Origin of Units.
Whole numbers and fractions (pure numbers without units) may appear in
counting as number or multipliers to describe many items are present.
For example, 256 represent the idea that a set of objects
may be grouped into 2 sets of one hundred, 5 sets of 10 and 6 objects left
over - a set of six perhaps. Two people may reach the same decimal
description or count 256 via groupings the object differently. The
object that appears in one person's first group of one hundred may appear in
another person's second group or in one of the sets of 10 or 6 the other forms
in the count, sets formed explicitly or not. But we assume (a counting
principle or practice) that any two people counting a set of 256 objects will
reach the same decimal description 256, but not necessarily with the same
grouping of the objects into sets of 100, 10 and the 6 leftover.
Simple units of
measurement may appear to identify what we a counting.
5 apples, 10 oranges, 10.5 centimeters, 5.34 kilograms, 10
degrees Celsius (temperature measure), 90 degrees (angle measure)
A number times (or written besides) a unit of measurement is called a quantity.
In daily life, science and technology, there are systems of
measurements for length, mass, time, money, and so on
Measurement systems in the physical involve units of length, mass and time:
- cgs system: centimeters, grams and seconds
- mks system: meters, kilograms and seconds
- imperial system(?): feet, slugs and seconds.
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 rates and further proportionality constants.
Pure mathematics deals only with pure numbers in what is called dimensionless or
context-free manner that leads to a separation of mathematics from
motivations and considerations that may lead to false conclusions. That
is not say, the logic in pure mathematics is perfect. Problems still remain.
You can investigate them if you become a mathematician.
Addition and Subtraction of Quantities
(Symbolic or Algebraic shorthand description/form)
When we have 5 apples and 6 bananas and 1 orange in a bag, the
expression
5 apples + 6 bananas + 1 orange
represent this collection of objects or fruit. The units here apples, bananas
and oranges. We write units in singular or plural form in accordance with
language rules. But in writing expressions, we do not care or distinguish units
written in singular or plural form. So in our calculation with units, we
write 5 penny means the same as 5 pennies. The expression
looks like a sum. Now from bag of 5 apples and 6 bananas and 1 orange in a bag,
we may remove 2 apples, 3 bananas and 1 orange. The result would be
3 apples, 3 bananas and zero oranges. We may write the foregoing in
shorthand form (algebraic or symbolic form) as
(5 apples + 6 bananas + 1 orange) - (2 apples + 3 bananas + 1 orange)
= (5-2) apples + (6-3) bananas + (1-1) oranges
= 3 apples + 3 bananas + 0 oranges
= 3 apples + 3 bananas
In this subtraction we are combining like terms: those involving apples,
bananas and oranges, respectively.
On the other hand if I have 10 dimes and 4 pennies and you have 6
dimes and 20 pennies, together we have
(10 dimes + 4 pennies) + (6 dimes + 20 pennies) = 16
dimes + 24 pennies.
This addition combines like terms - terms with the same units. A dime
is coin worth ten pennies. Changing dimes in pennies or vice-versa is optional
here.
In general, for a units of a quantity plus another b units
of the same quantity together give (a+b) units of the same quantity. Symbolically, we write
a units + b units = (a+b) units
provided of course the unit of measurement in all terms are identical. In the
same circumstances,
a units - b units = (a-b) units
Thee grouping represents the distributive law for working with numbers
and quantities.
Remark: In primary school mathematics, student
may have learnt or accepted that 2 + 3 = 5 from the question of how to
describe the result of combing 2 units and 3 units gives 5 units, simply by
counting how may units there are in total. Student met examples like the
following drawn instead of written
- 2 rabbits plus 3 rabbits give 5 rabbits (by
counting)
- 2 dots plus 3 dots give 5 dots (by counting)
- 2 pies plus 3 pies give 5pies (by counting)
Many examples like this for the pair of digits (multipliers)
2 and 3 may lead students to accept or proclain that 2 units + 3 units = 5
units for like units, or 2 ones plus 3 ones = 5 ones (here one serve as a
pronoun for a unit - so we could also write 2 its + 3 its =5 its) and
finally, arrive at 2 + 3 = 5. Similar considerations lead use as
young students to fill in the addition table for all pairs of digits 0 to
9. That process along with decimal value notation leads to addition, comparison,
subtraction and comparision of quantities and pure numbers. See the
development of arithmetic skills and concept in site pages.
Examples with like units
- 5 kilogram + 4.5 kilograms = 9.5 kilograms
- 4 hours + 8 hours = 12 hours
- 20 seconds - 16 seconds = 4 seconds
- 8 centimeters + 6 centimeters + 2 centimeters = (8+6 +2) centimeters =
16 centimeters
- 10 meters - 18 meters = - 8 meters
The last makes sense if 1 meter represented one step to the right and
-1 meter represented one step to the left.
Examples with unlike units (Read + as and)
- (8 apples + 4 pennies) + ( 3 pennies + 2 apples) = (8+2) apples +
(4+2) pennies = 10 apples + 7 pennies
- (4 oranges + 3 bananas + 2 lemons) + (2 oranges + 3 lemons) = 6 oranges
+ 3 bananas + 5 lemons
These calculations symbolically represent the addition of stocks of different
kinds of fruit in one calculation involving unlike units instead of separate
calculations involving like units. That being said, the counting of apples,
oranges and so on would as a matter of practice be done in separate calculations
(separate lines) where only one kind of unit appears.
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