Soalan 6:
Tentukan sama ada pernyataan berikut benar atau palsu.
(a) 1234 > 238
(b) 729 < 3415
(c) 4526 > 3348
Penyelesaian:
(a)
$$ \begin{aligned} 123_4 & =\left(1 \times 4^2\right)+\left(2 \times 4^1\right)+\left(3 \times 4^0\right) \\ & =16+8+3 \\ & =27_{10} \end{aligned} $$
$$ \begin{aligned} 23_8 & =\left(2 \times 8^1\right)+\left(3 \times 8^0\right) \\ & =16+3 \\ & =19_{10} \end{aligned} $$
$$ 27_{10}>19_{10} $$
$$ \text { Maka, } 123_4>23_8 \text { adalah benar } $$
(b)
$$ \begin{aligned} 72_9 & =\left(7 \times 9^1\right)+\left(2 \times 9^0\right) \\ & =63+2 \\ & =65_{10} \end{aligned} $$
$$ \begin{aligned} 341_5 & =\left(3 \times 5^2\right)+\left(4 \times 5^1\right)+\left(1 \times 5^0\right) \\ & =75+20+1 \\ & =96_{10} \end{aligned} $$
$$ 65_{10}<96_{10} $$
$$ \text { Maka, } 72_9<341_5 \text { adalah benar } $$
(c)
$$ \begin{aligned} 452_6 & =\left(4 \times 6^2\right)+\left(5 \times 6^1\right)+\left(2 \times 6^0\right) \\ & =144+30+2 \\ & =176_{10} \end{aligned} $$
$$ \begin{aligned} 334_8 & =\left(3 \times 8^2\right)+\left(3 \times 8^1\right)+\left(4 \times 8^0\right) \\ & =192+24+4 \\ & =220_{10} \end{aligned} $$
$$ 176_{10}<220_{10} $$
$$ \text { Maka, } 452_6>334_8 \text { adalah palsu } $$
Tentukan sama ada pernyataan berikut benar atau palsu.
(a) 1234 > 238
(b) 729 < 3415
(c) 4526 > 3348
Penyelesaian:
(a)
$$ \begin{aligned} 123_4 & =\left(1 \times 4^2\right)+\left(2 \times 4^1\right)+\left(3 \times 4^0\right) \\ & =16+8+3 \\ & =27_{10} \end{aligned} $$
$$ \begin{aligned} 23_8 & =\left(2 \times 8^1\right)+\left(3 \times 8^0\right) \\ & =16+3 \\ & =19_{10} \end{aligned} $$
$$ 27_{10}>19_{10} $$
$$ \text { Maka, } 123_4>23_8 \text { adalah benar } $$
(b)
$$ \begin{aligned} 72_9 & =\left(7 \times 9^1\right)+\left(2 \times 9^0\right) \\ & =63+2 \\ & =65_{10} \end{aligned} $$
$$ \begin{aligned} 341_5 & =\left(3 \times 5^2\right)+\left(4 \times 5^1\right)+\left(1 \times 5^0\right) \\ & =75+20+1 \\ & =96_{10} \end{aligned} $$
$$ 65_{10}<96_{10} $$
$$ \text { Maka, } 72_9<341_5 \text { adalah benar } $$
(c)
$$ \begin{aligned} 452_6 & =\left(4 \times 6^2\right)+\left(5 \times 6^1\right)+\left(2 \times 6^0\right) \\ & =144+30+2 \\ & =176_{10} \end{aligned} $$
$$ \begin{aligned} 334_8 & =\left(3 \times 8^2\right)+\left(3 \times 8^1\right)+\left(4 \times 8^0\right) \\ & =192+24+4 \\ & =220_{10} \end{aligned} $$
$$ 176_{10}<220_{10} $$
$$ \text { Maka, } 452_6>334_8 \text { adalah palsu } $$
Soalan 7:
Hitung nilai 1102 + 2789 + 1324 dalam asas sepuluh.
Penyelesaian:
$$ \begin{aligned} 110_2 & =\left(1 \times 2^2\right)+\left(1 \times 2^1\right)+\left(0 \times 2^0\right) \\ & =4+2+0 \\ & =6_{10} \end{aligned} $$
$$ \begin{aligned} 278_9 & =\left(2 \times 9^2\right)+\left(7 \times 9^1\right)+\left(8 \times 9^0\right) \\ & =162+63+8 \\ & =233_{10} \end{aligned} $$
$$ \begin{aligned} 132_4 & =\left(1 \times 4^2\right)+\left(3 \times 4^1\right)+\left(2 \times 4^0\right) \\ & =16+12+2 \\ & =30_{10} \end{aligned} $$
$$ \begin{aligned} 110_2+278_9+132_4 & =6_{10}+233_{10}+30_{10} \\ & =269_{10} \end{aligned} $$
Hitung nilai 1102 + 2789 + 1324 dalam asas sepuluh.
Penyelesaian:
$$ \begin{aligned} 110_2 & =\left(1 \times 2^2\right)+\left(1 \times 2^1\right)+\left(0 \times 2^0\right) \\ & =4+2+0 \\ & =6_{10} \end{aligned} $$
$$ \begin{aligned} 278_9 & =\left(2 \times 9^2\right)+\left(7 \times 9^1\right)+\left(8 \times 9^0\right) \\ & =162+63+8 \\ & =233_{10} \end{aligned} $$
$$ \begin{aligned} 132_4 & =\left(1 \times 4^2\right)+\left(3 \times 4^1\right)+\left(2 \times 4^0\right) \\ & =16+12+2 \\ & =30_{10} \end{aligned} $$
$$ \begin{aligned} 110_2+278_9+132_4 & =6_{10}+233_{10}+30_{10} \\ & =269_{10} \end{aligned} $$
Soalan 8:
Hitung nilai 2314 – 1102 dalam asas sepuluh.
Penyelesaian:
$$ \begin{aligned} 231_4 & =\left(2 \times 4^2\right)+\left(3 \times 4^1\right)+\left(1 \times 4^0\right) \\ & =32+12+1 \\ & =45_{10} \end{aligned} $$
$$ \begin{aligned} 110_2 & =\left(1 \times 2^2\right)+\left(1 \times 2^1\right)+\left(0 \times 2^0\right) \\ & =4+2+0 \\ & =6_{10} \end{aligned} $$
$$ \begin{aligned} 231_4-110_2 & =45_{10}-6_{10} \\ & =39_{10} \end{aligned} $$
Hitung nilai 2314 – 1102 dalam asas sepuluh.
Penyelesaian:
$$ \begin{aligned} 231_4 & =\left(2 \times 4^2\right)+\left(3 \times 4^1\right)+\left(1 \times 4^0\right) \\ & =32+12+1 \\ & =45_{10} \end{aligned} $$
$$ \begin{aligned} 110_2 & =\left(1 \times 2^2\right)+\left(1 \times 2^1\right)+\left(0 \times 2^0\right) \\ & =4+2+0 \\ & =6_{10} \end{aligned} $$
$$ \begin{aligned} 231_4-110_2 & =45_{10}-6_{10} \\ & =39_{10} \end{aligned} $$