Download PERCENTAGES AND DECIMALS

Real Numbers
Another Look at the Decimals.
We start with the integers: ...-3,-2,-1,0,1,2,3,...
Between each pair we put nine new numbers: 2, 2.1,2.1,2.3,2.4,2.5,2.6,2.7,2.8,2.9,3
Just to be consistent we give each old number an alternative name: 0.0, 1.0, 2.0, etc.
Now we repeat this process. Between each pair of numbers we put nine new ones:
2.1,2.11,2.12,2.13,2.14,2.15,2.16,2.17,2.18,2.19,2.2
To be consistent we give every existing number a new name with two digits after the
decimal: 0.00, 2.00, 3.50, etc.
2
2.0
2.1
2.00
2.10
2.11
now has 3 names
now has 2 names
so far has 1 name
Now we repeat the process, infinitely often. At the end of this, every number has infinitely
many other numbers between it and what was created as its neighbour. Every number has
infinitely many names, differing only in the number of zeros after the decimal point. All
the names are finite in length.
2
=
2.0
2.1
=
=
2.00 =
2.10 =
2.11
=
2.000 =
2.100 =
2.110 =
2.111 =
2.0000 ...
2.1000 ...
2.1100 ...
2.1110 ...
...
We try to find a name for 1/3 in this number line – but we can’t. 0.3, 0.33, 0.333, ... are all
too small and 0.4, 0.34, 0.334, ... are all too big. The only rational numbers we can find
have denominators of the form 2a5b.
We can find the “address where 1/3 ought to live”, though. It has the form 0.333...
where the 3’s go on forever. As long as we keep writing down 3’s we get closer to 1/3, but if
we ever write down a 4 we’re past it.
We cannot write down all the digits, and while “0.333...” suggests continuing for
ever, it is an equally valid approximation to 0.333131313 and many other things. To specify
the decimal expansion unambiguously, we can either write out a rule (“all the digits are 3”)
or use an overbar which
is understood to mean “repeat this forever”: 0.3
(Technically this overbar is called a “vinculum”, from the Latin word meaning “chain” or
“fetter”)
Every rational number that isn’t represented by a terminating decimal has a repeating
decimal form which can be found by long division; and every repeating decimal represents a
rational number.
Once more, we can add a new name to the old numbers to keep them consistent. Any
terminating decimal can be written either with a repeating 0 or a repeating 9:
1.2 = 1.20 = 1.19
When we looked to see where 1/3 fit into the terminating decimals, we found that it
“lived in a gap”, between better and better decimal approximations:
3 x 0.3 = 0.9, 3 x 0.33 = 0.99, 3 x 0.333 = 0.999...
Let’s try to find 2 in the same way.
12 = 1 <
2
<
4 = 22
1.42 = 1.96 < 2 < 2.25 = 1.52
1.412 =1.9881 < 2 < 2.0164 = 1.422
Continuing, IF 2 exists, it’s 1.4121....Does this repeat?
(Is it rational)?
NO! Proof (dating back to Ancient Greece):
Suppose 2 has a rational representation. Then it must have one in lowest terms, p/q. The
numerator and denominator have no common factor.
Then (p/q)2 = p2/q2 = 2
p2 = 2 q2
Then p must be even, so p2 is divisible by 4; its prime factorization contains 2 2 or a higher
power.
But q has no “2” in its prime factorization. So 2q2 has only a single “2” in its prime
factorization.
But then p2 and 2 q2 can’t be equal (contradiction!) We must have been wrong when we
assumed 2 was rational.
If we want 2 to have a square root (or a unit square’s diagonal to have a length) we
need to extend our number system again! There are various ways in which we can do this.
The simplest (but not the smallest) involves assigning a number to EVERY infinite decimal
expansion. This time we don’t have any equivalence relation except the one we’ve already
seen that 0.999... = 1.000... (etc) which only affects terminating decimals anyhow.
Finding irrational real numbers isn’t easy!
-roots of numbers that aren’t perfect squares
-trig functions of most angles
-e (the base of the natural logs)
-logs of most numbers
-
-decimals designed not to repeat:
0.1101001000100001000001....
Nevertheless: There are LOTS of real numbers! In fact, there are more of them than
there are rational numbers.
There exists a bijection between the rational numbers and the whole numbers (“we
can count the rationals”)
There is no bijection between the real numbers and the whole numbers (“we cannot
count the reals”)
However, most of them can’t be described!
Significant Figures and Precision
In some contexts when we use the number “3” we mean “exactly 3, no more and no less”.
*three apples, three pens, three people...
In other contexts, we mean “about 3”.
*three kilograms, three litres, three cups of flour...
If we mean “three kilograms as opposed to four” we write 3 kg
If we mean “three kilograms as opposed to 3.1” we write 3.0 kg
Similarly we can distinguish between 3.5 kg, 3.50 kg, and 3.500 kg.
We need a good way to distinguish between 1000 kg (as opposed to 2000kg), 1000 kg (as
opposed to 1100 or 900) or 1000 kg (as opposed to 1010 or 990).
Solution: scientific notation.