Exercise DET-01.01 – Writing numbers in different number systems (Solution)

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) ${E5B2}_{16}$

◀️ Back to the task

---

Solution

Given

Given explicitly:

• Binary numbers: $1101_2$, $10110110_2$
• Octal numbers: $16_8$, $327_8$, $5274_8$, $147251_8$
• Decimal numbers: $12_{10}$, $214_{10}$, $3517_{10}$, $48059_{10}$
• Hexadecimal numbers: ${\mathrm{B}}_{16}$, ${3C}_{16}$, ${A07}_{16}$, ${E5B2}_{16}$

Known:

• Value of a number in a positional number system (DET-01): $N={\sum }_{i=0}^{n-1}{d}_i\cdot b^i$, where $d_i$ is the digit at position $i$, counted from the right and starting with 0, $n$ is the number of digits and $b$ is the base
• Weights: the powers of the base, i.e. $2^i$ in the binary, $8^i$ in the octal, $10^i$ in the decimal and $16^i$ in the hexadecimal number system
• Values of the hexadecimal digits (DET-01): A = 10, B = 11, C = 12, D = 13, E = 14, F = 15

Required

In each of the four number systems: the number as the sum of its digits multiplied by their weights

Binary number system

The weights are the powers of two. A number with $n$ digits has the weights $2^{n-1}$ down to $2^0=1$ at the rightmost position. A digit 0 contributes nothing, but it keeps its position: it is written as $0\cdot 2^i$, so that every position appears in the sum.

a) $1101_2$

$$1101_2=1\cdot 2^3+1\cdot 2^2+0\cdot 2^1+1\cdot 2^0$$

The number has 4 digits, so the exponents run from 3 down to 0 ✓

b) $10110110_2$

$$\begin{array}{rl}10110110_2& =1\cdot 2^7+0\cdot 2^6+1\cdot 2^5+1\cdot 2^4\\ & +0\cdot 2^3+1\cdot 2^2+1\cdot 2^1+0\cdot 2^0\end{array}$$

The number has 8 digits, so the exponents run from 7 down to 0 ✓

Octal number system

The weights are the powers of eight, from $8^0=1$ at the rightmost position upwards.

a) $16_8$

$$16_8=1\cdot 8^1+6\cdot 8^0$$

The number has 2 digits, so the exponents run from 1 down to 0 ✓

b) $327_8$

$$327_8=3\cdot 8^2+2\cdot 8^1+7\cdot 8^0$$

The number has 3 digits, so the exponents run from 2 down to 0 ✓

c) $5274_8$

$$5274_8=5\cdot 8^3+2\cdot 8^2+7\cdot 8^1+4\cdot 8^0$$

The number has 4 digits, so the exponents run from 3 down to 0 ✓

d) $147251_8$

$$147251_8=1\cdot 8^5+4\cdot 8^4+7\cdot 8^3+2\cdot 8^2+5\cdot 8^1+1\cdot 8^0$$

The number has 6 digits, so the exponents run from 5 down to 0 ✓

Decimal number system

The decimal number system is a positional number system too: its weights are the powers of ten, from $10^0=1$ at the rightmost position upwards.

a) $12_{10}$

$$12_{10}=1\cdot 10^1+2\cdot 10^0$$

The number has 2 digits, so the exponents run from 1 down to 0 ✓

b) $214_{10}$

$$214_{10}=2\cdot 10^2+1\cdot 10^1+4\cdot 10^0$$

The number has 3 digits, so the exponents run from 2 down to 0 ✓

c) $3517_{10}$

$$3517_{10}=3\cdot 10^3+5\cdot 10^2+1\cdot 10^1+7\cdot 10^0$$

The number has 4 digits, so the exponents run from 3 down to 0 ✓

d) $48059_{10}$

$$48059_{10}=4\cdot 10^4+8\cdot 10^3+0\cdot 10^2+5\cdot 10^1+9\cdot 10^0$$

The number has 5 digits, so the exponents run from 4 down to 0 ✓

Hexadecimal number system

The weights are the powers of sixteen, from $16^0=1$ at the rightmost position upwards. The letters count with their values.

a) ${\mathrm{B}}_{16}$

$${\mathrm{B}}_{16}=11\cdot 16^0$$

The number has a single digit, so its only weight is $16^0=1$; B = 11 ✓

b) ${3C}_{16}$

$${3C}_{16}=3\cdot 16^1+12\cdot 16^0$$

The number has 2 digits, so the exponents run from 1 down to 0; C = 12 ✓

c) ${A07}_{16}$

$${A07}_{16}=10\cdot 16^2+0\cdot 16^1+7\cdot 16^0$$

The number has 3 digits, so the exponents run from 2 down to 0; A = 10 ✓

d) ${E5B2}_{16}$

$${E5B2}_{16}=14\cdot 16^3+5\cdot 16^2+11\cdot 16^1+2\cdot 16^0$$

The number has 4 digits, so the exponents run from 3 down to 0; E = 14, B = 11 ✓