What is a Mode in Math?

Math teacher, Ms Grace assigned the following number of problems for homework on 8 different days; 10, 12, 8, 10, 7, 10, 8, 10. Let us arrange the data in the increasing order, 7,8,8,10,10,10,10,12. What do you observe in the data? 10 is the number which is seen most number of times. This number 10, which occurs most often, is called the Mode in Math or Mode Math. So, Mode Definition Math would be, the Mode of a set of data is the value in the set that occurs most often (most number of times)

By definition, Statistics Mode or statistical mode is the value that occurs most frequently in the data set. For example, what would be the mode of the data set, 12, 11, 13, 9, 11, 8, 6, 11. Let us first arrange the data in the ascending order. That gives us, 6, 8, 9, 11, 11, 11, 12, 13. As you can see the value or the number 11, occurs most number of times than any other number in the data and hence 11 is the mode of the given data set.

To learn the steps involved in finding mode of a given set, let us consider an example. Let us assume that there is a basket ball match going on and the scores of the game are as listed below. Let us determine the mode of the scores.

Scores: 19, 6, 3, 22, 19, 9

Step1: list in the scores in the ascending order
3, 6, 9, 19, 19, 22
Step2: identify the number which is occurs most often, 19
  Hence,  19 is the mode of the score

So, the steps involved in finding mode are, first we need to order the list in ascending order and then identify the number or value which occurs most often that will give us mode.

Note: If there is no number which occurs most often, then we can say that the given data has no mode
On a cold winter day in December, the temperature of 7 cities in North America is recorded in Fahrenheit. How do we find the mode of these temperatures?  Temperatures recorded:  -9,-12, 0, -6,      - 10, -3, 5
To find the mean, the first step would be to order the given temperatures in ascending order and then identify the temperature which occurs most often
-12, -10, -9, -6,-3, 0, 5
From the above ordered list, we can see that there is no value or number which occurs most often. So, we can conclude that there is no mode for the given temperatures.

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Definition of Absolute Value

Definition of Absolute Value
The absolute value is defined as the distance of ‘a’ from zero and is denoted by the symbol |a|. The absolute value only tells how far from zero and not in which direction is the location of a.

From the above figure, the absolute value of |3| = 3 since it is 3 units from the right side of the zero, and |-3|= 3 since it is 3 unit from the left side of the Zero. So the absolute unit should not have the negative sign, and is always positive or zero.

Absolute value equation

To solve absolute value equations, first split the equation into two equations. Then solve the equation to get the solution of absolute value equation.
Example 1:
|x| = 7
X = 7 x = -7
The solutions are {7, -7}
Example2:
|3x-4| = 5
3x-4  = 5 3x-4 = -5
3x = 9 3x = -1
X = 3 x = -1/3
The solutions are {1/3, 3}


Absolute Value Inequality
To solve absolute value inequalities first isolate the one side on the inequality symbol. Then write the first equation with out absolute sign and solve the inequalities and write second equation without absolute sign, reverse the inequalities and then solve the problem. The absolute value should be greater than any negative number and it should not be less than a negative number, sincethe absolute value should be always positive.
Example1:  (greater than)
|x-20| > 5
x-20> 5 X-20 < -5
x> 25 x < 15
The solution is 15 > x > 25
Example2: (less then or equal to)
|X-3| = 4
X-3 = 4 X-3 = -4
X = 7 X = -1
The solution is -1 = x = 4

Absolute value of a complex number
The definition for absolute value ofa real number is not directly generalized for definition for complex numbers, since the complex numbers are not ordered. But the geometric interpretation of the absolute value of a real number is as its distance from ‘0’ to be generalized.

The definition for absolute value ofa complex number is, it is a distance in the complex plane from the origin ‘0’ by using of Pythagorean Theorem. The absolute value of the difference of the complex number is equal to the difference between them.
The definition for absolute value ofa complex number is,
Z = x + iy
Where,
Z - Absolute value or modulus
x, y -  Real numbers
|z| = v(x^2+y^2 )
In above equation, the absolute value of a real number is x, when the complex part becomes 0.
In polar form the complex number z is expressed like z = re i?
If the absolute value is (r = 0, ?-real)
|z| = r
So, the absolute value of a complex number in the complex analogue equation is,

The properties of absolute value of a complex number are as same as absolute values of a real numbers.

Give example how to solve the absolute value equations and inequalities
Example 1: absolute value equation
|x - 7| = |2x-2|
Write in to two equations with out absolute symbol.
X-7 = 2x-2 x-7 = -(2x-2)
X-2x = -2+7 x+2x = 7 + 2
-x = 5 3x = 9
X = -5 x = 9
The solutions are { 9, -5}
Example 2: Absolute value inequalities
|2x-3| > 5
Write in to two inequalities with out absolute symbol.

2x-3 > 5 2x-3 < - 5
2x > 8 2x< -2
X < 4 x < -1
The solution is -1 > x < 4

Equation of a line which is passing through two points

Question :-

Find the equation of the line which is passing through two points (-3,7)(5,-1)

Answer:-

We have to use the point formula to find the equation of the line,this is much similar to midpoint formula

y-y1     x-x1
------ = ------
y2-y1     x2-x1

We have 2 points

( -3 , 7 )  and ( 5 , -1 )
x1  y1          x2  y2

So the equation is

y-7      x-(-3)
------ = ------
-1-7      5-(-3)

y-7      x+3
------ = ------
-8        8

We can further simplify it by cross multiplication.which is a part of indices maths

similarly we can find all points having an x-coordinate of 2 whose distance from the point 2 1 is 5