Soalan 10:
$$ \text { Susun nombor berikut mengikut urutan menurun. } $$
$$ \text { (a) } 111101_2, 1213_4, 81_9 $$
$$ \text { (b) } 123_4, 73_8, 313_5 $$
$$ \text { (c) } 253_6, 161_7, 222_3 $$
Penyelesaian:
(a)
$$ \begin{aligned} 111101_2 & =\left(1 \times 2^5\right)+\left(1 \times 2^4\right)+\left(1 \times 2^3\right)+\left(1 \times 2^2\right)+\left(0 \times 2^1\right)+\left(1 \times 2^0\right) \\ & =32+16+8+4+0+1 \\ & =61 \end{aligned} $$
$$ \begin{aligned} 1213_4 & =\left(1 \times 4^3\right)+\left(2 \times 4^2\right)+\left(1 \times 4^1\right)+\left(3 \times 4^0\right) \\ & =64+32+4+3 \\ & =103 \end{aligned} $$
$$ \begin{aligned} 81_9 & =\left(8 \times 9^1\right)+\left(1 \times 9^0\right) \\ & =72+1 \\ & =73 \end{aligned} $$
$$ \text { Susun nombor mengikut urutan menurun }=1213_4, 81_9, 111101_2 $$
(b)
$$ \begin{aligned} 123_4 & =\left(1 \times 4^2\right)+\left(2 \times 4^1\right)+\left(3 \times 4^0\right) \\ & =16+8+3 \\ & =27 \end{aligned} $$
$$ \begin{aligned} 73_8 & =\left(7 \times 8^1\right)+\left(3 \times 8^0\right) \\ & =56+3 \\ & =59 \end{aligned} $$
$$ \begin{aligned} 313_5 & =\left(3 \times 5^2\right)+\left(1 \times 5^1\right)+\left(3 \times 5^0\right) \\ & =75+5+3 \\ & =88 \end{aligned} $$
$$ \text { Susun nombor mengikut urutan menurun }=313_5, 73_8, 123_4 $$
(c)
$$ \begin{aligned} 253_6 & =\left(2 \times 6^2\right)+\left(5 \times 6^1\right)+\left(3 \times 6^0\right) \\ & =72+30+3 \\ & =105 \end{aligned} $$
$$ \begin{aligned} 161_7 & =\left(1 \times 7^2\right)+\left(6 \times 7^1\right)+\left(1 \times 7^0\right) \\ & =49+42+1 \\ & =92 \end{aligned} $$
$$ \begin{aligned} 222_3 & =\left(2 \times 3^2\right)+\left(2 \times 3^1\right)+\left(2 \times 3^0\right) \\ & =18+6+2 \\ & =26 \end{aligned} $$
$$ \text { Susun nombor mengikut urutan menurun }=253_6, 161_7, 222_3 $$
$$ \text { Susun nombor berikut mengikut urutan menurun. } $$
$$ \text { (a) } 111101_2, 1213_4, 81_9 $$
$$ \text { (b) } 123_4, 73_8, 313_5 $$
$$ \text { (c) } 253_6, 161_7, 222_3 $$
Penyelesaian:
(a)
$$ \begin{aligned} 111101_2 & =\left(1 \times 2^5\right)+\left(1 \times 2^4\right)+\left(1 \times 2^3\right)+\left(1 \times 2^2\right)+\left(0 \times 2^1\right)+\left(1 \times 2^0\right) \\ & =32+16+8+4+0+1 \\ & =61 \end{aligned} $$
$$ \begin{aligned} 1213_4 & =\left(1 \times 4^3\right)+\left(2 \times 4^2\right)+\left(1 \times 4^1\right)+\left(3 \times 4^0\right) \\ & =64+32+4+3 \\ & =103 \end{aligned} $$
$$ \begin{aligned} 81_9 & =\left(8 \times 9^1\right)+\left(1 \times 9^0\right) \\ & =72+1 \\ & =73 \end{aligned} $$
$$ \text { Susun nombor mengikut urutan menurun }=1213_4, 81_9, 111101_2 $$
(b)
$$ \begin{aligned} 123_4 & =\left(1 \times 4^2\right)+\left(2 \times 4^1\right)+\left(3 \times 4^0\right) \\ & =16+8+3 \\ & =27 \end{aligned} $$
$$ \begin{aligned} 73_8 & =\left(7 \times 8^1\right)+\left(3 \times 8^0\right) \\ & =56+3 \\ & =59 \end{aligned} $$
$$ \begin{aligned} 313_5 & =\left(3 \times 5^2\right)+\left(1 \times 5^1\right)+\left(3 \times 5^0\right) \\ & =75+5+3 \\ & =88 \end{aligned} $$
$$ \text { Susun nombor mengikut urutan menurun }=313_5, 73_8, 123_4 $$
(c)
$$ \begin{aligned} 253_6 & =\left(2 \times 6^2\right)+\left(5 \times 6^1\right)+\left(3 \times 6^0\right) \\ & =72+30+3 \\ & =105 \end{aligned} $$
$$ \begin{aligned} 161_7 & =\left(1 \times 7^2\right)+\left(6 \times 7^1\right)+\left(1 \times 7^0\right) \\ & =49+42+1 \\ & =92 \end{aligned} $$
$$ \begin{aligned} 222_3 & =\left(2 \times 3^2\right)+\left(2 \times 3^1\right)+\left(2 \times 3^0\right) \\ & =18+6+2 \\ & =26 \end{aligned} $$
$$ \text { Susun nombor mengikut urutan menurun }=253_6, 161_7, 222_3 $$
Soalan 11:
$$ \text { Hitung beza nilai digit } 5 \text { antara nombor } 1576_8 \text { dengan } 125_7 \text {. } $$
Penyelesaian:
$$ \begin{aligned} 1 \underline{5} 76_8 & =\left(1 \times 8^3\right)+\left(5 \times 8^2\right)+\left(7 \times 8^1\right)+\left(6 \times 8^0\right) \\ 12 \underline{5}_7 & =\left(1 \times 7^2\right)+\left(2 \times 7^1\right)+\left(5 \times 7^0\right) \end{aligned} $$
$$ \begin{aligned} &\text { Beza nilai digit } 5 \text { antara nombor }\\ &\begin{aligned} & =\left(5 \times 8^2\right)-\left(5 \times 7^0\right) \\ & =320-5 \\ & =315 \end{aligned} \end{aligned} $$
$$ \text { Hitung beza nilai digit } 5 \text { antara nombor } 1576_8 \text { dengan } 125_7 \text {. } $$
Penyelesaian:
$$ \begin{aligned} 1 \underline{5} 76_8 & =\left(1 \times 8^3\right)+\left(5 \times 8^2\right)+\left(7 \times 8^1\right)+\left(6 \times 8^0\right) \\ 12 \underline{5}_7 & =\left(1 \times 7^2\right)+\left(2 \times 7^1\right)+\left(5 \times 7^0\right) \end{aligned} $$
$$ \begin{aligned} &\text { Beza nilai digit } 5 \text { antara nombor }\\ &\begin{aligned} & =\left(5 \times 8^2\right)-\left(5 \times 7^0\right) \\ & =320-5 \\ & =315 \end{aligned} \end{aligned} $$