A multiply-and-add scheme that converts a number written in binary, octal, duodecimal or hexadecimal into a decimal number without writing out the powers of the base.

The previous lesson converted numbers by writing out the value of every position. There is a shorter way: multiply the first digit by the base, add the next digit, multiply by the base again, and so on to the end. It is the same calculation, only arranged differently - less writing and fewer chances to slip. The fractional part works the same way, and at the end you divide the result by the base raised to the number of digits after the point.

Converting numbers from different number systems to decimal using an alternative method

\text{1)}\quad 357.121_{8} = 239.1582_{10}

Integer part

\begin{aligned}3 \cdot 8 &= 24 \\24 + 5 &= 29 \\29 \cdot 8 &= 232 \\232 + 7 &= 239\end{aligned}

Fractional part

\begin{aligned}1 \cdot 8 &= 8 \\8 + 2 &= 10 \\10 \cdot 8 &= 80 \\80 + 1 &= 81\end{aligned}

81 \cdot 8^{-3} = 81 \cdot \frac{1}{512} = \frac{81}{512} = 0.1582

correct to four decimal places

\text{2)}\quad 245.136_{12} = 341.1076_{10}

Integer part

\begin{aligned}2 \cdot 12 &= 24 \\24 + 4 &= 28 \\28 \cdot 12 &= 336 \\336 + 5 &= 341\end{aligned}

Fractional part

\begin{aligned}1 \cdot 12 &= 12 \\12 + 3 &= 15 \\15 \cdot 12 &= 180 \\180 + 6 &= 186\end{aligned}

186 \cdot 12^{-3} = 186 \cdot \frac{1}{1728} = \frac{186}{1728} = 0.1076

correct to four decimal places

\text{3)}\quad 11011.1011_{2} = 27.6875_{10}

Integer part

\begin{aligned}1 \cdot 2 &= 2 \\2 + 1 &= 3 \\3 \cdot 2 &= 6 \\6 + 0 &= 6 \\6 \cdot 2 &= 12 \\12 + 1 &= 13 \\13 \cdot 2 &= 26 \\26 + 1 &= 27\end{aligned}

Fractional part

\begin{aligned}1 \cdot 2 &= 2 \\2 + 0 &= 2 \\2 \cdot 2 &= 4 \\4 + 1 &= 5 \\5 \cdot 2 &= 10 \\10 + 1 &= 11\end{aligned}

11 \cdot 2^{-4} = 11 \cdot \frac{1}{16} = \frac{11}{16} = 0.6875

\text{4)}\quad 4C5.2B8_{16} = 1221.1699_{10}

Integer part

\begin{aligned}4 \cdot 16 &= 64 \\64 + 12 &= 76 \\76 \cdot 16 &= 1216 \\1216 + 5 &= 1221\end{aligned}

Fractional part

\begin{aligned}2 \cdot 16 &= 32 \\32 + 11 &= 43 \\43 \cdot 16 &= 688 \\688 + 8 &= 696\end{aligned}

696 \cdot 16^{-3} = 696 \cdot \frac{1}{4096} = \frac{696}{4096} = 0.1699

correct to four decimal places

Review exercises

Express each of the following numbers in the decimal number system, correct to three decimal places.

\text{1)}\quad 11001.11_{2} = 25.750_{10}

Integer part

\begin{aligned}1 \cdot 2 &= 2 \\2 + 1 &= 3 \\3 \cdot 2 &= 6 \\6 + 0 &= 6 \\6 \cdot 2 &= 12 \\12 + 0 &= 12 \\12 \cdot 2 &= 24 \\24 + 1 &= 25\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} = 0.750

correct to three decimal places

\text{2)}\quad 776.143_{8} = 510.193_{10}

Integer part

\begin{aligned}7 \cdot 8 &= 56 \\56 + 7 &= 63 \\63 \cdot 8 &= 504 \\504 + 6 &= 510\end{aligned}

Fractional part

\begin{aligned}1 \cdot 8 &= 8 \\8 + 4 &= 12 \\12 \cdot 8 &= 96 \\96 + 3 &= 99\end{aligned}

99 \cdot 8^{-3} = 99 \cdot \frac{1}{512} = \frac{99}{512} = 0.193

correct to three decimal places

\text{3)}\quad 4X9.2\Lambda 5_{12} = 705.246_{10}

Integer part

\begin{aligned}4 \cdot 12 &= 48 \\48 + 10 &= 58 \\58 \cdot 12 &= 696 \\696 + 9 &= 705\end{aligned}

Fractional part

\begin{aligned}2 \cdot 12 &= 24 \\24 + 11 &= 35 \\35 \cdot 12 &= 420 \\420 + 5 &= 425\end{aligned}

425 \cdot 12^{-3} = 425 \cdot \frac{1}{1728} = \frac{425}{1728} = 0.246

correct to three decimal places

\text{4)}\quad 6F8.3D5_{16} = 1784.240_{10}

Integer part

\begin{aligned}6 \cdot 16 &= 96 \\96 + 15 &= 111 \\111 \cdot 16 &= 1776 \\1776 + 8 &= 1784\end{aligned}

Fractional part

\begin{aligned}3 \cdot 16 &= 48 \\48 + 13 &= 61 \\61 \cdot 16 &= 976 \\976 + 5 &= 981\end{aligned}

981 \cdot 16^{-3} = 981 \cdot \frac{1}{4096} = \frac{981}{4096} = 0.240

correct to three decimal places

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Frequently asked questions

What is the alternative method of converting to the decimal system?

Take the first digit, multiply it by the base and add the second digit. Multiply the result by the base again and add the third digit, and so on to the last one. For 357 in base eight that gives 3 · 8 = 24, 24 + 5 = 29, 29 · 8 = 232, 232 + 7 = 239.

How do you handle the part after the point with this method?

Run the digits after the point through the same scheme, then multiply the result by the base raised to minus the number of those digits. For 121 in base eight that gives 81, and 81 · 8⁻³ is 81/512, which is 0.1582 correct to four decimal places.

Why is this method faster than writing out the powers?

Because you never need to know or compute the place values. Instead of multiplying each digit by a different power of the base, you only multiply by the base itself and add - operations you can do in your head.

Is the result always exact?

The integer part always is. The fractional part is sometimes a fraction with a non-terminating expansion - then we give it to a fixed accuracy, for example four decimal places. When the denominator is a power of two, as in 11/16, the expansion terminates and the equality is exact.

A lesson from the PhiBoard knowledge base: phiboard.com/en/knowledge-base/from-zero-to-engineer/number-systems-alternative-method. On the lesson page you will also find games and practice exercises - free and without an account.

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Number systems: binary, octal, duodecimal and hexadecimal

Converting a decimal number to another number system