Category: Maths Notes PPT

Correlation refers to the process of establishing a relationship between two variables. To identify or to understand whether a relationship exists between two variables or not, you plot the points on a scatter plot. There are many ways in which you can relate the variables – like the ordinal level of measurement or higher level of measurement, but the most commonly used approach is a correlation.

 

Correlation in Statistics

In this section, you will be learning how to interpret correlation coefficients and calculate correlation coefficients for interval level scales as well as the original level scales. A correlation coefficient is a single number which is summarized by the relationship between 2 numbers using methods of correlation. The reason behind scaling correlation coefficient is to make sure that it always lies between +1 and -1. If the coefficient is close to 0 then the relation between the relationship between the two numbers is less and when the relationship is far away from 0 then the relationship is strong between the two variables. 

The usual symbols given to these variables are X and Y. To show how these variables are related to each other, the values are illustrated by drawing them on the scatter diagram and then graph the combinations of the variables X and Y. First, the scatter diagram is drawn. Next, the method to determine Pearson’s r is performed. Initially, small samples are taken to represent it and then larger sizes of samples are used. 

Types of correlation

Now that we know that the scatter plots are used to explain the correlation between two numbers or variables, let us study about correlation and its types. The relationship between the two variables can be compared using three different types of correlation: positive correlation, negative correlation, or no correlation.

• Positive Correlation:  This situation occurs if the value of one variable increases the value of the variable also increases

• Negative Correlation:   This situation occurs if the value of one variable increases the value of the decreases also decreases

• No Correlation: In this situation, the variables are not dependent on each other

Pearson’s Correlation Coefficient Formula

The most commonly used formula to find the linear dependency of two sets of data is Pearson’s Correlation Coefficient Formula. The value of Pearson’s Correlation Coefficient lies between positive 1 and a negative 1. When the value of the coefficient is above +1 and less than – 1, the data is considered to be unrelated to each other. Data sets are considered to be in positive correlation if their coefficient is +1 and the data sets are considered to be in a negative correlation if their coefficient is -1.

r = [frac{n(sum{xy})-(sum{x})(sum{y})}{sqrt{[nsum{x^{2}-(sum{x})^{2}}][nsum{y^{2}-(sum{y})^{2}]}}}]

Here,

n = It is the quantity of information that is available

Σx = The total value of the first variable

Σy = The total value of the second variable

Σxy = It is the product of the sums of the first and the second value

Σx² = It is the square of the sum of the first value

Σy² = It is the square of the sum of the second value

 

Linear Correlation Coefficient Formula

The formula for the linear correlation coefficient is given below:

[frac{nsum_{i=1}^{n}{x_i}{y_i}-sum_{i=1}^{n}{x_i}sum_{i=1}^{n}{y_i}}{sqrt{nsum_{i=1}^{n}{x_i}^{2}-(sum_{i=1}^{n}{x_i})^{2}}sqrt{nsum_{i=1}^{n}{y_i}^{2}-(sum_{i=1}^{n}{y_i})^{2}}}]

Sample Correlation Coefficient Formula

The sample correlation coefficient formula is: rab = Sab / SaSb

Here, = is the sample standard deviation

Sa = is the sample standard deviation

Sb = is the sample standard deviation

Sab = is the sample covariance

 

Population Correlation Coefficient Formula

The formula for population correlation coefficient is:

 

rab = σab/σaσb

 

Here,

σa = is the population standard deviation

σb = is the population standard deviation

σab = is the population covariance

 

Solved Problem

The number of years of education received and The age of entering the workforce will give us the years of formal education one has received.  In the table below, you’ll see the years of education (A) a person has received and the age at which he entered the workforce (B). The survey was done among 12 people and all these people were aged above 30 years or more. 

 

 

Here, you can notice that people started their formal education early and you can also notice that the relationship between the number of years of schooling and the age at which they entered the workforce. For example: see Person 11. They had just 8 years of formal education but they entered the workforce at the age of 18. The scatter diagram helps you understand the relationship between the number of years of schooling and the age at which they entered the workforce.

