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Floor and Ceiling Functions Floor: For every real number x, the floor of x, x x, is defined = that unique integer n such that n ≤ x < n + 1. ( x = The greatest integer less than or equal to x.) ex. 2 2, 2.5 2, 1.1 2, Ceiling: For every real number x, the ceiling of x, x = 15 7 2 x , is defined that unique integer n such that n –1 < x ≤ n. ( x = The smallest integer greater than or equal to x.) ex. 2 2, 2.5 3, 15 1.1 1, 2 8 ex. Data stored on a computer disk or transmitted over a data network are usually represented as a string of bytes. Each byte is made up of 8 bits. How many bytes are required to encode 100 bits of data? ex. In asynchronous transfer mode (ATM), data are organized into cells of 53 bytes. How many ATM cells can be transmitted in 1 minute over a connection that transmits data at the rate of 500 kilo bits / second? Prove: For all real numbers x and integers m, x m x m . Proof: Prove: For all real numbers x, 1 2x x x 2 ex. Prove: x x Quotient Remainder Theorem Let n be any nonnegative integer and d be a positive integer. Then there exist integers q and r, such that n = dq + r, 0 ≤ r < d. n q , d n r n d d n n n divd , n mod d n d d d ex. n n 1 2 2 Do: Prove: For all odd integers n, .