Problems from System of decimal, binary and hexadecimal numeration

Find the equivalents of the following numbers in base $10$ :

$(101010)_{2}$

$(2220)_{3}$

$(246)_{7}$

$(B A)_{12}$

$(21 G)_{16}$

$(358)_{8}$

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Development:

To solve the exercises it is enough to decompose the numbers in power.

The first one is a binary number of $6$ digits, then its powers will decrease from $5$ to $0$ :

$(101010)_{2} = 1 \cdot 2^{5} + 0 \cdot 2^{4} + 1 \cdot 2^{3} + 0 \cdot 2^{2} + 1 \cdot 2^{1} + 0 \cdot 2^{0} =$

$= 32 + 0 + 8 + 0 + 2 + 0 = 42$

The second case, is a ternary number of $4$ digits, so its exponents will decrease from $3$ to $0$ :

$(2220)_{3} = 2 \cdot 3^{3} + 2 \cdot 3^{2} + 2 \cdot 3^{1} + 0 \cdot 3^{0} = 54 + 18 + 6 + 0 = 78$

The third one is a number in base $7$ :

$(246)_{7} = 2 \cdot 7^{2} + 4 \cdot 7^{1} + 6 \cdot 7^{0} = 98 + 28 + 6 = 132$

In the fourth number, it is necessary to remember that letters replace numbers of more than two digits, so that $A = 10$ and $B = 11$ :

$(B A)_{12} = ((11) (10))_{12} = 11 \cdot 12^{1} + 10 \cdot 12^{0} = 132 + 10 = 142$

The fifth number is a trick:

$(21 G)_{16}$

The number is expressed in the hexadecimal system which, as has already been said, uses letters, but G is not one of them, since a number in base $16$ uses $16$ symbols: $10$ from $0$ to $9$ and $6$ from A to F.

The solution of the problem is, therefore, that this number does not make any sense, so that it is not possible to translate it into the decimal system.

Something similar happens with the last number:

$(358)_{8}$

As in the previous case, a number in octal base uses $8$ symbols: from $0$ to $7$ . $8$ is not included, that is the reason why this number does not make sense either, and it is not possible to calculate its decimal equivalent.

Solution:

$42$

$78$

$132$

$142$

It does not make sense.

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