In algebra, Cramer’s rule is defined as an explicit formula or method used to solve a system or series of linear equations. It applies to those linear equations, having as many unknown variables as values and when a unique solution is probable. The Cramer rule method uses determinants to find the values of the variables given in a linear equation. The rule expresses the solution in terms of determinants by arranging the matrix in the form of Ax=B, where:

• A represents the coefficient matrix and contains all the numerical values.

• X represents the matrix of variables.

• B represents the matrix with all the constants on the right-hand side of the equation.
  

History of Cramer’s Rule

Cramer’s rule was formed and developed by the great mathematician Gabriel Cramer in the 1750s. He used Cramer’s method to find the solution of a system of equations with n number of variables and the same number of equations. 

Cramer’s Rule Formula

Cramer’s rule formula is easy to understand, and it helps us solve the AX=B form of the matrices. With the help of the formula, we can find the values of n number of variables in a linear equation. To solve using Cramer’s rule the equation AX=B, we need to follow the steps below:

• To solve by Cramer’s rule, we need to find the determinants for matrix A, x1, x2, x3, …., xn. 

• Each of these determinant matrices is denoted by D, such as D = |A|, Dx1, Dx2, Dx3, …, Dxn. 

• Here the D represents the determinant values of |A|, and Dxn represents the determinant values of the x matrix, where values of B replace the nth column. 

• Once we get all the determinant values for the equations, we apply the formula:

• Xn=  Dxn/D. We can apply this formula to as many equations as present in the matrices. Always remember that the value of D should never be equal to zero (D≠0).

• Hence, with the formula mentioned above, we can derive the value of the variables using Cramer’s method.

With Cramer’s rule formula, we can solve the system of two equations in two variables and three equations in three variables. If you are wondering how below, we will discuss the method in detail. 

Cramer’s Rule 2×2

Using Cramer’s rule formula, we will solve two equations in two variables. Let’s discuss the steps to solve two equations in two variables using Cramer’s rule matrix. For Cramer’s rule 2×2, let’s consider two equations in 2 variables, x and y:

Let the equation be:

p1x + q1y = r1 and,

p2x + q2y = r2.

Now, follow the steps below to solve using Cramer’s rule:

A =   $begin{bmatrix}p₁ &q₁ \ p₂ &q₂ end{bmatrix}$

x =   $begin{bmatrix}X\ Yend{bmatrix}$

And,

B =    $begin{bmatrix}r₁\ r₂ end{bmatrix}$

• Step 2 – Here, we need to find the determinants of the equations, where D = |A|, and Dx and Dy.

For D or |A|, solve :

$begin{vmatrix}p₁ &q₁\ p₂ &q₂  end{vmatrix}$

For DX, replace the first column of matrix A with values of B; hence we get Dx:

$begin{vmatrix}r₁ &q₁\ r₂ &q₂  end{vmatrix}$

For Dy, replace the second column of matrix A with values of B and solve

$begin{vmatrix}p₁ &r₁\ p₂ &r₂  end{vmatrix}$

• Step 3 – Once we get the determinants of A, Dx, and Dy, we will find values of x and y using:

X = DX/D

Y = Dy/D.

Cramer’s Rule 3×3

Above, we discussed the methods and steps to solve 2×2 equations using Cramer’s rule. Now, we will discuss solving three equations in three variables by Cramer’s method. For solving Cramer’s rule, 3×3 equations consider an equation with three variables x, y, and z.

Let the equation be:

p1 x + q1y + r1z = s1

P2x + q2y + r2z = s2

p3x + q3y + r3z = s3

Now, follow the steps mentioned below to solve by Cramer’s rule:

A =   $begin{bmatrix}p₁ &q₁  &r₁ \ p₂ &q₂  &r₂ \ p₃ &q₃  &r₃ end{bmatrix}$

X =    $begin{bmatrix}X\ Y\ Zend{bmatrix}$

B =    $begin{bmatrix}s₁ \ s₂\ s₃ end{bmatrix}$

• As per Cramer’s rule, determinant value is important to find the value of variables. Here also we need to find D|A|, Dx, Dy, and Dz. Where:

D|A| =  $begin{vmatrix}p₁  &q₁   &r₁  \ p₂ &q₂  &r₂ \ p₃ &q₃  &r₃end{vmatrix}$

For Dx, replace the first column of matrix A with values of B.

