4² = 4 x 4 = 16

2⁶ =2 x 2 x 2 x 2 x 2 x 2 = 64

3⁴ = 3 x 3 x 3 x 3 = 81

10² = 10 x 10 = 100

Therefore, the increasing order of the numbers are 16, 49, 64, 81, 100 and 216 and will be represented as  : 4² < 7² < 2⁶ < 3⁴ < 10² < 6³

Ascending order for class 1

Ascending order for class 1 examples are given below. You can also practice the same to have a better understanding of the concepts.

Example: Write these numbers in increasing order: 8, 10, 92, 1, 27.

Solution: The following is the increasing order of the given numbers:

1 < 8 < 10 < 27 < 92

Now, Practice the following on your own.

• Write these numbers in increasing order: 90, 34, 92, 1, 35.
• Write these numbers in increasing order: 80, 1, 12, 10, 72.
• Write these numbers in increasing order: 18, 11, 67, 19, 07.
• Write these numbers in increasing order: 7, 15, 90, 81, 56.

Ascending Order and Descending Order

Descending order is the contradiction of ascending order. That means it is the opposite process of writing the numbers in increasing order. Therefore, it can be mentioned as a decreasing order. In the case of descending order, for a given set of numbers, the highest valued number is written first, and the lowest valued number is written at last. It is denoted by the symbol ‘>’.

Example: Write 3, 7, 8, 2, 10, 28, 15 in descending order.

Solution: 28 > 15 > 10 > 8 > 7 > 3 > 2 is the descending order of the given set of numbers.

Ascending order Alphabets

Same as numbers you can also arrange alphabets in ascending order and descending orders. For example : a < b < c < d < e < f < g < h < i < j <  k < l < m < n < o < p < q < r < s < t < u < v < w < x < y < z (For small alphabets). 

You can reverse the order of the alphabets in the case of descending order.

Ascending Order for Fractions

There are two methods involved in finding the ascending order of the fractional numbers. Both the method will give the same solution

Method 1:

• Step 1: For a given series of fractions, first convert it into decimal numbers.
• Step 2: Find the increasing order of the decimal numbers.
• Step 3: Finally, replace the decimal values with the respective fractional numbers.

Example: Find the increasing order of the following fractions: 8/6, 2/3, 10/12 and 9/6.

Solution:

Given series: 8/6, 2/3, 10/12 and 9/6.

Step 1: Converting fractions into decimals

8/6 = 1.33

2/3 = 0.67

10/12 = 0.83

9/6 = 1.5

Step 2: Arranging the increasing order of the decimal values

0.67 < 0.83 < 1.33 < 1.5

Step 3 : Replacing the decimal values with fraction values

2/3 < 10/12 < 8/6 < 9/6

Therefore, Increasing order of the given fractions are 2/3, 10/12, 8/6 and 9/6.

Method 2 :

• Step 1: Find the L.C.M of the denominators.
• Step 2: Divide the L.C.M value by the denominator of the fraction.
• Step 3: Multiply both the numerator and denominator of the fraction with the resultant value of step 2.
• Step 4: As a result of step 2 and step 3, compare the like fractions.
• Step 5: Since the denominators are the same, compare the numerator values of like fractions.
• Step 6: Finally, arrange the fractions in increasing order with its respective fractions given in the problem.

Example: Consider the same example given above: 8/6, 2/3, 10/12 and 9/6.

Solution:

Step 1: L.C.M of denominator = L.C.M of 6, 3, 12, 6 =3 x 2 x 2 =12

Step 2: For a fraction 8/6, its denominator is 6, then divide the L.C.M by 6,

We get, 12/6 = 2

Step 3: Multiply both the numerator and denominator by 2,

i.e.,(2 x 8) / (6 x 2) =16/12

Similarly for other fractions repeat step 2 and step 3, we get

For 2/3 =12/ 3 = 4 ,so that (4 x 2) / (3 x 4) = 8/12

For 10/12 = 12/12 = 1 , so that (10 x 1) / (12 x 1)=10 / 12

For 9/6 = 12/6 = 2 , so that (2 x 9 ) / (6 x 2 ) = 18/12

Step 4: Now, the like fractions are 16/12 , 8/12, 10/12 and 18/12

Step 5: Compare the numerator values 8 < 10 <16 <18

It becomes 8/12 < 10/12 < 16/12 < 18/12

Step 6: Respective ascending order fractions are 2/3 < 10/12 < 8/6 < 9/6

Therefore, the increasing order of the given fractions are 2/3 , 10/12, 8/6 and 9/6.

Practice Problems

Solve the following problems on the ascending order given below: 

1. 1. Write the numbers in increasing order: 90, 34, 92, 1, 35.
   2. Write the numbers in ascending order using the symbol: 80, 1, 12, 10, 72.
   3. Rearrange the numbers in increasing order: 18, 11, 67, 19, 07.
   4. Write the numbers in increasing order: 7, 15, 90, 81, 56.

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