Interpret the number 101010 in bases 2, 10, 16. For each, convert to the other two bases. You need to show all 6 conversions: a) When interpreting 101010 as a base 2 number, convert to bases 10 and 16; b) When interpreting 101010 as a base 10 number, convert to bases 2 and 16; c) When interpreting 101010 as a base 16 number, convert to bases 2 and 10;

a.

101010 base 2 = 42 base 10

101010 in base 2 = 2A in base 16.

b.

101010 in base 10 = 11000101010010010 base 2

101010 in base 10 = 18A92

c.

101010 base 16 = 1,052,688 base 10

101010 base 16 = 100000001000000010000 base 2

Explanation:

Conversion to Base 10

The general steps for converting a base 10 or "normal" number into another base are: First, divide the number by the base to get the remainder. This remainder is the first, ie least significant, digit of the new number in the other base. Then repeat the process by dividing

Conversion from Base 10 to any base

The general steps for converting a base 10 or "normal" number into another base are:

First, divide the number by the base to get the remainder. This remainder is the first, ie least significant, digit of the new number in the other base

Then repeat the process by dividing the quotient of step 1, by the new base. This time, the remainder is the second digit, ie the second least significant.

Repeat this process until your quotient becomes less than the base. This quotient is the last digit, ie the most significant digit.

Conversion between bases other than base 10

First, we convert to base 10 then we convert from base 10 to the base we intend to convert to

Now, we proceed to questions

a. 101010 in base 2.

Conversion to base 10

101010 = 1 * 2^5 + 0 * 2⁴ + 1 *2³ + 0 * 2² + 1 * 2¹ + 0 * 2°

101010 = 1 * 32 + 0 + 1 *8 + 0 + 1 * 2 + 0

101010 = 32 + 8 + 2

101010 base 2 = 42 base 10

Conversion to base 16

First, we convert to base 10 42 (Calculated)

Then we convert to base 16

42/16 = 2 Remainder 10

2/16 = 0 Remainder 2

We start writing the number from bottom

2(10) .

10 in hexadecimal is represented by letter A

So, we have 2A

101010 in base 2 = 2A in base 16.

b. 101010 in base 10

Conversion to base 2

101010/2 = 50505 Remainder 0

50505/2 = 25252 Remainder 1

25252/2 = 12626 Remainder 0

12626/2 = 6313 Remainder 0

6313/2 = 3156 Remainder 1

3156/2 = 1578 Remainder 0

1578/2 = 789 Remainder 0

789/2 = 394 Remainder 1

394/2 = 197 Remainder 0

197/2 = 98 Remainder 1

98/2 = 49 Remainder 0

49/2 = 24 Remainder 1

24/2 = 12 Remainder 0

12/2 = 6 Remainder 0

6/3 = 3 Remainder 0

3/2 = 1 Remainder 1

1/2 = 0 Remainder 1

We start counting from the bottom

11000101010010010 base 2

Conversion to Base 16

101010/16 = 6313 Remainder 2

6313/16 = 394 Remainder 9

394/16 = 24 Remainder 10

24/16 = 1 Remainder 8

1/16 = 0 Remainder 1

We start counting from the bottom

18(10)92

10 is represented with A

18A92

c. 101010 in base 16

Conversion to Base 10

101010 = 1 * 16^5 + 0 * 16⁴ + 1 * 16³ + 0 * 16² + 1 * 16¹ + 0 * 16°

101010 = 1,048,576 + 0 + 4,096 + 0 + 16 + 0

101010 = 1,052,688 base 10

Conversion to base 2

1,052,688/2 = 526344 Remainder 0

526344/2 = 263172 Remainder 0

263172/2 = 131586 Remainder 0

131586/2 = 65793 Remainder 0

65793/2 = 32896 Remainder 1

32896/2 = 16448 Remainder 0

16448/2 = 8224 Remainder 0

8224/2 = 4112 Remainder 0

4112/2 = 2056 Remainder 0

2056/2 = 1028 Remainder 0

1028/2 = 514 Remainder 0

514/2 = 257 Remainder 0

257/2 = 128 Remainder 1

128/2 = 64 Remainder 0

64/2 = 32 Remainder 0

32/2 = 16 Remainder 0

16/2 = 8 Remainder 0

8/2 = 4 Remainder 0

4/2 = 2 Remainder 0

2/2 = 1 Remainder 0

1/2 = 0 Remainder 1

We start from the bottom

= 100000001000000010000 base 2

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