7 Arithmetic and Fractions with Units
1 Addition and Subtraction with Units
2 Equality and Units
3 Multiplying Units and Numbers
4 Fractions with Units
5 Reciprocals and Division for Fractions with Units
6 Simplification of Fractions with Units
7 Converting or Changing Units
Notes
In my school days, mathematics courses were too pure to
mention or show how to work with units in arithmetic and algebraic
calculations. So calculations with units in chemistry and physic course
appeared without sanction from my mathematics courses. But mathematics
courses should support the use of units, and calculations with units
may appear in the application of trigonometry and the development of
calculus. This subsection of the fraction folder provides a
remedy.
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Arithmetic and algebra with units represents alternate title for this
subject of fractions with units.
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An operational command of calculations with units could be
sufficient for further use in the representation of rates and
proportionality constants and for further use in calculation in
chemistry, physics and money matters. In this arithmetic with units
of measurement, products and quotients of monomials may be formed
and simplified with monomials that contain units to unlike powers.
In contrast, sums and difference of monomials to have a physical
context may only be formed and simplified only when the monomials
are real multiples of each other. Inclusion of this topic will help
later in examples of exponent addition and subtraction with
monomials in one to several variables x, y, z, ... and in their
products or quotients.
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Unit of measurement are part of applied mathematics and external to pure
mathematics. Yet calculations involving units of weight, mass, length,
time, money (hand it over) and so on appear in the measurements and
calculations of daily life alone or as part of calculations with rates
and proportionality constants.
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Origin and
Addition of Some Units: Measurements system appear in daily life,
science and technology. They involve calculation with units. This
lesson introduce standalone units, and indicates how to add subtract
multiplies of them in a way that resembles the forward and backward use
of a distributive law for counting and measuring.
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Units and Equal Signs:
Mathematical practice requires the equality sign to have the "forward
and backward" reflective property that a = b when and only when b = a.
Here may read the equality sign to mean the same as or is equivalent
to. Yet in daily life, the statement 3 apples + 4 oranges is, gives or
equals 7 fruits, departs from this practice. We are not going change
the habits of daily life, but should be aware of the difference here
between the technical and common use of the word equals or the symbol =
for it.
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Products of Units
and Numbers: The first part of this lesson shows how to add
subtract numerical multiplies of them in a way that resembles the
forward and backward use of a distributive law for counting and
measuring. Carries and borrows in addition and subtraction provide a
unit-free form of conversion. In counting and arithmetic, conversions
are further present in expressing counts or numbers in groups of ones,
tens, hundreds and so on, and in adding, subtracting and multiplying
such counts. The second part of this lesson talks about
changing and converting the units used to measure or keep track
of quantities. Your fortune may be measured in pennies. or in dollars?
The third part show how to form products of quantities. The
operations here are similar to work with monomials where the product of
4xy with 5 xy3z is 20 x2y4z. Instead
of using letters x, y and z, we use measurement units and counting
units as well, and could be more meaningful for students.
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Fractions With Units.
This lesson introduces fractions with units and extends the concept of
equivalent fractions to fractions with units. These fractions with
units are analogous to ratios of monomials and their
simplification:
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12 xy3
8 x2y2z
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=
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3 y
2 xz
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Simplification
of Fractions with Units. Simplification of fractions with units is
analogous to to simplification of ratios of monomials (fractions with
monomials) in one to several variables.
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24 yw8
9 w2y4z22
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=
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8
3
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w6
y3z22
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Reciprocals,
Division and Compound Fractions for Fractions with Units. This
two part lesson shows how to (i) write the Reciprocals of Fractions
with Units, and how to (ii) divide Fractions with Units, and how to
(iii) evaluate the corresponding compound fractions where numerators
and denominators are fractions with units. Equivalent fractions
reappear here as part of the simplification of results. The operations
here analogous to calculating reciprocals of ratios of monomials in
variables x, y and z, etc; to dividing such ratios and to evaluating
compound fractions with those ratios as numerators and denominators.
three part lesson shows how to (i) write the Reciprocals of Fractions
with Units, and how to (ii) divide Fractions with Units, and how to
(iii) evaluate the corresponding compound fractions where numerators
and denominators are fractions with units. Equivalent fractions
reappear here as part of the simplification of results. The operations
here anonymous to calculating reciprocals of ratios of monomials in
variables x, y and z, etc; to dividing such ratios and to evaluating
compound fractions with those ratios as numerators and
denominators.
- 50167_Converting_or_Changing_Units.html">Conversion of Units in
Fractions With Units: A physical quantity may be described in different
units. Here we show how speed (it is written as a fraction with units)
written in distance per hour may be expressed in distance per minute. The
trick of multiplying by a fraction with value 1 (a fraction with units
which is equivalent to one) appears here.
To Learn More about how about how fractions with units are treated and
occur in daily life and physics, see the Units in
Calculations pages in site Volume 3: [7 Velocity]
[7
Varying Velocity Example] [7. Velocity
Calculation] [7 Changing
Units] [7 Same Velocity
Motions] [10 Slopes without
Units.] [10 Units &
Slopes] [10 Units in Cost
vs. Quantity] [10 How Units
Appear] [10 Unit
Elimination] [10 Partial
Elimination][10 Interest &
Units] [12 More on
Units]
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In general quantities in daily life involve basic
units of measure, that is of time, length, mass (or weight) and of
money too - an artificial concept. The units and their numerical
multiples may be multiplied together form monomials - expressions
equal to a real number times a product of units to nonnegative
powers. Writing these monomials as numerators and denominators in
fraction like expression (they look like fractions) gives fractions
with units. This section on arithmetic and algebra with units shows
how form and simplify expression for monomials form by products of
units and fractions formed by taking monomials for numerators and
denominators.
Advanced Students: The presentation here is
informal, but it could be codified in a formal way. See the first
or second chapter of Henri Cartan's work below for a model. . That
codification coupled with invariance ideas leads in the physical
sciences to similarity analysis and similarity requirements for
solutions of equations.
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End Notes for Teachers
- In applied mathematics calculations with proportionality, multiples
of simple and compound units will serve as proportionality constants and
rates with physical or social meaning. An operational command of
calculations with units could be sufficient for further use in the
representation of proportionality constants and for further use in
calculation in chemistry, physics and money matters where units
appear.
- Operations with monomials involving units and their quotients are
similar to operations on monomials in variables x, y, z etc and their
quotients. The latter too may represent formal operations on expressions
that have no meaning for students other being marks on paper. In
contrast, the previous note implies a role for polynomial like operations
on units in calculations.
- In the modern axiomization of mathematics as seen in school and
colleges, the codification is limited to pure numbers. Units are not
discussed. But as a service to the physical and social sciences, money
matters included, the algebraic role of units in their computations can
be codified or modeled. That can be done informally. Henri Cartan's work
Elementary theory of analytic functions of one or several complex
variables, translation of THEORIE ELEMENTAIRE DES FONCTIONS ANALYTIC
D;UNE OU PLUSIEURS VARIABLES COMPLEX, could be extended to provide a
formal codification of operation with units.
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