Prove that if x is a real number then [2x] = [x] + [x + ½ ], Mathematics

Assignment Help:

Prove that if x is a real number then

[2x] = [x] + [x + ½ ]

Ans: Let us consider x be any real number. It comprises two parts: integer and fraction. With no loss of any type of generality, fraction part can all time be made +ve. For instance, -1.3 can be written as -2 + 0.7.

Here now write x = a + b, and [x] = a (integer part only of the real x). The fraction part b requires to considered in two cases: 

0 < b < 0.5 and 0.5 ≤ b < 1.

Case 1: 0 < b < 0.5; In this case [2x] = 2a, and [x] +[x + .5] = a + a = 2a

Case 2: 0.5 ≤ b < 1; In this case [2x] = 2a + 1, and [x] +[x + .5] = a + (a + 1) = 2a +1

Hence [2x] = [x] + [x + .5]


Related Discussions:- Prove that if x is a real number then [2x] = [x] + [x + ½ ]

Sampling distribution p-hat, Caterer verifies that 87% of people who sample...

Caterer verifies that 87% of people who sampled the food thought it was delicious. A random sample of 144 out of population of 5000 taken. The 144 are asked to sample the food. If

Matrix addition and subtraction, What is Matrix addition and subtraction? I...

What is Matrix addition and subtraction? Illustrate the procedure of Matrix addition and subtraction.

Calculus online, need help completing my online text. can provide login det...

need help completing my online text. can provide login details

Derive the probability distribution of the completion times, Derive the pro...

Derive the probability distribution of the completion times: a. The following probability distributions relate to the completion times, in weeks, T A and T B of two independ

Sequences and series - calculus, Sequences and Series In this section ...

Sequences and Series In this section we will be taking a look at sequences and infinite series.  In fact, this section will deal approximately exclusively with series.  Though

Area of a circle, How do you find the area of a circle given the diameter?

How do you find the area of a circle given the diameter?

What is addition rule of probability, Q. What is Addition Rule of probabili...

Q. What is Addition Rule of probability? Ans. Suppose there are 17 girls and 15 boys in your stats class. There are 17 + 15 = 32 ways for your teacher to pick one student

Give the example of exponents, Give the example of Exponents? When a nu...

Give the example of Exponents? When a number is multiplied several times, it is easier to write it as an exponent. For example, four multiplied to itself three times, is writte

Definition of minimum and maximum values, Definition 1.   We say that f...

Definition 1.   We say that f(x) consist an absolute (or global) maximum at x = c if f ( x ) ≤ f (c ) for every x in the domain we are working on. 2.  We say that at x = c ,

Number system, NATURAL NUMBERS The numbers 1, 2, 3, 4.... Are called as...

NATURAL NUMBERS The numbers 1, 2, 3, 4.... Are called as natural numbers, their set is shown by N. Hence N = {1, 2, 3, 4, 5....} WHOLE NUMBERS The numbers 0, 1, 2, 3, 4

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd