Multiplication tables

In this article we are going to see about multiplication tables.  Multiplication facts table lets you learn and practice the basic time tables using the numbers zero through twelve. Children are no longer memorizing their multiplication facts table. There are many tricks to teach children multiplication facts table in mathematics.

Simplest Method for Multiplication Facts Table

Remember, work on the 2's, 5's and 10's  first; then the doubles (6x6, 7 x7, 8x8); then move to each of the fact families: 3's, 4,s, 6's, 7's, 8's and 9's, 11's and 12's. Don’t move to a different fact family without mastering the previous one.

Simplest method for multiplication facts table:

x	0	1	2	3	4	5	6	7	8	9	10	11	12
0	0	0	0	0	0	0	0	0	0	0	0	0	0
1	0	1	2	3	4	5	6	7	8	9	10	11	12
2	0	2	4	6	8	10	12	14	16	18	20	22	24
3	0	3	6	9	12	15	18	21	24	27	30	33	36
4	0	4	8	12	16	20	24	28	32	40	40	44	48
5	0	5	10	15	20	25	30	35	40	45	50	55	60
6	0	6	12	18	24	30	36	42	48	54	60	66	72
7	0	7	14	21	28	35	42	49	56	63	70	77	84
9	0	8	16	24	32	40	48	56	64	72	80	88	96
9	0	9	18	27	36	45	54	63	72	81	90	99	108
10	0	10	20	30	40	50	60	70	80	90	100	110	120
11	0	11	22	33	44	55	66	77	88	99	110	121	132
12	0	12	24	36	48	60	72	84	96	108	120	132	144

Your life will be lot easier when you can simply remember the multiplication facts table. Please train your memory.  First, you can use the multiplication facts table above to start putting the answers into your memory. Use it a few times per day for about 10 minutes each, and you will learn your multiplication tables.

Graph of Y Square Root of X

Generally a function is described in the form y = f(x), y being the value the function at any general point x in the domain of the function. The letter y is chosen because the values of the function is represented by the y-axis in a graph.

The graph of the function gives a visual presentation of the function which helps us to study the function at a glance.

The values of square root of positive real numbers is expressed as function f(x) = square root of x or simply y = square root of x. The graph of y = square root of x is an interesting study and let us take a closer look.


Concept of a Square Root Function:

Consider a simple square function y = x2. It is a special  form of a quadratic function. We know that the graph of a quadratic function is a parabola. In this case it is a vertical, concave up parabola with origin as vertex. The graph is shown below.



Let see what is the inverse of the function y = x2

y = x2, or, √y = x and by interchanging the variable we get the inverse function as,

y = √x

Thus the square root function is nothing but the inverse of the square function.

We had seen that the graph of y = = x2 is symmetrical over y –axis. Therefore, the graph of y = square root of x has to be symmetrical over x-axis. However, since negative values of root of x have no meaning, the graph exists only in the first quadrant of the Cartesian system. It is shown below.




Example Problems on Graph of Y Square Root of X

From the graph of y square root of x, find the value of square root 3 to the nearest tenths. (Assume the sale to be 1unit on both the axes)



Since the scale of the graph is 1 unit each small division represents 0.2 unit. It is seen from the graph, when x = 5, the value of y is 1.7. Hence, square root of 3 is 1.7, nearest to tenths.

Use the graph of y square root of x, to evaluate 2( square root 6) to the nearest tenths. (Assume the sale to be 1unit on both the axes)



Since the scale of the graph is 1 unit each small division represents 0.2 unit. It is seen from the graph, when x = 6, the value of y is 2.5. Hence, 2(square root of 6) = 2(2.5) = 5

Solve Cumulative Probability Distribution Function

Computation of probability of events is called as probability distribution.  It can give all possible outputs in a single process.  Cumulative probability distribution function of a random variable X is defined as F(x) = P(X <= x) = sum_(x)I <= x)p(x_i): (-oo < x oo).  Let us discuss about the topic of “solve cumulative probability distribution function” in the following below with some related problems.

Example Problems – Solve Cumulative Probability Distribution Function

Example problem 1

A random variable Y has the following probability mass function.  Find the value of b and evaluate `P(Y lt= 2)` .

