## 4.10. Exercises

1. What are the decimal and hexadecimal representations for the value 0b01001010?

2. What are the binary and hexadecimal representations for the value 389?

3. As a five-armed creature, Sally the starfish prefers to represent numbers using a base 5 number system. If Sally gives you the base 5 number 1423, what is the equivalent decimal value?

### Solutions

1. 0b01001010 in decimal is:

(0 * 27)    +    (1 * 26)    +    (0 * 25)    +    (0 * 24)    +    (1 * 23)    +    (0 * 22)    +    (1 * 21)    +    (0 * 20)

=    0 + 64 + 0 + 0 + 8 + 0 + 2 + 0    =    74

```0100 1010
4    A  ->  0x4A```
2. Converting 389 to decimal…​

Using powers of two:

• 256 fits into 389, so d8 should be a 1. That leaves 389 - 256 = 133.

• 128 fits into 133, so d7 should be a 1. That leaves 133 - 128 = 5.

• 64 does not fit into 5, so d6 should be a 0.

• 32 does not fit into 5, so d5 should be a 0.

• 16 does not fit into 5, so d4 should be a 0.

• 8 does not fit into 5, so d3 should be a 0.

• 4 fits into 5, so d2 should be a 1. That leaves 6 - 5 = 1.

• 2 fits does not fit into 1, so d1 should be a 0.

• 1 fits into 1, so d0 should be a 1. That leaves 1 - 1 = 0.

Thus, decimal 389 corresponds to 0b110000101.

Using repeated division:

• 389 is odd, so d0 should be a 1.

• 389 / 2 = 194, which is even, so d1 should be a 0.

• 194 / 2 = 97, which is odd, so d2 should be a 1.

• 97 / 2 = 48, which is even, so d3 should be a 0.

• 48 / 2 = 24, which is even, so d4 should be a 0.

• 24 / 2 = 12, which is even, so d5 should be a 0.

• 12 / 2 = 6, which is even, so d6 should be a 0.

• 6 / 2 = 3, which is odd, so d7 should be a 1.

• 3 / 2 = 1, which is odd, so d8 should be a 1.

• 1 / 2 = 0, so any digit numbered nine or above will be 0.

Thus, decimal 389 corresponds to 0b110000101.

```0001 1000 0101