Square Numbers - III

Some interesting facts

• There are 2n non-perfect square numbers between the squares of two consecutive natural numbers n and (n+1) e.g. between 2² and 3² , there are  

         2 x 2 = 4 , non-perfect square numbers.

•  The square number of an odd natural number n , can be expressed as the sum of two consecutive natural numbers (n²-1)/2 and (n²+1)/2 e.g.  

          5² = (5²-1)/2 + (5²+1)/2.

• A triplet of three natural numbers a , b and c forms a Pythagorean Triplet , if a²+b²=c² e.g. (3,4,5) is a Pythagorean Triplet. For any natural number p greater than 1 , (2p , p²-1 , p²+1) is a Pythagorean triplet. 
• The squares of numbers which have all the digits as 1 , exhibit the following pattern

 

• Addition of two consecutive triangular numbers exhibits the following pattern :

             1 + 3 = 4 = 2²
             3 + 6 = 9 = 3²
             6 + 10 = 16 = 4²
This shows that if we add two consecutive triangular numbers we get a square number.  

Square Numbers - II

Properties of Square Numbers

• A number that ends in 2 , 3 , 7 or 8 is never a perfect square. 
• The ones digit in the square of a number can be determined if the ones digit of the number is known. Look at the following table

                                 

• The number of zeroes at the end of a perfect square is always even e.g.                40² = 1600 , 200² = 40000
• The square of an even number is always an even number and square of an odd number is always an odd number. e.g. 12² = 144 , 23² = 529
• If n is a perfect square then 2n can never be a perfect square. e.g. 100 is a perfect square , but 2 x 100 = 200 is not a perfect square number. 
• The difference between the squares of the two consecutive numbers is equal to their sum or twice the smaller number plus 1. e.g. 

                     (m+1) ² - m² = [(m+1) + m] [(m+1) – m] = (2m +1)

• If a number is a square number , it has to be the sum of successive odd numbers starting from 1. e.g. 

               1 + 3 = 4 = 2²                          [Sum of first two odd numbers]   
               1 + 3 + 5 + 7 + 9 = 25 = 5²       [ Sum of first five odd numbers]

• The square of a number , either negative or positive is always positive.          e.g. (-3) ² = (-3) x (-3) = 9 
• The square of a natural number other than 1 is either a multiple of 3 or exceed the multiple of 3 by 1. Thus we can express square of a number (other than 1) as 3m or 3m+1 for some natural number m. e.g.

                      5² = 25 = 3 x 8 + 1 
                    12² = 144 = 2 x 36 

• The square of a natural number other than 1 is either a multiple of 4 or exceed the multiple of 4 by 1. Thus we can express square of a number (other than 1) as 4m or 4m+1 for some natural number m. e.g.

                    7² = 49 = 4 x 12 + 1 

                 14² = 196 = 4 x 49

Square Numbers - I

We know that the area of a square = side x side. Let us study the following table :

What is special about the numbers 4 , 9 , 16 , 36 and other such number ?

Since 4 can be expressed as 2 x 2 = 2² , 16 can be expressed as 4 x 4 = 4² , all such numbers can be expressed as the product of the numbers with itself.
Such numbes are known as square numbers.

In general , if a natural number p can be expressed as q² , where q is also a natural number , then p is a square number. Square numbers are also called perfect squares.

How can we check whether a number is perfect square of not , let see the following square numbers and their prime factors :

 4 = 2 x 2       4² = 16 = 2 x 2 x 2 x 2 = 2² x 2²
 6 = 2 x 3      6² = 36 = 2 x 2 x 3 x 3 = 2²x 3² 

Thus we can see that in the prime factorization of a perfect square , every prime number occurs two time. We can define the following algorithm to check a number is perfect square or not. 

Step 1 – Find the prime factors of the given number. 
Step 2 – Group the factors into pairs of like factors. 
Step 3 – If all the factors can be paired , then the given number is a perfect square , otherwise it is not.

In my next post I will discuss some properties and interesting facts about the square number. Happy Reading.

Domain and Range - I

Conversion - Polar and Cartesian Coordinates

Refer figure above , from trigonometry note that cos θ = x / r so that x = r cos θ. Similarly sinθ = y / r , so that y = r sin θ. 

Hence , if we know the polar coordinates of a point (r , θ) , we can find its cartesian coordinates. 

Alternatively , using Pythagoras theorem , we note that r = √(x²+y²).Further tanθ = y/x , so that θ=tan^-1 (y/x) 

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Polar Coordinate System

The (x, y) co-ordinates of a point in the plane are called its Cartesian co-ordinates.But there is another way to specify the position of a point,and that is to use polar co-ordinates(r, θ). 

A polar coordinate system consists of a fixed point (called the pole or origin)and a ray from the origin (called the polar axis). The polar axis is usually horizontal and directed toward the right.