Y	0	1	2	3	4
P(Y = y)	10b	20b	30b	40b	50b

Solution:

P(Y = y) is the probability mass function `sum_(y = 0)^4 (Y = y) = 1`

P(Y=0) + P(Y = 1) + P(Y = 2) + P(Y = 3) + P(Y = 4) = 1

10b + 20b + 30b + 40b + 50b = 1

150b = 1

b = `1/150`

`P(Y lt= 2) = P(Y = 0) + P(Y = 1)`

                = `10/150 + 20/150 + 30/150`

                = `(10 + 20 + 30)/150`

                = `60/150`

                = `2/5`

Answer:

The value of b is `1/150`

`P(Y lt= 2) = 2/5`

Example problem 2

Find out probability mass function and cumulative distribution function for getting ‘4’s when two dice are thrown.

Solution:

Let us consider Y be the random variable of getting number of ‘4’s.

First, we are going to find the probability mass function

Therefore, Y can take the values 0, 1, 2.

P(no ‘4’) = P(Y = 0) = `25/36`

P(one ‘4’) = P(Y = 1) = `10/36`

P(two ‘4’) = P(Y = 2) = `1/36`

Probability mass function

y	0	1	2
P(Y = y)	25/36	10/36	1/36

Now, we are going to find the cumulative probability distribution function

Formula:  `F(x) = sum_(x_i = -oo)^x P(X = x_i)`

Y = 0, F(0) = P(Y = 0) = 25/36

Y = 1, `F(1) = sum_(i = -oo)^1 P(Y = y_i)`

                = P(Y = 0) + P(Y = 1)

                = `25/36 + 10/36`

                = `35/36`

Y = 2, `F(2) = sum_(i = -oo)^2 P(Y = y_i)`

                = P(Y = 0) + P(Y = 1) + P(Y = 2)

                = `25/36 + 10/36 + 1/36`

                = `36/36`

                = 1

Cumulative distribution function

y	0	1	2
F(Y)	25/36	35/36	1

Answer:

Probability mass function

y	0	1	2
P(Y = y)	25/36	10/36	1/36

Cumulative distribution function

y	0	1	2
F(Y)	25/36	35/36	1

Practicing Problems – Solve Cumulative Probability Distribution Function

Practicing problem 1

Find the cumulative distribution function for the given probability mass function

y	0	1	2	3
P(Y = y)	2a	4a	6a	8a

Find the value of a and `P(Y lt= 3)`

Answer:

The value of a is `1/20`

`P(Y lt= 3) = 1`

Practicing problem 2

Find the cumulative distribution function for getting number of heads when three coins are tossed once.

Answer:

Cumulative distribution function

y	0	1	2	3
F(y)	1/8	1/2	7/8	1

Triangle Geometry Cosine

Look at this geometry figure:



In all these right angled triangles of different sizes, <A has the same measure. Since <B = 90o, we have <C = 90o - <A, in each of these triangles. So, all these triangles have equal angles and hence are similar. Thus even though the lengths of the sides BC, CA, AB are different in these, the numbers `(BC)/(AC)` and `(AB)/(AC)` are the same for all these triangles.

In all these triangles, AC is the hypotenuse. The side BC is the opposite side of <A. The third side AB is called the adjacent side of<A.    In a right angled triangle with <A as one of its angles, the number obtained by dividing the adjacent side of  <A by the hypotenuse is called the cosine of <A and its written cos A. Thus

cos A = `(adjacent- side- of- angleA) / ( hypotenuse )`

Note that cos A is a number indicating the size of <A alone and not depend on the size of the right angled triangle containing <A.

Examples for Triangle Geometry Cosine:

Example 1:

Determine the cosine of angle A or cos A; If <A =45o in the right angled triangle ABC, as in the figure below



then, as seen earlier, we have

`(BC)/(AC)` = `(AB )/(AC)` =1/`sqrt(2)`

which in our new terminology can be written

cos 45o = 1/`sqrt(2)`

Example 2:

Determine the cos30o, if <A =30o in the right angled triangle ABC, as in the figure below



Solution:

By definition, we know that   cos A = `(AB)/(AC)`

Therefore,   cos 30 = `sqrt(3)` / 2


Example 3:

Measure the cos C, if <C =60o in the right angled triangle ABC, as in the figure below





Solution:

Given:

`Delta` ABC , AB = `sqrt(3)` , BC = 1 and AC = 2.

