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E Reflections on Numbers
In this section, you investigated the properties of real numbers. Here
are the names of all the different real numbers.
Natural Numbers
Natural numbers are numbers 1, 2, 3, 4, and so on. The counting
numbers is another name for natural numbers.
, 65, 234, 2.57 106
Examples: 1, 36
Whole Numbers
All natural numbers and zero make up the whole numbers.
, 54, 2.54 104
Examples: 0, 1, 49
Integers
All natural numbers, zero, and the opposite of the natural numbers,
make up integers.
, 0, Examples: –34, –5, 16
1
00 , 32, 99, 2.67 105
Rational Numbers
All numbers that you can write as a ratio of two integers are rational
numbers.
1 , –6, – 1 , –3.7 10-4 , 0, 3.7 10-4 ,
Examples: –22.7, –9 4, – 4
3
1, 1
2 4,
– 2
25 , 22.7, 3.7 104
Irrational Numbers
Numbers that cannot be expressed as rational numbers are called
irrational numbers.
) cannot be written as an integer
Examples: The square root of 10 ( 10
or as a fraction; it is approximately equal to 3.16.
1.
Pi (π) is approximately equal to 3.14 or 3 7
Real Numbers
The rational and irrational numbers combined make up real numbers.
1 , –6, – 1 , –3.7 10 -4, 0,
, – Examples: –22.7, –9 10
4, – 4
3
1, 1
3.7 10 -4, 2 4, π,
52 Revisiting Numbers
, 10
2
25 ,
22.7, 3.7 104
1. Is it possible to end up with a smaller number when you multiply?
Explain and give an example.
2
2. a. Is 3 a rational or irrational number?
1 a rational or irrational number?
b. Is 1
2
c. Is 2 a rational or irrational number?
d. Would the numbers in 2a and 2b be rational if they were
negative?
3. Consider a circle with a radius of 5 centimeters. On his calculator,
Pablo computes the area of this circle as follows.
π 25 Josie uses her calculator to compute:
3.14 25
a. Will Pablo and Josie get the same answer? Why or why not?
b. Will either of them get an exact answer? Why or why not?
a rational or irrational number?
23
4. a. Is a rational or irrational number?
b. Is 49
5. Do negative irrational numbers exist? Explain your answer and
give an example if appropriate.
How are ratios and rational numbers the same? How are they different?
Section E: Reflections on Numbers 53