Download Natural Numbers

1.
1
3
(30x  2) 
5
5
2. -15 – 6 – [-4 – (-6)]
3. a2 + b – c
when a = -6, b = 5, c = -3
1
6x 
5
 23
44
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Math 8H
1-8 Number Systems
Algebra 1
Glencoe McGraw-Hill
JoAnn Evans
What are real numbers?
Real Numbers
Natural
Numbers
Natural Numbers:
“Counting Numbers”
1, 2, 3, 4, 5, …
Whole
Numbers
Natural
Numbers
Whole Numbers:
Natural Numbers and 0
0, 1, 2, 3, 4, …
Integers
Whole
Numbers
Natural
Numbers
Integers:
Positive and Negative
Whole Numbers
…-3, -2, -1, 0, 1, 2,…
Rational Numbers
Integers
Whole
Numbers
Natural
Numbers
Rational Numbers
can be written as:
a where a & b are
b int egers and (b  0)
When written as
decimals they
either repeat or
terminate.
1
 0.333... repeats
3
1
 .25 ter min ates
4
Rational Numbers
Integers
Whole
Numbers
Natural
Numbers
Irrational Numbers
Ex:  and 2
These must be
represented by a
symbol (ex: ),
or as a rounded
number, or in
radical form because
the decimal doesn’t
repeat or terminate
(stop).
Rational and Irrational Numbers are…
Real Numbers
Rational Numbers
Integers
Whole
Numbers
Natural
Numbers
Irrational Numbers
So what isn’t a real number?
•
•
When you divide by zero and get no
solution 
1 = i
(imaginary numbers)
Do you remember the closure property?
A set of numbers is CLOSED
under an operation if the
result of the operation (the
answer) is in the same number
set as the two numbers used in
the operation.
Determine whether each set of numbers
is closed under the indicated operation:
Is the set of whole numbers closed under
the operation of multiplication?
(When you multiply a
whole number times another whole number,
is the answer always a whole number?)
Remember, the whole numbers are 0, 1, 2, 3, …
Closed
Is the set of integers closed under
the operation of division?
(When you divide an
integer by another integer,
is the answer always an integer?)
Remember, integers are all positive and negative whole
numbers, including 0.
4
Not closed. A counterexample:
 0.8
5
Is the set of irrational numbers closed under
the operation of multiplication?
(When you multiply an
irrational number times
another irrational number,
is the answer always an irrational number?)
Remember, irrational numbers are numbers like pi or nonrepeating and non-terminating decimals.
Not closed. A counterexample to try on
your calculator:
2 2  2
Name the set of numbers to which each real number belongs.
Example 1
6
11
Example 2
rational
irrational
 2
Example 3
81
Example 4
5

1
natural
rational
whole
number
integer
integer
rational
Square roots are written with a radical
symbol
. The number or expression
inside a radical symbol is called the
radicand.
radical
symbol
9
radicand
A square root is one of two equal
factors of a number.
33=9
The positive square root of 9 is 3.
93
-3  -3 = 9
The negative square root of 9 is -3.
 9  3
4  2


4  2
225  15
The positive square root of 225 is 15.
225  15
The negative square root of 225 is -15.
0  0
4

9

The positive square root of 4 is 2.
The negative square root of 4 is -2.
4
 
9
Zero has only one square root!
4
2

9
3
4
2
 
9
3
4
2
The positive square root of
is .
9
3
2
4
The negative square root of
is  3 .
9
7
Some square roots aren’t whole numbers. The
square roots of numbers that aren’t perfect
squares are IRRATIONAL numbers.
In the case of irrational
numbers, approximate
the square root by
rounding the result to
two decimal places and
replacing the equal sign
with a  sign.
7  2.65
Approximate the square roots of:
11
5
2
43
Negative real numbers do not have square roots
because two negative numbers multiplied produce a
positive number.
9 =
undefined
The square root of a negative radicand is
undefined!
Evaluate each expression:
36
 36
 36
6
-6
6
 36
undefined
To graph a set of numbers means to draw, or plot,
the points named by those numbers on a number
line. The number that corresponds to a point on a
number line is called the coordinate of that point.
-5
-4
•
-3
-2
•
-1
0
1
•
2
3
4
Graph – 4, 5 , – 6 and 0 on a number line.
Order the numbers from least to greatest.
•
-6
-5
•
-4
-3
-2
-1
•
0
5 is an irrational number.
Find its approximation on your
calculator, then place it on the
number line.
– 6, – 4, 0, 5
1
•
2