Dx =  $begin{vmatrix}s₁  &q₁   &r₁  \ s₂&q₂ &r₂\ s₃&q₃ &r₃end{vmatrix}$

For Dy, replace the second column of matrix A with values of B.

Dy, = $begin{vmatrix}p₁ &s₁  &r₁ \ p₂ &s₂ &r₂ \ p₃ &s₃  &r₃end{vmatrix}$

For Dz, replace the third column of matrix A with the values of B.

Dz =  $begin{vmatrix}p₁ &q₁  &s₁ \ p₂ &q₂  &s₂ \ p₃ &q₃  &s₃ end{vmatrix}$

X = Dx/D

Y = Dy/D

Z = Dz/D

These are the steps to solve three equations in three variables using Cramer’s rule.

Rules for Using Cramer’s Rule in Matrix

Above, we discussed how to solve equations using Cramer’s rule formula, but before using Cramer’s rule in the matrix, one should know the rules to be followed to get the correct solution for the equations. Below are some rules for using Cramer’s rule in the matrix:

• While calculating the determinant value, one should note that D should never be equal to 0. If D = 0, the equation does not have any solution, or it can have infinite solutions.

• If there are n number of variables and n equations in the matrix, we must calculate the value of  (n+1) determinants.

• You can only find a solution using Cramer’s rule if D is not equal to zero.

The symbol of a square root is √ while the cube root of a number is denoted by ∛. That said, the cube root of a number y is that number whose cube results in y. Thus, we denote the cube root of y by ∛y. As an example of the cube root of 64, we denote it as 64 that gives 4³ = 4 ×4 × 4 = 64. For example:

(i) Since (4 × 4 × 4) = 64, we have ∛64 = 4, similarly

(ii) Since (6 × 6 × 6) = 216, we have ∛216 = 6

[sqrt[3]{64}]

What is Cubing?

It might take you by surprise, but the process of cubing is the same as squaring, just that the number is multiplied three times instead of two. The exponent or power of the factor used for cubes is 3, which is also represented by the superscript³. As an example the cube root of 64 is 4 and it is mathematically expressed as 4³ = 4 ×4 × 4 = 64 or 7³ = 7 × 7 × 7 = 343.

Introduction to a Cubic Function

The cubic function states a one-to-one function. You might be wondering how and why this is so. This is due to the reason 4 cubing negative numbers result in an outcome different from that of cubing its positive coordinate. Also, when we multiply the three negative numbers together, two of the negatives are canceled off, but one remains, so the result obtained is also negative. 9³ = 9 ×9 × 9 = 729 = (-9) × (-9) × (-9) = -729. In a similar as a perfect square, a cube number, or a perfect cube is an integer that results from cubing another integer. 729 and -729 are examples of perfect cubes.

Factorization Method to Find the Cube Root

We can easily find the cube root of a given number by using the method of factorization. To determine the cube root of a given number, proceed as given below:

Step I. write the given number as the product of primes.

Step II. Create a group in triplets of the same prime.

Step III. Identify the product of primes, selecting one from each triplet.

Step IV. This product obtained is the needed cube root of the given number.

Remember that if the grouping in triplets of the same prime factors cannot complete, then we cannot find a perfect cube root.

(Image to be added soon)

How to Calculate Prime Number Factors?

If you think that finding prime or natural number factors is a tough task, then it is not. Simply, to obtain the number that you are factoring, you just need to multiply whatever number in the set of whole numbers with another within the same set. As an example:

For example, the number 3 has two factors 1 and 3. Number 8 has four factors 1, 2, 4, and 8 itself. It is very easy to factor numbers in a set of natural numbers since all numbers carry a minimum of two factors (one and itself). However, to find other factors, you will have to begin dividing the number starting from 2 and keep continuing with dividers increasing until reaching the number that was divided by 2 in the starting.

