Root 2 is an Irrational Number: Proof, Explanations & Examples

Root 2 is the first irrational number whose value is equivalent to 1.414. When root 2 is multiplied by itself, it will give a value equivalent to 2. 

• The concept of root 2 was first introduced by the Greeks.
• According to the Greek philosopher, a square of side 1 unit cannot have a diagonal whose value is rational.
• By using Pythagoras theorem, the value of the diagonal is equivalent to the root of 2.
• The value of fraction 99/70 is also sometimes equivalent to the value of √2.
• It is a transcendental number that can be used as a rational approximation.
• The most common algorithm to determine the value of square root is the Babylon method.
• Its value can also be determined by the long division method.

Read More:- Complex Numbers and Quadratic Equations

Key Terms: Irrational Number, Perfect Square, Pythagoras Theorem, Rational Number, Long Divison Method, Euler Number, Real Numbers

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What is an Irrational Number?

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Irrational Numbers are the real numbers that cannot be expressed in a fractional form of (p/q); otherwise, the remaining are called rational numbers. It can also be expressed in the form of decimal numbers. 

• Irrational Numbers can also be written in the form of P – Q, which defines the relationship between the set of real numbers and the set of rational numbers.
• These numbers are neither recurring nor terminating.
• It will obey all rules and properties of a real number.
• When an irrational number and a rational number are added, the result is always irrational.
• Examples of Irrational Numbers: √2, √5, √7, pi are some examples of Irrational Numbers.
• Here, these root values cannot be expressed in fractional forms if expressed, so they give floating values.

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Prove that Root 2 Is an Irrational Number

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To prove that root 2 is an irrational number, follow a negation approach that first assumes a root is a rational number.

AIM: To prove that Root 2 is an Irrational Number.

PROOF: From the contradiction approach, also known as the Euclidian approach, let us assume that root 2 is a rational number.

• This means that √2 can be expressed in (p/q) form where p and q are not co-primes and q is not equal to 0.

√2 = p/q -------------------(equation1)

• Now, squaring both sides of equation 1,

(√2)² = (p/q)²

• The above equation can be written as,

2 = (p/q)²

• The equation can be rewritten as

2q² = p² or p²/2 = q²  -------------------(equation 2)

From the above equations, it is clear that 2 divides p and q in the multiple of 2, so let us assume that

P = 2n

p² = 4n² -------------------(equation 3)

• By comparing the equation 2 and 3, we get

2q² = 4n²

• q² = 2 n²
• q² is a multiple of 2.

Hence, q is a multiple of 2.

Then, our assumption fails from this one, and it is a clear contradiction.

Thus proving root 2 is an irrational number.

CONCLUSION: From the above set of assumptions, we prove that root 2 is an irrational number.

Read More: Multiplication Theorem on Probability

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Determining the value of Root 2 by long division method

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The first thing involved in calculating the value of root 2 is determining whether it is a perfect square or not. Since numbers like 2, 3, 5, and 7 are not perfect squares, their value is determined using the long division method.

The steps for determining the value of root 2 by the long division method are as follows:

• First, choose an estimated value by selecting a value equivalent to a perfect square.
• Divide the number by the value of the perfect square.
• Repeat the division process until a satisfactory value is obtained.

Read More: Difference Between Fraction and Rational Numbers

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Uses of Irrational Numbers

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Some of the uses of irrational numbers are as follows:

• The irrational number pi is used to calculate the area of different geometrical shapes in real life, and predicting the correct distances between two points.
• An Euler's number is used to derive many physics formulas and prove many theorems.
• Irrational numbers cannot be expressed in the form of hexadecimal, decimal, binary, or any other format.
• Most mathematical domains, logarithms, and algebraic equations can be solved using irrational numbers.
• Many Engineering innovations and civil constructions are achieved by using irrational numbers, as even to construct a swimming pool, you need to measure the radius of the site.
• Irrational Numbers are indirectly used in day-to-day life to solve several complex problems.
• Some common types of irrational numbers are Pi, Euler's number, Golden ratio, and many others.
• The lowest common multiple of two irrational numbers may or may not exist in all cases.
• Natural logarithms with base e are considered irrational numbers.

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Things to Remember

• Root 2 is proved to be an irrational number by using the contradiction method.
• The product of rational and irrational numbers is sometimes rational or irrational.
• The value of an irrational number can be calculated using the long division method.
• The addition of rational and irrational numbers is always an irrational number. Ex: 2 + √5
• Most common examples of root 2 include Euler's number, Golden ratio, etc.