Converting scientific notation to engineering notation, review exercises on powers and roots, and giving a result to the appropriate number of significant figures when the data comes from measurement.
Engineering notation is scientific notation with one extra condition: the exponent must be a multiple of three. That way the power matches a prefix - kilo, mega, milli, micro - and the result reads straight off in units. The second part of the lesson is about accuracy: if the data comes from measurement, the result cannot be more precise than the data.
Engineering notation
1) Convert the numbers in scientific notation to engineering notation
a)8.236⋅107=82.36⋅10−1⋅107=82.36⋅106
b)1.624⋅10−4=162.4⋅10−2⋅10−4=162.4⋅10−6
c)4.827⋅104=48.27⋅10−1⋅104=48.27⋅103
d)6.243⋅105=624.3⋅10−2⋅105=624.3⋅103
e)3.274⋅10−2=32.74⋅10−1⋅10−2=32.74⋅10−3
f)5.362⋅10−7=536.2⋅10−2⋅10−7=536.2⋅10−9
In engineering notation, the exponent is restricted to a multiple of 3, for example 10³, 10⁶, 10⁻³, 10⁻⁶ …
2) (4.72 · 10²) · (8.36 · 10⁵)
a)4.72⋅8.36⋅107=39.4592⋅107=3.94592⋅108
scientific notation
b)394.592⋅10−2⋅108=394.592⋅106
engineering notation
Review exercises
Write each of the following numbers as a power
a)58⋅52=510
b)64:66=6−2
c)(74)3=712
d)(19−8)0=190=1
Find the value of each of the following numbers, correct to three decimal places
a)1641=±2
b)33=331=1.442
c)(−8)51=−1.516
d)(−7)41
does not exist
Write each of the following expressions as a single decimal number
a)1.0521⋅103=1052.1
b)0.0135:10−3=0.0135⋅103=13.5
c)123.456⋅10−2=1.23456
d)165.21:104=165.21⋅10−4=0.016521
Write each of the following numbers in scientific notation
a)125.87=1.2587⋅102
b)0.0101=1.01⋅10−2
c)1.345=1.345⋅100
d)10.13=1.013⋅101
Write each of the following numbers in engineering notation
a)1.3204⋅105=132.04⋅10−2⋅105=132.04⋅103
b)0.0101=10.1⋅10−3
c)1.345=1.345⋅100
d)9.5032⋅10−8=95.032⋅10−1⋅10−8=95.032⋅10−9
In each of the following cases, the numbers were obtained by measurement. Give the result of each calculation to the appropriate degree of accuracy
a)13.6:0.012⋅7.63−9015=−367,(6)=−370
to two significant figures
b)13.60.003⋅194=0.042794=0.04
to one significant figure
c)19.3⋅1.042.00=20.87488=20.9
to three significant figures
d)8.6⋅2.9+5.7⋅9.218⋅2.1−3.6⋅0.54=0.46338=0.46
to two significant figures
Frequently asked questions
How does engineering notation differ from scientific notation?
In scientific notation the mantissa lies between one and ten and the exponent can be anything. In engineering notation the exponent must be a multiple of three, so the mantissa can be larger - for example 8.236 times ten to the seventh is written as 82.36 times ten to the sixth.
What are significant figures and why limit a result to them?
They are the digits that carry information about the value, counted from the first non-zero digit. When data comes from measurement the result cannot be more precise than the measurement - so 0.042794 is given as 0.04 if the data had one significant figure.
Why does the fourth root of a negative number not exist?
Because the degree is even, and any real number raised to an even power is non-negative. With an odd degree there is no problem - the fifth root of minus eight is about minus 1.516.
What is a number raised to the power of zero?
Always one, no matter what the base is. That is why nineteen to the power of minus eight, all raised to the power of zero, gives one.
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Engineering notation and accuracy of calculations | PhiBoard