From Zero to Engineer

Number systems - review exercises

A set of exercises covering all four systems: converting to decimal with the Horner scheme, converting a decimal number to duodecimal with the ladders, and the route through binary to hexadecimal.

The three previous lessons gave three tools: the multiply-and-add scheme, the division and multiplication ladders, and the grouping of binary digits. Here we use them together, on numbers written in binary, octal, duodecimal and hexadecimal. Nothing new to memorise - what counts is recognising which tool is the shortest route to the answer in each exercise.

Review exercises

Express the following numbers in the decimal number system.

\text{a)}\quad 1110.11_{2} = 14.75_{10}

Integer part

\begin{aligned}1 \cdot 2 &= 2 \\2 + 1 &= 3 \\3 \cdot 2 &= 6 \\6 + 1 &= 7 \\7 \cdot 2 &= 14 \\14 + 0 &= 14\end{aligned}

Fractional part

\begin{aligned}1 \cdot 2 &= 2 \\2 + 1 &= 3\end{aligned}

3 \cdot 2^{-2} = 3 \cdot \frac{1}{4} = \frac{3}{4} = 0.75

\text{b)}\quad 507.632_{8} = 327.801_{10}

Integer part

\begin{aligned}5 \cdot 8 &= 40 \\40 + 0 &= 40 \\40 \cdot 8 &= 320 \\320 + 7 &= 327\end{aligned}

Fractional part

\begin{aligned}6 \cdot 8 &= 48 \\48 + 3 &= 51 \\51 \cdot 8 &= 408 \\408 + 2 &= 410\end{aligned}

410 \cdot 8^{-3} = 410 \cdot \frac{1}{512} = \frac{410}{512} = 0.801

\text{c)}\quad 345.2\Lambda 7_{12} = 485.247_{10}

Λ = 11

Integer part

\begin{aligned}3 \cdot 12 &= 36 \\36 + 4 &= 40 \\40 \cdot 12 &= 480 \\480 + 5 &= 485\end{aligned}

Fractional part

\begin{aligned}2 \cdot 12 &= 24 \\24 + 11 &= 35 \\35 \cdot 12 &= 420 \\420 + 7 &= 427\end{aligned}

427 \cdot 12^{-3} = 427 \cdot \frac{1}{1728} = \frac{427}{1728} = 0.247

\text{d)}\quad 2B4{,}CA3_{16} = 692.790_{10}

A = 10, B = 11, C = 12

Integer part

\begin{aligned}2 \cdot 16 &= 32 \\32 + 11 &= 43 \\43 \cdot 16 &= 688 \\688 + 4 &= 692\end{aligned}

Fractional part

\begin{aligned}12 \cdot 16 &= 192 \\192 + 10 &= 202 \\202 \cdot 16 &= 3232 \\3232 + 3 &= 3235\end{aligned}

3235 \cdot 16^{-3} = 3235 \cdot \frac{1}{4096} = \frac{3235}{4096} = 0.790

Express 427.362 in duodecimal form

427.362_{10} = 2\Lambda 7.442_{12}

		remainder
427	: 12	7
35	: 12	Λ
2	: 12	2
0		↑

427_{10} = 2Λ7_{12}

	0.362	· 12
4	.344	· 12
4	.128	· 12
1	.536	· 12
6	.432

we multiply only the fractional part

0.362_{10} = 0.442_{12}

The fourth digit decides how the third is rounded: 6 is half of twelve, so 0.4416 is written as 0.442.

Convert 139.825 to its equivalent forms in the octal, binary and hexadecimal systems

139.825_{10} = 213.646_{8} = 10001011.110100110_{2} = 8B{,}D3_{16}

		remainder
139	: 8	3
17	: 8	1
2	: 8	2
0		↑

139_{10} = 213_{8}

	0.825	· 8
6	.600	· 8
4	.800	· 8
6	.400	· 8
3	.200

we multiply only the fractional part

0.825_{10} = 0.646_{8}

010

001

011

110

100

110

0000

1000

1011

1101

0011

0000

Frequently asked questions

What does the symbol Λ mean in the duodecimal system?

Eleven. The duodecimal system needs two symbols beyond the digits 0 to 9: X stands for ten and Λ for eleven. In the number 345.2Λ7 the symbol Λ is the second digit after the point and has the value 11.

Which method should you use to convert a number to decimal?

The scheme from Part 11: multiply by the base and add the next digit. It is shorter than writing out the place values and works the same way in every system, including when the digits are letters.

Why is 0.4416 in the duodecimal system written as 0.442?

Because we round to three digits and the fourth digit is 6 - exactly half of twelve, so the third one goes up from 1 to 2. In the duodecimal system the rounding boundary is 6, not 5 as in decimal.

How do you get a hexadecimal form from a decimal number?

First to octal with the ladders, then each octal digit into three binary digits, and those grouped in fours are read as hexadecimal. For 139.825 that gives 213.646 and then 8B.D3.

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Octal as an intermediate step: from decimal to binary and hexadecimal

Arithmetic - review exercises