Every point in the polar coordinate system is described by an ordered pair of real numbers, (r, θ). The first coordinate describes the point's distance from the pole and the second coordinate describes the angle formed with the polar axis. 

Following conventions are used while using polar co-ordinates : 

• θ is measured in radians. 
• θ is measured from the polar axis ,and is measured as positive in an anticlockwise direction from the it, or negative in a clockwise direction from the polar axis.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Cartesian Coordinate System

A cartesian coordinate system usually consists of two perpendicular number lines, one horizontal say XX’ and one vertical say YY’. We combine both the lines in such a way that the two lines cross each other at their zeroes or origins. The horizontal line XX’ is called the x-axis and the vertical line is called the y-axis. The point of intersection of these two lines is called the origin usually denoted by letter O.

The points of the x-axis to the right (OX) of the origin represent positive numbers and the points of the x-axis to the left (OX’) of the origin represent negative numbers . Similarly, the points of the y-axis above (OY) the origin represent positive numbers and the points of the y-axis below(OY’) the origin represent negative numbers.

The x –axis and the y – axis divide the plane into four parts, these parts are called the quadrants. Quadrants are numbered I , II , III and IV anticlockwise from OX.

Every point in the rectangular plane is described by an ordered pair of real numbers, (x,y). The x and y are called the coordinates of the point. The x - coordinate of a point is its perpendicular distance from the y – axis measured along the x – axis. Similarly the y-coordinate of a point is its perpendicular distance from x–axis measured along y–axis.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Polynomial and its Zeroes

A polynomial in one variable is an expression of the form: p(x) = a_nxⁿ+ a_n-1x^n-1+……..+ a₂x² + a₁x + a_{0 ,} where a_n ≠ 0 and a_n …. a₀ are real numbers , and n is a non-negative number.
Following are the important points about this definition 

• The numbers a₀, a₁,……… a_n are called the coefficients of the polynomial.
• The degree of the polynomial p is n, the highest power of x 
• A polynomial is linear , quadratic or cubic if its degree is 1,2 or 3 respectively.
• a is a zero of the polynomial if and only if p(a) = 0. 
• Zeroes of a polynomial p(x) can be obtained by factorizing it. 
• Graphically zero of a polynomial is the x – coordinate of point(s) where the graph of a polynomial intersects with x – axis. 
• A polynomial of degree n has atmost n zeroes.

 Enter any polynomial of one variable in the input box below.
This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Straight Line - Distance Form

The equation of a straight line passing through (x₁,y₁) and making an angle θ with the positive direction of x – axis is 
                              (x-x₁)/cosθ = (y-y₁)/sinθ=r , 
where r is the distance of the point (x,y) on the line from the point (x₁,y₁).
  This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Straight Line - Normal Form

The equation of a straight line upon which the length of the perpendicular from the origin is p and this perpendicular makes an angle α with x- axis is x cosα + y sinα = p.


This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Equations of a Circle

Let C be the centre of a circle with coordinates (h,k). If radius of the circle is R , then equation of such a circle is given by (x-h)² + (y-k)² = R² . Here we will explore the changes in circle with the changes made in the values of h , k and R. 

• When the centre of the circle coincides with the origin i.e. h = k = 0 . Equation of the circle is x² + y² = R² 
• When the circle passes through the origin i.e. R² = h² + k² . Equation of the circle is (x-h)² + (y-k)² = h² + k² or x² + y² – 2hx – 2 ky = 0. 
• When the circle touches x – axis i.e. R = k. Equation of the circle is (x-h)² + (y-R)² = R2 or x² + y² – 2hx – 2Ry + h² = 0 
• When the circle touches y - axis i.e. R = h. Equation of the circle is (x-R)² + (y-k)² = R² or x² + y² – 2Rx – 2ky + k² = 0 
• When the circle touches both the axes i.e. R = h = k. Equation of the circle is (x-R)² + (y-R)² = R² or x² + y² – 2Rx – 2Ry + R² = 0 
• When the circle passes through the origin and centre lies on x-axis i.e. k = 0 and h = R. Equation of circle is (x-R)² + (y-0)² = R² or x² + y² – 2xR = 0 
• When the circle passes through the origin and centre lies on y-axis i.e. h = 0 and k = R. Equation of circle is (x-0)² + (y-R)² = R² or x² + y² – 2yR = 0 

You can enter the equations of above seven types in the input box at the near bottom of the following applet and see the changes in circle. This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Coordinates of a Point in Space

We have already plotted a point on the xy-plane by an ordered pair that consists of two real numbers, (x-coordinate , y-coordinate) , which denote signed distances along the x-axis and y-axis, respectively, from the origin (0,0). These axes, which are collectively referred to as the coordinate axes, divided the plane into four quadrants. 

