What number differs from its reciprocal by 1?

Algebra is one of the branches of mathematics used to represent problems in the form of mathematical expressions. For example (X² + 2X + 3) is the quadratic equation find the roots like that is one of the examples for this algebraic expression (AX² + BX + C), here A = 1, B = 2, C = 3. We can use variables like alphabets (a, b, c… x, y, z). By using this variable we can frame mathematical questions by using addition, subtraction, and divisions. Some topics in mathematics like trigonometry, Calculus, and geometry involve using algebra. Example: Expressions like 8X + 3 = 1 are one equation that is in the form of variables, constants, and operators, where X is the variables, ‘+’ is the operator, and 1, 3 are constants.

Finding the Reciprocal of an algebraic expression

The reciprocal of an algebraic expression can easily be found by following simple two steps. Below are the two steps that are required to be followed for the same,

• Step1: Reverse (if 2 is the number then reverse is 1/2) the number using 1.
• Step 2: Calculate 1 divided by a number using a calculator.

What is that number that differs from its reciprocal and by 1?

Answer: 

To find whether there is a number that differs from its reciprocal by 1 we have to know the reciprocal of a number or variable. Let’s suppose X is the variable. To find the reciprocal of X we have to inverse X as (1/X). In reciprocal X should be greater the 0 otherwise it should be infinity. If X > 0 for example X = 1, 2, 3… we can able to find a reciprocal of X. 

Example: If X = 2 then what is the reciprocal of (X) which is (1/X) equal to 1/X,

= 1/2 

= 0.5 [Reciprocal is less than 1]

The perfect answer to the question (Is there a number that differs from its reciprocal by 1?) all-natural numbers except “1”, real numbers, and all whole numbers except  “0, 1” can differ from the original number and reciprocal number.

Sample Questions

Question 1: Calculate the reverse of the number “2”

Solution:   

The reverse of 2 is, 1/2

Calculation of 1/2 is, 0.5

So thereby, the reverse of 2 is 1/2 = 0.5 (less than 1).

Question 2: Calculate the reverse of X²  if x = 2.

Solution:  

Calculate the reverse of X² = 1/X²

Given that X = 2, So 1/X² is equal to 1/(2²) which is equal to 1/4

Calculation of 1/X² if X = 2 is 1/4 which is equal to 0.25

So thereby, the reverse of 1/X² if X = 2 is 0.25 which is also less than 1.

Question 3: Calculate the reverse of a quadratic equation X² + 0.2X + 3 if X = 1.

Solution: 

Calculate the reverse of X² + 0.2X + 3 = 1/(X² + 0.2X + 3)

Substitute the X = 1 given in the problem,   

1/(X² + 0.2X + 3) = 1/(1² + 0.2 × 1 + 3)

=1/(1 + 0.2 + 3)

= (1/4.2)

= 0.238

So thereby, the reverse of 1/(X² + 0.2X + 3) if X = 1 is 0.238 which is also less than 1.

Question 4: Calculate the reverse of a logarithmic equation which is log₁₀(X² + 8) when X = 2.

Solution: 

The reverse of log₁₀(X² + 8) = 1/ log₁₀(X² + 8)

Substitute the X = 2 

1/log₁₀(X² + 8) = 1/log10(2² + 8)

= 1/log10(4 + 8)

= 1/log10(12)

= 1/1.070

= 0.9345

So thereby, the reverse of 1/log₁₀(X² + 8) if X = 2 is 0.9345 which is also less than 1 only if logarithmic is greater than 1.

Question 5: Calculate the reverse of a logarithmic equation which is ln_e(X² + 2) when X = 3.

Solution:

The reverse of ln_e(X² + 2) = 1/ ln_e(X² + 2)

Substitute the X = 3 

1/ln_e(X² + 2) = 1/ln_e(3² + 2)

= 1/ln_e(9 + 2)

= 1/ln_e(11)

= 1/2.39

= 0.41

So thereby, the reverse of 1/ln_e(X² + 2) if X = 3 is 0.41. 

Question 6: Calculate the reverse of a logarithmic equation which is a² + b² when a, b = 3.3.

Solution:

Calculate the reverse of a² + b² = 1/ a² + b²

Substitute the a, b = 3.3

1/a² + b² = 1/a² + b²

= 1/3.3² + 3.3²

= 1/10.89 × 2

= 1/21.78

= 0.045

So thereby, the reverse of a² + b² which is 1/a² + b² if a, b = 3.3 is 0.045.