Perform the following conversions in the stated number bases;
(a) AB1\(_{16}\) to Octal system
(b) 1110111.101\(_2\) to Denary system
(c) 507\(_2\) to Binary system.
Explanation
(a) ABI\(_{Hex}\) to octal
A=1010 B=1011 1=0001 -------- Step 1
= 101010110001\(_2\)
= 101=5 010=2 110=6 001=1 ---------- Step 2
=5261\(_{octal}\) ------ Step 3
= AB1\(_{16}\)
= Ax16\(^2\) + Bx16\(^1\) + 1x16\(^0\)
= 10 X 16\(_2\) + 11 X 16\(_1\) + 1 X 16\(_0\)
= 2560 + 176 + 1
= 2737\(_{10}\)
It has to be converted to octal
| 8 | 2737 | |
| 8 | 342 | R1 |
| 8 | 42 | R6 |
| 8 | 5 | R2 |
| 0 | R5 |
= 5261
(b) 1110111.101\(_2\) to Denary
64=1 32=1 16=1 8=0 4=1 2=1 1=1
64+32+16+0+4+2+1
=96+16+7
=96+23
=119
2nd step fractional part
-1 -2 -3
.1 0 1
=1x2\(^{-1}\) + 0x2\(^{-2}\) + 1x2\(^{-3}\)
= \(\frac{1}{2^1}\) + \(\frac{0}{2^2}\) + \(\frac{1}{2^3}\)
= \(\frac{1}{4}\) + 0 + \(\frac{1}{8}\)
= \(\frac{5}{8}\)
= 0.625
: 1110111.101\(_2\) = 119.625
(c) 507\(_{octal}\) to Binary
5=101 0=000 9=111
=101000111\(_2\)
OR
507\(_8\) to decimal
=5x8\(^2\) + 0x8\(^1\) + 7x8\(^0\)
= 5x64+0+7
=327\(_{10}\)
Conversion to binary
327\(_{10}\) to binary
| 2 | 327 | |
| 2 | 163 | R1 |
| 2 | 81 | R1 |
| 2 | 40 | R1 |
| 2 | 20 | R0 |
| 2 | 10 | R0 |
| 2 | 5 | R0 |
| 2 | 2 | R1 |
| 2 | 1 | R0 |
| 0 | R1 |
=101000111\(_2\)