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 ]

Determine the length of the diagonal, A box is 30 cm long, 8 cm wide and 12...

A box is 30 cm long, 8 cm wide and 12 cm high. Determine the length of the diagonal AB ? Round to the nearest tenth. a. 34.5 cm b. 32.1 cm c. 35.2 cm d. 33.3 cm

What is stem-and-leaf plots, Q. What is Stem-and-Leaf Plots? Ans. ...

Q. What is Stem-and-Leaf Plots? Ans. A stem-and-leaf plot is a table that provides a quick way to arrange a set of data and view its shape, or distribution. Each data val

Circles, alternate segment theorum

alternate segment theorum

Describe the properties of inequalities, Describe the Properties of Inequal...

Describe the Properties of Inequalities ? Postulate In comparing two quantities, say a and b, there are exactly three possibilities. (1) a is less than b. (a b)

Intervals of validity, I've termed this section as Intervals of Validity si...

I've termed this section as Intervals of Validity since all of the illustrations will involve them. Though, there is many more to this section. We will notice a couple of theorems

Find the lesser of two consecutive positive even integers, Find the lesser ...

Find the lesser of two consecutive positive even integers whose product is 168. Let x = the lesser even integer and let x + 2 = the greater even integer. Because product is a k

What is the greatest value of the number, Five more than the quotient of a ...

Five more than the quotient of a number and 2 is at least that number. What is the greatest value of the number? Let x = the number. Notice that quotient is a key word for div

Las leyes de kepler, la expresión que permite calcular el radio medio de la...

la expresión que permite calcular el radio medio de la órbita de cada planeta es?

Functions, The figure shows the sketch graphs of the functions

The figure shows the sketch graphs of the functions

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