Question

Переведите числа из одной системы счисления в другую: 1) 3710 →А2; 2) 1011012 →А10; 3) 12510 →А8; 4) 5178 →А10. С объяснениями, пожалуйста.

Answer 1

Ответ:

1) 100101₂

2) 45₁₀

3) 175₈

4) 335₁₀

Объяснение:

Для перевода числа из 10 системы счисления  в любую другую систему счисления надо нацело делить число на основание новой системы счисления (пока целая часть не станет равной 0) и записывать получившиеся остатки от деления. Число в новой системе счисления записывается в виде остатков деления, начиная с последнего

1) 37₁₀ → А₂

37/2 = 18 и остаток 1

18/2 = 9 и остаток 0

9/2 = 4 и остаток 1

4/2 = 2 и остаток 0

2/2 = 1 и остаток 0

1/2 = 0 и остаток 1

37₁₀ = 100101₂  

3) 125₁₀ → А₈

125/8 = 15 и остаток 5

15/8 = 1 и остаток 7

1/8 = 0 и остаток 1

125₁₀ = 175₈

Для перевода числа из любой системы счисления в 10 систему счисления используется развернутая запись числа - сумма произведений цифр числа на основание системы счисления числа в степени разряда цифры числа

2) 101101₂ → А₁₀

101101₂ = 1 * 2⁵ + 0 * 2⁴ + 1 * 2³ + 1 * 2² + 0 * 2¹ + 1 * 2⁰ = 32 + 0 + 8 + 4 + 0 + 1 = 45₁₀

4) 517₈ → А₁₀

517₈ =  = 5 * 8² + 1 * 8¹ + 7 * 8⁰ = 320 + 8 + 7 = 335₁₀

Answer 2

Converting Numbers from One Number System to Another

To convert numbers from one number system to another, we need to understand the base or radix of each number system. The base determines the number of digits used in the number system and the value of each digit.

Let's go through each conversion step by step:

1) Converting 37 (base 10) to base 2 (binary):

To convert a decimal number to binary, we repeatedly divide the decimal number by 2 and note down the remainders until the quotient becomes zero. The binary number is then obtained by writing the remainders in reverse order.

In this case, to convert 37 to binary:

- Divide 37 by 2: quotient = 18, remainder = 1 - Divide 18 by 2: quotient = 9, remainder = 0 - Divide 9 by 2: quotient = 4, remainder = 1 - Divide 4 by 2: quotient = 2, remainder = 0 - Divide 2 by 2: quotient = 1, remainder = 0 - Divide 1 by 2: quotient = 0, remainder = 1

The remainders in reverse order are 100101. Therefore, 37 (base 10) is equal to 100101 (base 2).

2) Converting 101101 (base 2) to base 10 (decimal):

To convert a binary number to decimal, we multiply each digit of the binary number by the corresponding power of 2 and sum the results.

In this case, to convert 101101 to decimal:

- Multiply the leftmost digit (1) by 2^5 (32) - Multiply the next digit (0) by 2^4 (16) - Multiply the next digit (1) by 2^3 (8) - Multiply the next digit (1) by 2^2 (4) - Multiply the next digit (0) by 2^1 (2) - Multiply the rightmost digit (1) by 2^0 (1)

Summing the results, we get: 32 + 8 + 4 + 1 = 45. Therefore, 101101 (base 2) is equal to 45 (base 10).

3) Converting 125 (base 10) to base 8 (octal):

To convert a decimal number to octal, we repeatedly divide the decimal number by 8 and note down the remainders until the quotient becomes zero. The octal number is then obtained by writing the remainders in reverse order.

In this case, to convert 125 to octal:

- Divide 125 by 8: quotient = 15, remainder = 5 - Divide 15 by 8: quotient = 1, remainder = 7 - Divide 1 by 8: quotient = 0, remainder = 1

The remainders in reverse order are 175. Therefore, 125 (base 10) is equal to 175 (base 8).

4) Converting 517 (base 8) to base 10 (decimal):

To convert an octal number to decimal, we multiply each digit of the octal number by the corresponding power of 8 and sum the results.

In this case, to convert 517 to decimal:

- Multiply the leftmost digit (5) by 8^2 (64) - Multiply the next digit (1) by 8^1 (8) - Multiply the rightmost digit (7) by 8^0 (1)

Summing the results, we get: 64 + 8 + 7 = 79. Therefore, 517 (base 8) is equal to 79 (base 10).

I hope this explanation helps! Let me know if you have any further questions.