Exercise DET-01.01 – Writing numbers in different number systems

Prof. Dr. Thorsten Jungmann (last updated 2026-11-02)

Relates to DET-01 Introduction to the module and its content

Task

In a positional number system, each digit is weighted with a power of the base. Write each of the following numbers as the sum of its digits multiplied by their weights. For hexadecimal digits, use their values, i.e. A = 10 to F = 15. You do not need to calculate the sums.

Binary number system: a) $1101_2$, b) $10110110_2$

Octal number system: a) $16_8$, b) $327_8$, c) $5274_8$, d) $147251_8$

Decimal number system: a) $12_{10}$, b) $214_{10}$, c) $3517_{10}$, d) $48059_{10}$

Hexadecimal number system: a) ${\mathrm{B}}_{16}$, b) ${3C}_{16}$, c) ${A07}_{16}$, d) ${E5B2}_{16}$

Solution

Binary number system:
a) $1101_2=1\cdot 2^3+1\cdot 2^2+0\cdot 2^1+1\cdot 2^0$
b) $10110110_2=1\cdot 2^7+0\cdot 2^6+1\cdot 2^5+1\cdot 2^4+0\cdot 2^3+1\cdot 2^2+1\cdot 2^1+0\cdot 2^0$

Octal number system:
a) $16_8=1\cdot 8^1+6\cdot 8^0$
b) $327_8=3\cdot 8^2+2\cdot 8^1+7\cdot 8^0$
c) $5274_8=5\cdot 8^3+2\cdot 8^2+7\cdot 8^1+4\cdot 8^0$
d) $147251_8=1\cdot 8^5+4\cdot 8^4+7\cdot 8^3+2\cdot 8^2+5\cdot 8^1+1\cdot 8^0$

Decimal number system:
a) $12_{10}=1\cdot 10^1+2\cdot 10^0$
b) $214_{10}=2\cdot 10^2+1\cdot 10^1+4\cdot 10^0$
c) $3517_{10}=3\cdot 10^3+5\cdot 10^2+1\cdot 10^1+7\cdot 10^0$
d) $48059_{10}=4\cdot 10^4+8\cdot 10^3+0\cdot 10^2+5\cdot 10^1+9\cdot 10^0$

Hexadecimal number system:
a) ${\mathrm{B}}_{16}=11\cdot 16^0$
b) ${3C}_{16}=3\cdot 16^1+12\cdot 16^0$
c) ${A07}_{16}=10\cdot 16^2+0\cdot 16^1+7\cdot 16^0$
d) ${E5B2}_{16}=14\cdot 16^3+5\cdot 16^2+11\cdot 16^1+2\cdot 16^0$

▶️ Full worked solution