Remember that all numbers not having the remainders are factors, including the divider itself. Take an example of factorization with the number 9. Number 9 is not divisible by 2 evenly, so we will skip it. Note the numbers 4 and 5 so you know where to stop later. Number 9 is divisible by 3, so add 3 to your factors. Keep going on until you arrive at 5 (9 / 2, rounded up). Finally, you have 1, 3 and 9 as a list of factors.

Solved Examples of Cube Root

Let’s Quickly Find the Cube Root Using Step by Step Procedure With an Explanation

Example1: Find Out the Cube Root of 216.

Solution1:

Step I. Express the given number as ∛216

Step II. Using the method of prime factorization, we have

 

This means: 2 × 2 × 2 × 3 × 3 × 3 = 216

Step III. Creating a group in triplets of the same prime, we have

= [2 × 2 × 2] × [3 × 3 × 3]

Thus, ∛216 = [2 × 3] = 6 

Example 2: Evaluate the Cube Root of 9261.

Solution 2:

Following the same procedure of factorization, we get

This means: 3 × 3 × 3 × 7 × 7 × 7 = 9261

(3 × 3 × 3) × (7 × 7 × 7)

3 × 7 = 21

Thus, ∛9261 = [3 × 7] = 21

In Mathematics, Statistics and Probability play a very important role in helping to calculate data sufficiency. The Cumulative Distribution Function is a major part of both these sub-disciplines and it is used in a number of applications. This function, also abbreviated as CDF, takes into account that a random variable valued at a real point, like X, is evaluated at x. In this case, the function holds that X will be of a lower value than x or will be valued the same as x. 

We mentioned that X is a random variable. What this means is that this variable explains the probable resulting values on an unexpected phenomenon. Understanding this is fundamental to understanding the Cumulative Distribution Function. 

What is a Cumulative Distribution Function?

CDF of a random variable ‘X’ is a function which can be defined as,

FX(x) = P(X ≤ x)

The right-hand side of the cumulative distribution function formula represents the probability of a random variable ‘X’ which takes the value that is less than or equal to that of the x. The semi-closed interval in which the probability of ‘X’ lies is (a.b], where a < b, 

P(a < X ≤ b) = F_x(b) – F_x(a)

Note that the ≤ sign which is used here is not conventionally used at all times, but it can be useful for discrete distributions. Depending on this, the right use of binomial and Poisson’s Distribution tables are employed. Many important formulas in Mathematics are totally dependent on the equal to or the lesser than sign, such as Paul Levy’s inversion formula. 

Understanding Cumulative Distributions

When random variables such as X, Y, and so on are solved, the letter that is used to subscript is the lower case of the same letter. This is done to avoid unnecessary confusion and mixups. However, the use of a subscript may not be necessary when a single random variable is being used. If the capital letter F is used for the cumulative distribution function then the lowercase letter f is used in the probability density and the probability mass functions. 

The continuous random variable probability density function can be derived by differentiating the cumulative distribution function. This is shown by the Fundamental Theorem of Calculus. 

[f(x) = frac{d}{dx} f(x) ]

The CDF of a continuous random variable ‘X’ can be written as integral of a probability density function. The ‘r’ cumulative distribution function represents the random variable that contains specified distribution.

[F_x(x) = int_{-infty}^{x} f_x(t)dt ] 

Understanding the Properties of CDF

In case any of the below-mentioned conditions are fulfilled, the given function can be qualified as a cumulative distribution function of the random variable: 

• Every CDF function is right continuous and it is non increasing. Where [limlimits_{x rightarrow -infty } F_x(x) = 0,  limlimits_{x rightarrow +infty } F_x(x) = 1 ] 

• If ‘X’ is a discrete random variable then its values will be x1, x2, …..etc and the probability Pi = p(xi) thus the CDF of the random variable ‘X’ is discontinuous at the points of xi. FX(x) = P(X ≤ x) = Σxi ≤ x P(X = xi) = Σxi ≤ x p(xi). 

