東京大学 情報理工学系研究科 創造情報学専攻 2026年2月実施 筆記試験 第1問

Author

itsuitsuki

Description (Memorized version, English)

Consider an 8-bit binary number (e.g., 0b10110111, where underscores _ can be freely added as separators without affecting the meaning).

• [7:4] (Bits 7, 6, 5, 4): represent the exponent E.
• [3:0] (Bits 3, 2, 1, 0): represent the mantissa M.

The floating-point value represented by this binary number is calculated using the formula: This is called EM notation.

Answer the following questions.

(1) How is 1.0 represented in EM notation?

(2) What is the decimal value of the binary number 0b1000_1000?

(3) What are the decimal values of the largest and the second largest numbers that can be represented?

We define functions as follows:

• : Input a decimal number ; output the largest decimal number strictly less than that can be exactly represented in EM notation.
• : Input a binary EM representation ; output the corresponding decimal value.
• : Input a decimal number ; output the EM representation (binary) of the largest number strictly less than that can be exactly represented in EM notation.

(4) Prove: For two binary numbers and , if , then .

(5) Prove: Among all numbers representable in EM notation, no two numbers have the same EM representation (i.e., the mapping is unique).

(6) Calculate the value of .

(7) Calculate the value of .

(8) Let and be numbers that can be exactly represented in EM notation, with . Let be the integer part of the decimal value of . Find a value for (provide the decimal value) such that the lower 4 bits (the Mantissa part, ) of correspond to the value for every .