Construction of Square root spiral.

Objective of the Spiral Root Activity:

The primary objective of the spiral root activity is to help students visualize the concept of square roots in a geometric manner. It serves as a bridge between the abstract world of numbers and the tangible reality of shapes and forms. By constructing a square root spiral, students can see the progression of square roots in a physical space, making the concept of irrational numbers more comprehensible.

here is the image of square root spiral and then the process is given :

 Aim: 

• To construct a square root spiral.

Materials Required

1. Adhesive
2. Geometry box
3. Marker
4. A piece of plywood

Prerequisite Knowledge

1. Concept of number line.
2. Concept of irrational numbers.
3. Pythagoras theorem.

Theory

1. A number line is a imaginary line whose each point represents a real number.
2. The numbers which cannot be expressed in the form p/q where q ≠ 0 and both p and q are integers, are called irrational numbers, e.g. √3, π, etc.
3. According to Pythagoras theorem, in a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of other two sides containing right angle. ΔABC is a right angled triangle having right angle at B. (see Fig. 1.1)
4. Therefore, AC² = AB² +BC²
   where, AC = hypotenuse, AB = perpendicular and BC = base

Procedure

1. Take a piece of plywood having the dimensions 30 cm x 30 cm.
2. Draw a line segment PQ of length 1 unit by taking 2 cm as 1 unit, (see Fig. 1.2)
   
3. Construct a line QX perpendicular to the line segment PQ, by using compasses or a set square, (see Fig. 1.3)
4. From Q, draw an arc of 1 unit, which cut QX at C(say). (see Fig. 1.4)
   
5. Join PC.
6. Taking PC as base, draw a perpendicular CY to PC, by using compasses or a set square.
7. From C, draw an arc of 1 unit, which cut CY at D (say).
8. Join PD. (see Fig. 1.5)
   
9. Taking PD as base, draw a perpendicular DZ to PD, by using compasses or a set square.
10. From D, draw an arc of 1 unit, which cut DZ at E (say).
11. Join PE. (see Fig. 1.5)

Keep repeating the above process for sufficient number of times. Then, the figure so obtained is called a ‘square root spiral’.

Demonstration

1. In the Fig. 1.5, ΔPQC is a right angled triangle.
   So, from Pythagoras theorem,
   we have PC² = PQ² + QC²
   [∴ (Hypotenuse)² = (Perpendicular)² + (Base)²]
   = 1² +1² =2
   => PC = √2
   Again, ΔPCD is also a right angled triangle.
   So, from Pythagoras theorem,
   PD² =PC² +CD²
   = (√2)² +(1)² =2+1 = 3
   => PD = √3
2. Similarly, we will have
   PE= √4
   => PF=√5
   => PG = √6 and so on.

Observations
On actual measurement, we get
PC = …….. ,
PD = …….. ,
PE = …….. ,
PF = …….. ,
PG = …….. ,
√2 = PC = …. (approx.)
√3 = PD = …. (approx.)
√4 = PE = …. (approx.)
√5 = PF = …. (approx.)

Result
A square root spiral has been constructed.

Application
With the help of explained activity, existence of irrational numbers can be illustrated.

Viva Voce

Question 1: Define a rational number.

Answer:
A number which can be expressed in the form of p/q, where q ≠ 0 and p, q are integers, is called a rational number.

Question 2: Define a irrational number.
Answer: A number which cannot be expressed in the form of p/q, where q ≠ 0 and p, q are integers, is called an irrational number.

Question 3: Define a real number.
Answer:
A number which may be either rational or irrational is called a real number.

Question 4: How many rational and irrational numbers lie between any two real numbers?
Answer:
There are infinite rational and irrational numbers lie between any two real numbers.

Question 5: Is it possible to represent irrational numbers on the number line?
Answer:
Yes, as we know that each point on the number line represent a real number (i.e. both rational and irrational), so irrational number can be represented on number line.

Question 6: In which triangle, Pythagoras theorem is applicable?
Answer:
Right angled triangle

Question 7: Give some examples of irrational numbers.
Answer:
Some examples of irrational numbers are √5, 3 – √7,2π, etc.

Question 8: Can we represent the reciprocal of zero on the number line.
Answer:
No, because reciprocal of zero is undefined term, so we cannot represent on number line.

Question 9: In a square root spiral, is it true that in each square root of natural number is equal to the square root of the sum of 1 and previous natural number (> 1)?
Answer:
Yes

Question 10: Is it possible that we make a square root spiral of negative numbers?
Answer:
No

Wish you a happy learning 👍