• If the CDF of a real-valued function is said to be continuous, then ‘X’ is called a continuous random variable F_x(b) – F_x(a) = P(a < X ≤ b) = ∫ab fX(x) dx.

The function fx = derivative of Fx is the probability density function of X.

Derived Functions

• Complementary Cumulative Distribution Function: It is also known as tail distribution or exceedance, it is defined as, F_x(x)=P(X>x)=1−FX(x)

• Folded Cumulative Distribution: When the cumulative distributive function is plotted, and the plot resembles an ‘S’ shape it is known as FCD or mountain plot.

• Inverse Distribution Function: The inverse distribution function or the quantile function can be defined when the CDF is increasing and continuous. F−1(p),pϵ[0,1]F−1(p),pϵ[0,1]F^{-1} (p), p epsilon [0,1] such that F(x) = p.

• Empirical Distribution Function: The estimation of cumulative distributive function that has points generated on a sample is called empirical distribution function.

Solved Example 1. 

1. What is the cumulative distribution function formula? 

Given the CDF F(x) for the discrete random variable X, 

Find: (a) P(X = 3) (b) P(X > 2)

Solution: 

CDF of a random variable ‘X’ is a function which can be defined as,

FX(x) = P(X ≤ x)(a) P(X = 3)

To obtain the CDF of the given distribution, here we have to solve till the value is less than or equal to three. From the table, we can obtain the value 

F(3) = P(X  3) = P(X = 1) + P(X = 2) + P(X = 3)

From the table, we can get the value of F(3) directly, which is equal to  0.67.

(b) P(X > 2)

P(X > 2) = 1 – P(X ≤ 2)

P(X > 2) = 1 – F(2)

P(X > 2) = 1 – 0.32P

(X > 2) = 0.87

2. What is the CDF of normal distribu
tion in r?

Given the probability distribution for a random variable x, 

find (a) P(x ≤ 4.5) (b) P(x > 4.5)

Solution: 

The CDF of the normal distribution can be denoted by ” φ ” the probability of a random variable that has a related error function.

(a) P(x ≤ 4.5) = F(4.5) = 0.8

(b) P(x > 4.5) = 1 – P(x ≤ 4.5)

(c) P(x > 4.5) = 1 – 0.8

(d) P(x > 4.5) = 0.2

To recall we know that a Fraction is formed up of two parts – Numerator and Denominator.

And expressed as –  Numerator/Denominator.

A Fraction or a Mixed Number in which the Denominator is a power of 10 such as 10, 100, 1000. etc. usually expressed by use of the Decimal point is termed as Decimal Fraction Math. Writing the Fraction in terms of Decimal makes it easier to carry on Mathematical Operations on them. For example any Fraction which has a Denominator as power of 10 like 53/100 can be written in Decimal as 0.53.

Examples of Decimal Fractions

4/100 = 0.04

57/10 = 5.7

53/100 = 0.53

The Right Way to Study about Decimals

Read the entire Number part first, followed by “and,” and then read the Fractional component in the same way as Whole Numbers, but with the last digit’s place value. Individual digits are always read as a Decimal Number. A Decimal Number of 145.367, for example, might be interpreted as one hundred forty-five point three six seven.

What is Decimal Fraction?

A Fraction where the Denominator i.e the bottom Number is a power of 10 such as 10, 100, 1000, etc is called a Decimal Fraction. You can write Decimal Fractions with a Decimal point and no Denominator, which make it easier to do calculations like addition, subtraction, division, and multiplication on Fractions.

Some of the Decimal Fractions examples are

1/10 th = read as one-tenth = written as 0.1 in Decimals.

6/1000 th = read as six-thousandths = written as 0.006 in Decimals

Operations on Decimal Fractions:

The given Numbers are so placed under each other that the Decimal points lie in one column below one another. The Numbers are now added or subtracted in the regular way.

