Test on Arithmetic in Different Number Bases

Question 1: The base of a number system, denoted as 'z', must be a natural number greater than 1.

A. Ano

B. Ne

Explanation: The definition states: 'Let z be any natural number greater than 1.' It further clarifies that the smallest number that can be a base is 2, meaning it cannot be 0, 1, or 01.

Question 2: Convert the number 154 base 6 to its equivalent in the decimal system.

A. 58

B. 70

C. 154

D. 64

Explanation: To convert a number from a non-decimal base (z) to the decimal system, we use the expanded form: a = a_n * z^n + ... + a_1 * z^1 + a_0 * z^0. For 154 base 6, this means: 1 * 6^2 + 5 * 6^1 + 4 * 6^0. Calculating this gives: 1 * 36 + 5 * 6 + 4 * 1 = 36 + 30 + 4 = 70. Therefore, 154 base 6 is 70 in the decimal system.

Question 3: The expanded notation algorithm for addition in base 10, when calculating 395 + 468, combines the powers of ten for each digit and then regroups the sums into the final result.

A. Ano

B. Ne

Explanation: The study materials show the 'algoritmus scitání (rozvinutý zápis)' for 395 + 468, which involves breaking down each number into its expanded form using powers of 10, summing the corresponding powers, and then regrouping carries to achieve the final sum of 863.

Question 4: When converting a number from base $z_1$ to base $z_2$ (where $z_1 eq 10$ and $z_2 eq 10$, and $z_1 eq z_2$), it is always necessary to first convert the number to the decimal system, regardless of whether $z_2 = z_1^k$ or $z_1 = z_2^1$ applies.

A. Ano

B. Ne

Explanation: The study materials state that conversion can be performed by first expressing the number in the decimal system and then converting to base $z_2$. However, it also explicitly notes that if $z_2 = z_1^k$ or $z_1 = z_2^1$, there is an alternative method to solve it directly, implying it's not always necessary to go through the decimal system.

Question 5: Using the division method described in the study materials, what is the binary representation of the decimal number 85?

A. 1010101₂

B. 1101010₂

C. 1010111₂

D. 1001001₂

Explanation: The division method involves repeatedly dividing the number by 2 and noting the remainders. For the number 85: 85 ÷ 2 = 42 remainder 1; 42 ÷ 2 = 21 remainder 0; 21 ÷ 2 = 10 remainder 1; 10 ÷ 2 = 5 remainder 0; 5 ÷ 2 = 2 remainder 1; 2 ÷ 2 = 1 remainder 0; 1 ÷ 2 = 0 remainder 1. Reading the remainders from bottom to top gives the binary representation: 1010101₂.