Express Base number

Question

Express (1 x 4³) + (3 x 4) + 2 as a number in

(i) base four.

(ii) base two.

Answer 1

1 x 4^3 = 64

64+12 + 2 = 78 in base 10

78 in base 4 is a number which never has a digit larger than 3. It has only 4 digits, 0,1, 2 and 3

78 = 64 + 14

64=4^3

14 = 12 + 2

12= 3x4^1

2 = 2x4^0

78 = 1 x 4^3 + 0 x 4^2 + 3 x 4^1 + 2 x 4^0

Look at the coefficients of the powers of 4

they're 1032, that's 78 in base 4

in base 2, you never have a digit higher than 1. All you have are 0's and 1's

78 = 1001110 in base 2

78 = 1x2^6 + 0 x 2^5 + 0 x 2^4 + 1 x 2^3 + 1 x 2^2 + 1 x 2^1 + 0 x 2^0

Collect all the coefficients of the powers of 2

1001110

2^6= 64

2^3 = 8

2^2 = 4

2^1 = 2

they sum to 78



Answer 2

1 x 4^3 = 64

64+12 + 2 = 78 in base 10

78 in base 4 is a number which never has a digit larger than 3. It has only 4 digits, 0,1, 2 and 3

78 = 64 + 14

64=4^3

14 = 12 + 2

12= 3x4^1

2 = 2x4^0

78 = 1 x 4^3 + 0 x 4^2 + 3 x 4^1 + 2 x 4^0

Look at the coefficients of the powers of 4

they're 1032, that's 78 in base 4

in base 2, you never have a digit higher than 1. All you have are 0's and 1's

78 = 1001110 in base 2

78 = 1x2^6 + 0 x 2^5 + 0 x 2^4 + 1 x 2^3 + 1 x 2^2 + 1 x 2^1 + 0 x 2^0

Collect all the coefficients of the powers of 2

1001110

2^6= 64

2^3 = 8

2^2 = 4

2^1 = 2

they sum to 78