Let us now generalize these concepts to three-dimensional space, or xyz-space. In this space, a point is represented by an ordered triple (x, y, z) that consists of three numbers, an x-coordiante, a y-coordinate, and a z-coordinate. These coordinates indicate the signed distance along the coordinate axes, the x-axis, y-axis and z-axis, respectively, from the origin, denoted by O, with coordinates (0, 0, 0). 

Three-dimensional space contains infinitely many planes, just as two-dimensional space consists of infinitely many lines. Three planes are of particular importance: the xy-plane, which contains the x-axis and y-axis; the yz-plane, which contains the y-axis and z-axis; and the xz-plane, which contains the x-axis and z-axis.

Alternatively, the xy-plane can be described as the set of all points (x, y, z)for which z=0. Similarly, the yz-plane is the set of all points of the form (0, y, z), while the xz-plane is the set of all points of the form (x, 0, z).

These three planes divide xyz-space into eight octants. Within each octant, all x-coordiantes have the same sign, as do all y-coordinates, and all z-coordinates. In particular, the first octant is the octant in which all three coordinates are positive.

The nth Roots of Unity

The equation xⁿ = 1 has n roots which are called the nth roots of unity. 
xⁿ = 1 = cos 0 + i sin 0      = cos 2kπ + i sin 2kπ    [ k is an integer]
x = (cos 2kπ + i sin 2kπ)^1/n 
  = (cos (2kpi/n) + i sin (2kpi/n)) where k = 0 , 1, 2 , 3 , 4 , ……… , (n-1) 

So each root of unity is cos[ (2kπ)/n] + i sin[(2kπ)/n] where 0 ≤ k ≤ n-1.

We know that complex numbers of the form x + iy can be plotted on the complex plane (Argand Diagram). If we compare each root of unity with x + iy ,  we get x = cos[(2kπ)/n] , y=sin[(2kπ)/n].
Now , we see that the values of x and y satisfy the equation of unit circle with centre (0,0) i.e. x² + y² = 1.

From above it can be concluded that the roots are located on the circumference of the unit circle with center at (0,0). If we join the points we always get a regular polygon of n sides.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Complex Numbers - Division

If z₁ = a+ib and z₂ = c+id are two complex numbers , then their division z₁ / z₂ can be expressed as [(ac + bd)+i(bc - ad)] / (c² + d²). This achieved by multiplying and dividing the numerator and denominator by conjugate of the denominator.

Let us see what happens geometrically when we divide two complex numbers? Let us express complex numbers in their polar form. With this approach we find the division of two complex numbers   w = r (cos α + i sin α) and z = s (cos β + i sin β) is given by w / z = r * [(cos (α- β) + i sin (α - β)] / s.

This means that, when dividing two complex numbers w and z, we divide their moduli and we subtract the angles which w and z make with the positive x-direction.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Complex Numbers - Multiplication

If z₁ = a+ib and z₂ = r+is are two complex numbers , then their multiplication           z₁ * z₂ can be expressed as [(ar - bs)+i(as+br)].

 Let us see what happens geometrically when we multiply two complex numbers? To answer this question, it is advantageous to express complex numbers in their polar form.

With this approach we find that the product of two complex numbers w = r (cos α + i sin α) and   z = s (cos β + i sin β) is given by w.z = r s [(cos (α+β) + i sin (α+β)].

This means that, when multiplying two complex numbers w and z, we multiply their moduli and we add the angles which w and z make with the positive x-direction.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Complex Numbers - Subtraction

When subtracting complex numbers, we subtract real parts to other real parts, and imaginary parts to other imaginary parts.

If z₁ = a+ib and z₂= r+is are two complex numbers , then their subtraction z₁ - z₂ can be expressed as  z₁ + (-z₂) i.e. (a + ib) - (r + is) = (a + ib) + (-r - is) where (-r - is) = -z₂.         Hence  z₁ – z₂ = (a – r) + i(b-s).

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Complex Numbers - Addition

Since complex numbers are legitimate mathematical entities, just like scalar numbers, they can be added, subtracted, multiplied, divided, squared, inverted,     and such, just like any other kind of number.

Addition of complex numbers in rectangular form (a + ib ) is easy. For addition, simply add up the real components of the complex numbers to determine the real component of the sum, and add up the imaginary components of the complex numbers to determine the imaginary component of the sum.

For example If z₁ = x+iy and z₂= r+is are two complex numbers, then their sum z₃ = z₁+z₂ can be expressed as (x+r)+i(y+s).

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Complex Number - Polar Form

The form a + i b is called the rectangular coordinate form of a complex number because to plot the number we imagine a rectangle of width a and height b.

Complex numbers can also be expressed in polar coordinate form, r ∠ θ. (This is spoken as “r at angle θ ”). The number r in front of the angle symbol is called the magnitude (modulus) of the complex number. The angle θ after the angle symbol is the direction of the complex number from the origin measured counterclockwise from the positive part of the real axis.

Polar form is sometimes called trigonometric form as well and written in the form               r(sin θ + icos θ) or r cis (θ)
This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com