To find the triangle geometry cosine C:

By definition, we know that,  cos C = `(BC)/(AC)`

That is                                     cos 60o = `1/2`

This is the required triangle geometry cosine values.

Example 4:

Measure the cosine of angle A or cos A; If <A =20o, AB = 4.7 and AC =5 in the right angled triangle ABC, as in the figure below



Solution:

Given:

In `Delta` ABC, <A = 20o , AB = 4.7 and AC =5.

To find the triangle geometry cosine of angle A:

We have found by actual measurement that in such a triangle as `Delta` ABC

`(AB)/(AC)` = `(4.7) /(5)`  = 0.94

This means in other words

cos 20o = 0.94

Quadratic Functions in the Real World

Quadratic functions have their general form as the one shown below:

f(x) = a.x2 + b.x + c


Here we can see how quadratic functions are used in in the real world.
In the coordinate Geometry when this equation is plotted we get a parabola. And the shape, size and other dimensions are decided by the coefficients of the variables in the quadratic function i.e on a, b and c. And the solution of the equation in the general form is given by:

x =  ( -b ± √(b2 - 4ac)) / (2a)

where discriminant  d = (b2 - 4ac)

A quadratic equation with real coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots
Quadratic Functions in the Real World- Physics and Maths

Quadratic Functions in Physics:

Projectile: A projectile is an object upon which the only force acting is gravity. A stone when thrown upwards with a velocity returns to ground after sometime. When the path of the projectile is traced it's a parabola. So when something is thrown up and allowed to fall down freely later we get the projective path as a parabola.
Basing on the properties of the parabola and the path traversed further observations are done for  studying the body motion in physics.

The volcanic eruptions also generally follow the projectile motion since the hot fluid masses or the lava is pumped out with great velocities. So Quadratic equations also have a dominant role in estimating the safe zones in case of an eruptions and for further assistance and analysis.

Quadratic Functions in Maths:

Non-normal distributions in probability cover good applications of parabolas when plotted. The distribution plots mostly have their distribution curves as parabolas.
Other real life applications would be parabolic mirror, shape of a spillway for a dam and sound reflector.

Another application of a quadratic equation used in higher math and engineering, where a second-order differential equation is solved for a spring-damper system. This sounds scary but actually has real-world application. This example also shows how the "imaginary" number "i" is used in a real-world application. The fact that the exponent is ^2 while the inputs are ^1 indicates that it would solve perimeter vs area (fencing) problems, or bill of materials issues for tanks and containment etc.

Quadratic Functions in the Real World -networking

Quadratic Functions in Computers Networking

Quadratic equations are used considerably in computer networking. They're used for all sorts of error checking, as well as encryption. They're also used in the mathematics of Quality of Service (Qos), the concepts of understanding how to dynamically allocate and share bandwidth.

In this way we can have lot many Quadratic Functions in the real World.

Siding Square Calculator

Let us see about siding square calculator.  The square feet are one of the measurements of siding the square. The measurement tool of square feet is denoted as sq ft. The square foot is the plural term of the square feet. The length of the siding square is calculated for the 2 dimensional objects.

Formula for the Siding Square:

Let us see the siding square formula’s which is used by the calculator. The formulas are described as below.

1. Siding square of the rectangle     = b * h.
2. Siding square of the square         = a * a.

These are all the formulas which are helped to find the siding square.

Examples of Siding Square:

Let us see some examples of siding square which are used by the calculator.

Example 1:

Find the siding square for the rectangle with the help of calculator, where the base has 8ft and the height is 7ft.

Solution:

Step 1: First, choose the siding square object.



Step 2: Enter the base length in the calculator.



Step 3: Enter the height of the rectangle.



Step 4: Choose the calculate option in the calculator. The calculation is done by the formula of b * h.



Step 5: Choose the exit button after getting the siding square.

These are procedures used to calculate the siding square of the rectangle.


Example 2:

Find the siding square for the square with the help of calculator, where the side’s lengths are 12ft.

Solution:

Step 1: First, choose the siding square object.



Step 2: Enter the side length of square in the calculator.





Step 3: Choose the calculate button in the calculator. The calculation is done by the formula of a * a.



Step 4: Choose the exit button after getting the siding square.

These are procedures used to calculate the siding square of the square.