From Zero to Engineer

Arithmetic - review exercises III

Roots as fractional powers, scientific and engineering notation, significant figures in calculations on measured values, and converting numbers between the decimal, binary, octal, duodecimal and hexadecimal systems.

The last part of the arithmetic review gathers what is used later in technical calculations: a root written as a fractional power, scientific and engineering notation, and the rule of significant figures. At the end the number systems come back - this time all of them at once, in both directions. One exercise deliberately has no answer: an even root of a negative number does not exist among the real numbers.

Arithmetic - review exercises III

Find the value of each of the following numbers, correct to three decimal places.

\text{a)}\quad 11^{\tfrac{1}{4}} = 1.821

\text{b)}\quad \sqrt[7]{3} = 3^{\tfrac{1}{7}} = 1.170

\text{c)}\quad (-81)^{\tfrac{1}{5}} = -2.408

\text{d)}\quad (-81)^{\tfrac{1}{4}}

does not exist

Express in scientific notation.

\text{a)}\quad 537.6 = 5.376 \cdot 10^{2}

\text{b)}\quad 0.364 = 3.64 \cdot 10^{-1}

\text{c)}\quad 4902 = 4.902 \cdot 10^{3}

\text{d)}\quad 0.000125 = 1.25 \cdot 10^{-4}

Convert to engineering notation.

\text{a)}\quad 6.147 \cdot 10^{7} = 61.47 \cdot 10^{-1} \cdot 10^{7} = 61.47 \cdot 10^{6}

\text{b)}\quad 2.439 \cdot 10^{-4} = 243.9 \cdot 10^{-2} \cdot 10^{-4} = 243.9 \cdot 10^{-6}

\text{c)}\quad 5.286 \cdot 10^{5} = 528.6 \cdot 10^{-2} \cdot 10^{5} = 528.6 \cdot 10^{3}

\text{d)}\quad 4.371 \cdot 10^{-7} = 437.1 \cdot 10^{-2} \cdot 10^{-7} = 437.1 \cdot 10^{-9}

Calculate the following product, giving the result in both scientific notation and engineering notation.

(6.43 \cdot 10^{3})(7.35 \cdot 10^{4}) = 6.43 \cdot 7.35 \cdot 10^{3} \cdot 10^{4} = 47.2605 \cdot 10^{7}

4.72605 \cdot 10^{1} \cdot 10^{7} = 4.72605 \cdot 10^{8}

scientific notation

472.605 \cdot 10^{-1} \cdot 10^{7} = 472.605 \cdot 10^{6}

engineering notation

Each of the following parts contains numbers obtained from measurements. Perform the calculations, maintaining the appropriate degree of accuracy.

\text{a)}\quad 18.4^{1.6} \cdot 0.01 = 105.6108448 \cdot 0.01 = 1.056108448 \approx 1

0.01 has one significant figure, so the result is given to one significant figure as well.

\text{b)}\quad \frac{7.632 \cdot 2.14 - 8.32 : 1.1}{16.04} = \frac{16.33248 - 7.56(36)}{16.04} = \frac{8.768844}{16.04} = 0.5466860349 \approx 0.55

The least accurate of the numbers has two significant figures, so the result is rounded to two significant figures.

Express the following numbers in the decimal number system

\text{a)}\quad 1111.11_{2} = 15.75_{10}

Integer part

\begin{aligned}1 \cdot 2 &= 2 \\2 + 1 &= 3 \\3 \cdot 2 &= 6 \\6 + 1 &= 7 \\7 \cdot 2 &= 14 \\14 + 1 &= 15\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 777.701_{8} = 511.877_{10}

Integer part

\begin{aligned}7 \cdot 8 &= 56 \\56 + 7 &= 63 \\63 \cdot 8 &= 504 \\504 + 7 &= 511\end{aligned}

Fractional part

\begin{aligned}7 \cdot 8 &= 56 \\56 + 0 &= 56 \\56 \cdot 8 &= 448 \\448 + 1 &= 449\end{aligned}

449 \cdot 8^{-3} = 449 \cdot \frac{1}{512} = \frac{449}{512} = 0.877

\text{c)}\quad 3\Lambda 3.9\Lambda 1_{12} = 567.827_{10}

Λ = 11

Integer part

\begin{aligned}3 \cdot 12 &= 36 \\36 + 11 &= 47 \\47 \cdot 12 &= 564 \\564 + 3 &= 567\end{aligned}

Fractional part

\begin{aligned}9 \cdot 12 &= 108 \\108 + 11 &= 119 \\119 \cdot 12 &= 1428 \\1428 + 1 &= 1429\end{aligned}

1429 \cdot 12^{-3} = 1429 \cdot \frac{1}{1728} = \frac{1429}{1728} = 0.827

\text{d)}\quad E02{,}FAB_{16} = 3586.979_{10}

A = 10, B = 11, E = 14, F = 15

Integer part

\begin{aligned}14 \cdot 16 &= 224 \\224 + 0 &= 224 \\224 \cdot 16 &= 3584 \\3584 + 2 &= 3586\end{aligned}

Fractional part

\begin{aligned}15 \cdot 16 &= 240 \\240 + 10 &= 250 \\250 \cdot 16 &= 4000 \\4000 + 11 &= 4011\end{aligned}

4011 \cdot 16^{-3} = 4011 \cdot \frac{1}{4096} = \frac{4011}{4096} = 0.979

Convert 19.872 to its equivalent forms in the octal, duodecimal and hexadecimal systems

19.872_{10} = 23.676_{8} = 13{,}DF_{16} = 17{,}X57_{12}

The octal system

		remainder
19	: 8	3
2	: 8	2
0		↑

19_{10} = 23_{8}

	0.872	· 8
6	.976	· 8
7	.808	· 8
6	.464	· 8
3	.712

we multiply only the fractional part

0.872_{10} = 0.676_{8}

The hexadecimal system through the binary form

010

011

110

111

110

0001

0011

1101

1111

0000

The duodecimal system

		remainder
19	: 12	7
1	: 12	1
0		↑

19_{10} = 17_{12}

	0.872	· 12
X	.464	· 12
5	.568	· 12
6	.816	· 12
9	.792

we multiply only the fractional part

0.872_{10} = 0.X57_{12}

The fourth digit decides how the third is rounded: 9 is more than half of twelve, so 0.X569 is written as 0.X57.

Frequently asked questions

Why does the fourth root of −81 not exist?

Because any real number raised to an even power gives a positive result, so no number has a fourth power of −81. An odd root of a negative number does exist: (−81) to the power of one fifth is −2.408.

How does engineering notation differ from scientific notation?

In scientific notation the number before the power is between 1 and 10, while in engineering notation the exponent has to be a multiple of three - which matches the prefixes kilo, mega, milli. That is why 6.147 · 10⁷ is written as 61.47 · 10⁶.

How many significant figures should the result of a calculation on measurements have?

As many as the least accurate of the numbers used. Multiplying by 0.01, a number with one significant figure, gives a result quoted to one significant figure - hence 1.056108448 is written as 1.

How do you convert a decimal number to several systems at once?

The shortest way is through octal: the ladders give the octal form, from which the binary form follows three digits at a time, and grouping those in fours gives hexadecimal. The duodecimal form is worked out separately, because 12 is not a power of two.

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Arithmetic - review exercises II

Algebra: the laws of algebra, like terms and removing brackets