Express the following octal numbers into their equivalent decimal numbers.
(i) 145
(ii) 6760
(iii) 455
(iv) 10.75

Question

Accepted Answer

> [!TLDR]
> Expand by powers of 8 (…64, 8, 1 before the point; 1/8, 1/64 after it) and add: (i) 101,
> (ii) 3568,
> (iii) 301,
> (iv) 8.953125.

**The idea.** An octal number's digits carry place values that are powers of 8: moving left from the octal point they are 8^0=1, 8^1=8, 8^2=64, 8^3=512; moving **right** they are 8^-1 = 1/8 = 0.125 and 8^-2 = 1/64 = 0.015625. Converting to decimal is expand-and-add.

**(i) (145)8**
```text
(145)8 = 1x8^2 + 4x8^1 + 5x8^0
       = 1x64  + 4x8   + 5x1
       = 64 + 32 + 5
       = (101)10
```

**(ii) (6760)8**
```text
(6760)8 = 6x8^3 + 7x8^2 + 6x8^1 + 0x8^0
        = 6x512 + 7x64  + 6x8   + 0
        = 3072 + 448 + 48 + 0
        = (3568)10
```

**(iii) (455)8**
```text
(455)8 = 4x64 + 5x8 + 5x1
       = 256 + 40 + 5
       = (301)10
```

**(iv) (10.75)8** — the digits after the octal point use negative powers of 8 (note: 7 and 5 are valid octal digits, and "0.75 octal" is NOT 0.75 decimal):
```text
Integer part:    (10)8  = 1x8 + 0x1 = 8
Fractional part: (.75)8 = 7x(1/8) + 5x(1/64)
                        = 0.875 + 0.078125 = 0.953125
Total: (10.75)8 = 8 + 0.953125 = (8.953125)10
``` …

Digit	Place value (power of 8)	Contribution
3	$8^2 = 64$	$3 \times 64 = 192$
4	$8^1 = 8$	$4 \times 8 = 32$
2	$8^0 = 1$	$2 \times 1 = 2$

Digit	Place value	Contribution
1	$8^2 = 64$	$1 \times 64 = 64$
0	$8^1 = 8$	$0 \times 8 = 0$
7	$8^0 = 1$	$7 \times 1 = 7$

Q.Express the following octal numbers into their equivalent decimal numbers.

Concept understanding — Octal To Decimal Conversion

What Does "Octal to Decimal" Even Mean?

The Core Idea: Positional Value in Base 8

Worked Example: Convert $(342)_8$ to Decimal

Another Example: $(107)_8$

Q.Express the following octal numbers into their equivalent decimal numbers.

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