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Is it Possible to Represent the Irrational Numbers on a Number Line?

Steffy Alen

All the numbers defined are broadly classified into Real Numbers and Complex Numbers. A complex number consists of two parts in the form a+ib in which i is the imaginary part, a and b are real numbers. A real number is either a rational or an irrational number.

An integer is a number with no decimal or fractional part, including sets of negative to positive numbers including zero. Natural numbers are all integers from one to positive infinity, that is, 1, 2, 3, 4, 5…, ∞. They do not include zero, fractions, decimals and negative numbers, hence, natural numbers are also called counting numbers. A rational number is a real number that can be expressed in the form p/q where p and q are integers and q is not zero. A rational number differs from a fraction when the numerator and denominator of a fraction are whole numbers. Whole numbers are all natural numbers including zero. The decimal notation of a rational number has the decimal expansion terminating or repeating numbers.

Irrational Numbers

Rational Numbers

Real Numbers

Integers

Transcendental Numbers

Natural Numbers

Whole Numbers

Number types of:

Integers are 21, -21, 0, 1, 2, …

Whole Numbers are 0, 21, 2, 73 and 46.

Natural numbers are 1, 2, 3, 4, …, ∞

Rational numbers are 0.5 or  , –  , 0.3 or  , 0.141414… or  , etc.

Fractions are 

All the other real numbers that do not classify as rational numbers are called irrational numbers. Irrational numbers cannot be expressed as a ratio of two integers because their real value is incommensurable. A π is an irrational number which expressed as the ratio of a circle’s circumference to its diameter. Euler’s number e, golden ratio φ and square roots of natural numbers are all irrational numbers. It is not inhibited to express irrational numbers in positional notation, that is a decimal number. But the only difference of a decimal number notation of the irrational number is the decimal expand to infinity and not ending with a repeating sequence. The decimal value of π is 3.1415926…, does not possess a finite number nor does it repeat numbers. Other than expressing π as a non-terminating continued fraction, it can also be expressed as a fraction value  .

Representation of Irrational numbers on the number line

It is quite simple to place all rational numbers on a one dimensional number line. Placing an irrational number on the number line is tricky and cannot be placed exactly. However, representing simple irrational number value on the number line requires to include the simple concept of Pythagoras Theorem. Pythagoras theorem states that the square of the hypotenuse of a right-angled triangle is equal to the sum of the square of the other two sides, that is the perpendicular and base. Using this theorem, a simple irrational number like  can be represented on the number line, and following steps for marking  must be followed:

  1. Draw a number line and mark -1, 0, 1 and 2 starting left side of the line and ending with 2 on the right end. The values less than 0 on the number line will be ignored here.
  2. The numbers are marked equidistant from one another.
  3. Using a divider take the distance 0 to 2 marking.
  4. From 1 marking, mark a point located perpendicular to the number and at a distance of length as measured in Step (3).
  5. Now join the marking at 0 and 1 on the number line which will be the base and equals to 1 unit.
  6. Join marking 1 on number line to the point as marked in Step (4) and this will be the height/ perpendicular of a right-angled triangle and equals to 2 units.
  7. Now joining point 0 on number line and top end of the perpendicular which will be the hypotenuse and its value is  .
  8. Now using the hypotenuse as a radius and point 0 as the centre, cut an arc on the number line to represent the length  on the number line.

Any other natural number can be represented on the number line by following the given steps:

Figure 1: Graphical representation of 

Using Figure 1.

  1. For  , AO=x and OB=1,

AB= x+1

AM=MB=

  1. Join CM and
  2. MO=MB-OB= =
  3. MC=MA=MB=
  4. CO2 = CM2– OM2= x  CO=
  5. Now CO as radius and point 0 as the centre, cut an arc on the number line AB to represent the length on the number line.

The answer to this question is sure one can represent the irrational number on a number line although not all of them with the help of a compass and a ruler, square root of 2 can be represented in a number line.

For more enquiries regarding this topic and others: Number System from Class 9 Maths

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