Show that 0.3333… = 0.3, can be expressed in the form of p/q

Before proceeding with the given question one must know what are rational numbers. Hence a brief definition of rational number is added below.

Rational Numbers Definition

A number is said to be rational if it can be represented in the form of p/q, where p and q are integers and q ≠ 0. All rational numbers are real numbers. All positive integers, positive fractions, zero, negative integers, and negative fractions are part of the set of rational numbers. 

Proof of this Concept

To prove the decimal property of rational numbers, we need to show that every rational number can be expressed as a decimal expansion that either terminates or repeats indefinitely in a periodic pattern.

Given: A rational number is a number that can be expressed as a fraction of two integers, where the denominator is not zero. That is, a rational number can be written as a/b, where a and b are integers, and b ≠ 0.

Proof:

Let r = a/b be a rational number, where a and b are integers, and b ≠ 0.

Step 1: Divide a by b using long division.

a = b × q + r

where q is the quotient, and r is the remainder (0 ≤ r < b).

Step 2: If r = 0, then a/b = q, which is a terminating decimal.

Step 3: If r ≠ 0, we can express a/b as:

a/b = q + r/(b × 10^n)

where n is the number of digits in b.

Step 4: We can repeat the division process for r/(b × 10ⁿ) and express it as:

r/(b × 10ⁿ) = q’ + r’/(b × 10⁽ⁿ⁺¹⁾)

where q’ is the quotient, and r’ is the remainder (0 ≤ r’ < b × 10^(n+1)).

Step 5: If r’ = 0, then a/b has a terminating decimal representation.

Step 6: If r’ ≠ 0, we can continue the division process indefinitely. Since there are only a finite number of possible remainders (less than b × 10⁽ⁿ⁺¹⁾), at some point, we will encounter a remainder that has appeared before. This means that the decimal expansion will begin to repeat in a periodic pattern.

Therefore, every rational number can be expressed as a decimal expansion that either terminates or repeats indefinitely in a periodic pattern, proving the decimal property of rational numbers.

Decimal Property of Rational Number

It is known that all rational numbers can be expressed in the form p/q where p and q are integers provided q ≠ 0. p divided q can result into an integer, terminating decimal, or repeating decimal.

Now, let’s solve the given problem statement,

Show that 0.3333… = 0 3, can be expressed in the form of rational number, i.e. p/q.

To show that 0.333… can be expressed in the form of rational number, i.e. p/q, we can use the concept of infinite geometric series.

Step 1: Let x = 0.333…. This means that is a repeating decimal where the digit 3 repeats infinitely.

Step 2: Multiply x by 10. This shifts the decimal point one place: 10x = 3.3333….

Step 3: Now, subtract x from 10x:

10x – x = 3.3333…. – 0.3333….

9x = 3

Step 4: Solve for x:

x = 3/9 = 1/3 (p/q where q ≠ 0)

So, 0.3333… = 1/3, which shows that 0.3333… can indeed be expressed in the form of a rational number p/q, where p = 1 and q = 3.

Similar Question

Q1: Express 0.40777777… in rational form.

Solution:

Let x = 0.40777777…     -(1)

Multiplying (1) by 100 we get

100x = 40.777777…     -(2)

Multiplying (2) by 10 we get

1000x = 407.777777..    -(3)

Now, subtracting (1) from (2) we get

900x = 367

⇒ x = 367/900

Hence, 0.40777777… in rational form is 367/900.

Q2: Express 1.0033333… in rational form.

Solution:

Let x = 1.0033333…     -(1)

Multiplying (1) by 100 we get

100x = 100.33333…     -(2)

Now, subtracting (2) from (1) we get

99x = 99.33

⇒ x = 99.33/99 = 9933/9900

Hence, 1.0033333… in rational form is 9933/9900.