Ascending Order

Any group of numbers, things, or anything arranged in order of its increasing value is termed as the ascending order in mathematics of that group. 

1.0Ascending Order in Maths

Ascending order means the arrangement of elements, including numbers, letters, and so on, from smallest to largest or in ascending order. This is extensively used for numbers and words. The order can be denoted by the sign “>” or “<” in between the values. The open end is pointed towards the bigger values, while the pointed end is towards the smaller values like 5 >3 or 3<5. 

Ascending order is quite commonly used in many domains, such as mathematics, sorting data, or arranging items systematically.

2.0Ascending Orders in Different Contexts

In Terms of Numbers

When arranging numbers in ascending order, we begin with the smallest number and move all the way to the largest.

For instance:

• 3, 5, 7, 9, 10 (in ascending order)
• 1, 4, 6, 8, 12 (in ascending order)

In mathematical contexts, this ordering is used to compare numbers or arrange them systematically for easier analysis.

In Terms of Words

Words or strings of characters can be arranged alphabetically in an ascending manner, too.

For instance:

• Apple, Banana, Cherry, Grapes, Orange (alphabetical order is one form of ascending order)
• Car, Cat, Dog, Elephant, Zebra- alphabetical order

3.0Steps to Arrange in Ascending Order

1. Identify the elements: Begin by identifying the numbers or words that have to be sorted.
2. Compare the values: For numbers, compare their size. For words, compare their alphabetical order.
3. Arrange in ascending order: Start from the lowest and increase to the largest.
4. Check your result: Ensure the arrangement is in the correct order.

4.0Ascending Order and Descending Order

While discussing the arrangement of numbers or words in maths, it can be arranged in both increasing as well as decreasing order, which are also known respectively as: 

• Ascending Order
• Descending Order  

5.0Ascending Order of Rational Numbers

Rational Numbers: These are the numbers that can be expressed in the form of $\frac{a}{b}$, where a and b are integers, b\ne0. For example: $\frac{1}{2},-\frac{5}{7},5,-2$.

Ascending order of Rational numbers refers to the arrangement of a given set of rational numbers from the smallest to the largest.

6.0Steps to Arrange Rational Numbers in Ascending Order

• Compare Using Common Denominators (when possible): Compare rational numbers by converting them to have a common denominator. This is helpful when comparing fractions that have different denominators.
• Convert to Decimal Form (when possible): Compare rational numbers by converting to decimal form.
• Compare the Values: Compare directly after converting fractions either to a common denominator or decimal form. The smallest value is the smallest rational number, and the largest value is the largest rational number.
• Consider Negative Rational Numbers: Negative rational numbers are ordered oppositely. The negative rational number with the lowest absolute value is the largest negative. Therefore, when arranging the negative rational numbers in ascending order, remember that the smallest absolute values are more significant but still negative in value.

7.0Solved Problems

Problem 1: Arrange the following rational numbers in ascending order:

$\frac{3}{5},\frac{7}{10},\frac{1}{2},\frac{2}{3}$

Solution: taking the LCM of All the denominators to get a common denominator: 

LCM = 30

The Number will be: 

$\frac{18}{30},\frac{21}{30},\frac{15}{30},\frac{20}{30}$

The Ascending order will be: 

$\frac{15}{30}<\frac{18}{30}<\frac{20}{30}<\frac{21}{30}$

Problem 2: Arrange in ascending order: $\frac{2}{3},\frac{1}{4},\frac{5}{6}$

Solution: Converting to decimals: 

$\frac{2}{3}=0.666$

$\frac{1}{4}=0.25$

$\frac{5}{6}=0.833$

The ascending order will be: 

$\frac{1}{4}<\frac{2}{3}<\frac{5}{6}$

Problem 3: Arrange the following rational numbers in ascending order:

$-\frac{1}{3},-\frac{5}{6},-\frac{1}{2},-\frac{3}{4}$

Solution: Converting the numbers to decimal form: 

$-\frac{1}{3}=-0.333$

$-\frac{5}{6}=-0.833$

$-\frac{1}{2}=-0.5$

$-\frac{3}{4}=-0.75$

Hence, the ascending order will be: 

$-\frac{1}{3}<-\frac{1}{2}<-\frac{5}{6}<-\frac{3}{4}$

8.0Also Read