Download Numbers - Dalton State

Number System
Do not train children to learning by
force and harshness, but direct them
to it by what amuses their minds, so
that you may be better able to
discover with accuracy the peculiar
bent of the genius of each.
~ Plato
Numbers
Numbers come from several different families.
The simplest are the NATURAL NUMBERS,
made up of normal counting numbers,
1, 2, 3, …
Numbers
What happens when we have no more?
ZERO !
These are called the WHOLE NUMBERS
So now we have 0, 1, 2, 3, etc. but we are not yet
quite complete.
Numbers
If we count down from, say, three, we get
3
2
1
0
-1
-2
-3
Now what?
Numbers
We often think of these numbers as arranged
along a line:
… -3 -2 -1 0 +1 +2 +3…
This line goes off as far as we like (to infinity) in
either direction.
We call all the numbers on this line the
INTEGERS
Numbers
There are lots of numbers which occur in between
the integers – all the fractions,
for example
½, -¾, 0.317, 2 1/3 etc.
All the fractions and integers together are called
RATIONAL NUMBERS, because they can all be
written as ratios of whole numbers. Ratio is just an
old word for fraction.
Numbers
There are other numbers which cannot be
represented by a fraction (unless we use an
infinite number of decimal places).
These are called IRRATIONAL NUMBERS and
some you will be familiar with:
√2 = 1.4142…
√3 = 1.732…
π = 3.14159… (This is special type of irrational,
called a transcendental number)
Numbers
All of these groups:
Naturals
N
Wholes
W
Integers
Z
Rationals Q
Irrationals I
when added together make up the
REAL NUMBERS
Numbers
There is one final class of numbers whose members
are not all in the REAL group and these are the
COMPLEX NUMBERS
which include things like √-1 or the square root of any
other negative number.
Numbers
You may not like complex numbers to start
with, but, like the real numbers, they are
extremely useful in calculations.
We would probably have no electricity or
certainly no electronic gadgets (cellphones,
computers etc.) if people did not use complex
numbers in their design.
Number Line
We can picture integers as equally spaced
points on a line called the number line.
0
1
2
3
4
5
A whole number is graphed by placing a dot on
the number line. The graph of 4 is shown.
Comparing Numbers
For any two numbers graphed on a number line,
the number to the right is the greater number, and
the number to the left is the smaller number.
0
1
2
3
4
5
2 is to the left of 5, so 2 is less than 5
5 is to the right of 2, so 5 is greater than 2
Comparing Numbers
0
1
2
3
4
5
2 is to the left of 5, so 2 is less than 5
5 is to the right of 2, so 5 is greater than 2
2 is less than 5
is written as
2<5
5 is greater than 2
is written as
5>2
Helpful Hint
One way to remember the meaning of the
inequality symbols < and > is to think of them
as arrowheads “pointing” toward the smaller
number.
For example,
2 < 5 and 5 > 2
are both true statements.
Comparing Numbers
The integer –5 is to the left of –3,
so –5 is less than –3
-5 < -3
Since –3 is to the right of –5,
–3 is greater than –5.
–6
–5
–4
–3
–2
–1
0
1
-3 > -5
2
3
4
5
6
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Absolute Value
The absolute value of a number is the number’s
distance from 0 on the number line. The symbol for
absolute value is | |.
2
–6
–6
–5
is 2 because 2 is 2 units from 0.
–4
–3
–2
–1
0
1
2
2
is 2 because –2 is 2 units from 0.
–5
–4
–3
–2
–1
0
1
2
3
3
4
4
5
5
6
6
Helpful Hint
Since the absolute value of a number is that
number’s distance from 0, the absolute value of
a number is always 0 or positive. It is never
negative.
0 =0
zero
6 =6
a positive number
Opposite Numbers
Two numbers that are the same distance from 0
on the number line but are on the opposite sides of
0 are called opposites.
5 units
–6
–5
–4
–3
–2
5 units
–1
0
1
2
3
5 and –5 are opposites.
4
5
6
Opposite Numbers
5 is the opposite of –5 and –5 is the opposite of 5.
is – 4 is written as
The opposite of 4
–(4) = –4
The opposite of – 4
–(– 4) =
is 4 is written as
4
–(–4) = 4
If a is a number, then –(– a) = a.
Opposites
Remember that 0 is neither positive nor
negative.
Therefore, the opposite of 0 is 0.