Convert the following numbers into decimal numbers.
(i) (110101)2
(ii) (1703)8
(iii) (COF5)16

Question

Accepted Answer

> [!TLDR]
> Expand each number by positional weights (powers of its base) and add: (110101)2 = 53, (1703)8 = 963, (C0F5)16 = 49397.

To convert **any** base to decimal, multiply each digit by its positional value — base^0 for the rightmost digit, then base^1, base^2, … moving left — and add the products.

**(i) (110101)2** — base 2, weights 32, 16, 8, 4, 2, 1:

```text
(110101)2 = 1x32 + 1x16 + 0x8 + 1x4 + 0x2 + 1x1
          = 32 + 16 + 0 + 4 + 0 + 1
          = (53)10
```

**(ii) (1703)8** — base 8, weights 512, 64, 8, 1:

```text
(1703)8 = 1x512 + 7x64 + 0x8 + 3x1
        = 512 + 448 + 0 + 3
        = (963)10
```

**(iii) (COF5)16** — first, a correction to the printed question: the letter **O is not a valid hexadecimal digit** (hex digits are 0–9 and A–F). The "O" in the book can only be the digit **0**, so the number is (C0F5)16. With C = 12 and F = 15, weights 4096, 256, 16, 1:

```text
(C0F5)16 = 12x4096 + 0x256 + 15x16 + 5x1
         = 49152 + 0 + 240 + 5
         = (49397)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.Convert the following numbers into 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.Convert the following numbers into decimal numbers.

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