For example: add 0.007 and 3.002

0. 0 0 0 7

                     + 3. 0 0 0 2

          ____________________

          3. 0 0 0 9

When a Decimal Fraction is multiplied by the powers of 10, shift the Decimal point to the right by as many places as is the power of 10.

For example, 5.9632 x 1000 = 5963.2;  

0.073 x 1000 = 730.

Multiply the given Numbers without a Decimal point. Now, in the product, the Decimal point is marked as many places of Decimal as is the sum of the Number of Decimal places in the given Numbers.

For example:  we have to find the product 0.2 x 0.02 x 0.0002

Consider the Number without Decimal points

Now, 2 x 2 x 2 = 8.

 Sum of Decimal places = (1 + 2 + 4) = 7.

Thus mark the Decimal point 7 places to the left that will be 0.0000008

 .2 x .02 x .002 = .0000008

Divide the given Number without the Decimal point, by the given Number. Now, in the quotient, mark the Decimal point as many places of Decimal as there are in the dividend.

For Example we have to find the quotient  for 0.0204 ÷ 17 

Now, 204 ÷ 17 = 12.

Dividend contains 4 places of Decimal. 

So, 0.0204 ÷ 17 = 0.0012

Multiply both the dividend and the divisor by a suitable power of 10 to make the divisor a whole Number.

Now, proceed as above.

How to convert Decimal to Fraction

You can convert a Decimal to a Fraction by following these three steps.

Let us convert 0.25 in Fraction

Step 1: Rewrite the Decimal Number over one as a Fraction where the Decimal Number is the Numerator and the Denominator is one.

0.25/1

Step 2: Multiply both the Numerator and the Denominator by 10 to the power of the Number of digits after the Decimal point. If there is one value after the Decimal point, multiply by 10, if there are two values after the Decimal point then multiply by 100, if there are three values after the Decimal point then multiply by 1,000, and so on.

For converting 0.25 to a Fraction, there are two digits after the Decimal point. Since 10 to the 2nd power is 100, we have to multiply both the Numerator and Denominator by 100 in step two.

0.25/1  x 100/100  = 25/100

Step 3: Express the Fraction in Decimal Fraction form and simplest form.

25/100  = ¼

By following these steps in the above Decimal Fraction questions, you can conclude that the Decimal 0.25, when converted to a Fraction, is equal to 1/4.

Let us solve Decimal questions.

Solved Examples

Decimal Fractions questions

Convert the given fractions into decimal fractions:

1. ½

Solution: ½ x 5/5  

= 5/10 

= 0.5

2. 10 ¼

Solution: 10 ¼

= 10 ¼ x 25/25

= 10 (25/100)

= 10.2

Quiz Time

1. Jim purchased 100 apples from a local fruit dealer, only to discover that five of them were rotting. Can you calculate the Fraction and Decimals of the rotten apples in relation to the total apples purchased by Jim?

Ans: Out of 100 apples, we have 5 rotten ones. As a result, the Percentage of rotten oranges is 5/100. Now we must convert this Fraction to a Decimal. We must divide the Numerator 5 by the Denominator 100 to achieve this. As a result, by addin
g two Decimal places to the Fraction 5/100, it can be converted to a Decimal. 0.05 is the Decimal answer. As a result, the rotten apples are 0.05 in Decimals.

2. In an 80-student class, 48 pupils chose ice cream as a snack, while the other students preferred soft drinks. Calculate the Percentage of students that choose a soft drink and give the result in Decimals.

Ans: There are 80 pupils in a class, 48 students who enjoy ice cream, and 80 – 48 = 32 students who enjoy soft drinks. Soft drinks are enjoyed by 32 percent of students out of 80. This Fraction is equivalent to 2/5 on simplification. Let’s convert this Fraction to a Decimal and then to a Percentage. To convert the Fraction to a Decimal, divide 2 by 5, and the result is 0.4. In order to convert 0.4 to a Percentage, we must multiply it by 100, which is 0.4 x 100 percent = 40%. As a result, the Percentage of students who enjoy soft drinks is 40%, and the Decimal equivalent is 0.4.

3. Write 1/4th in Decimals.

Ans: Let’s look at how to express 1/4 in Decimals. To get a 100 in the Denominator, multiply the Numerator and Denominator with a 25. We also need to convert this Fraction to a Decimal with a Denominator of 100.

0.25 = 1/4 x 25/25 = 25/100

Descending Order Definition

While arranging numbers from highest to lowest, it forms descending order. In case, we have been given 25, 34, 63, 38, 10, 5, 97. If we arrange them from highest to lowest in the form given by 97, 63, 38, 34, 25, 10, 5, then these are arranged in descending form. The arrangement is also called from largest to lowest. In simple words, descending order means to arrange the numbers from largest to lowest. Usually, we use this concept in the fractions, the amount of money, decimals and numbers.

Descending Order Meaning

Arrangement of numbers, quantities or lengths from highest to lowest is known as descending order. Also, you can name it in decreasing order.

Look at the example given below

Arrange the following numbers in decreasing sequence:

192, 168, 20, 10, 63, 25

We will get the answer for descending numbers as: 

192→168→63→25→20→10

Descending Order Symbol

For representing descending order, we use bigger to smaller signs, or we can say that greater than sign, i.e. ‘>.’

In the sequence for arranging descending order in maths for the first ten numbers, we will present it as

10 > 9 > 8 > 7 > 6 > 5 > 4 > 3 > 2 > 1 

Descending Order for Alphabets

We are already aware of the arrangement of numbers in descending sequence. However, it is not only limited to the number but also English alphabets. From alphabets A to Z, we always consider A as the smallest and Z as the largest. Hence the sequence for decreasing order will start from Z and last till A.

The descending order for English Alphabets will be represented as below:

Z> Y> X> W> V> U> T> S> R> Q> P> O> N> M> L> K> J> I> H> G> F> E> D> C> B> A

Similarly, we can arrange any quantity, decimal, real number, fractions, or whole numbers in the ascending or descending sequence. The below diagram can explain this concept of ordering in a better way.

IIn the above diagram, the descending order is explained with the help of a staircase example. There are a total of six steps numbered as sixth being the highest one and first being the lowest one—a man starting his movement from the highest step and moving towards the lowest step. 

How Can You Find a Descending Order in Maths?

Taking the example of the first ten numbers in a sequence from 1 to 10. For finding descending sequence we will keep subtracting one from each as below:

10-1 = 9, 9-1 = 8, 8-1 = 7, 7-1 = 6, 6-1 = 5, 5-1 = 4, 4-1 = 3, 3-1 = 2, 2-1 = 1

Thus the order formed is 

10→ 9→ 8→ 7→ 6→ 5→ 4→ 3→ 2→ 1

Rules to Form Descending Order

There are some basic concepts to understand and to form decreasing order:

• In the above example, we kept subtracting 1 from each number and formed the decreasing sequence. The case is not the same for other examples. The order is not followed in the same manner.

• Always pick the largest number first.

• Now ensure you are placing them all from largest to smallest.

• Your sequence must end with the smallest number from the given set of numbers.

Descending Order in Measurement of Length

Suppose we have to arrange people according to decreasing height. Simply, we can say that arranging them in descending sequence or the heights. Thus we arrange the person with a bigger height on the first number and keep arranging the next highest and so on. Taking the example of animals as below:

In the above example, Giraffe has the maximum height and hence placed on the first number. Cat having minimum height placed on the last position. Thus arranging height-wise from highest to lowest also forms the decreasing sequence.

Decreasing Order Dates

Now let us take an example of calendar dates of decreasing sequence. 

Suppose we have a collection of dates in the form of a table as given below. 

 

We need to form a descending order in the SQL table for the same. The arrangement will be as follows:

Fun Facts on Descending Numbers

Consider the case of a number line as given below:

In this case, starting from -10 and ending to 10, it forms ascending order. In the case of a number line, numbers are always arranged in ascending sequence. 

If we reverse the ascending order direction and place numbers from 10 to -10, it forms a descending sequence.