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# Special Number Sets

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by

Tweet## Jael Harris

on 6 September 2012#### Transcript of Special Number Sets

N: Is the set of Natural or Counting number 0,1,2,3,4

Z: Is the set of all Integers 0,+- 1, +-2, +-3

Q: Is the set of all Rational Numbers or numbers that can be written in the form p/q, where p and q are integers and q doesn't = 0.

R: Is the set of all Real Numbers, which are all numbers which can be placed on a number line. Z+: Is the set of all Positive Integers 1,2,3,4,5,......

Q+: Is the set of all Positive Rational Numbers

R+: Is the set of all Positive Real Numbers. *All terminating and recurring decimal numbers can be shown to be rational Ex: Explain why>

a. Any positive integer is also a rational number

b. -7 is rational number

-----------------------------------------------------------

a. We can write any positive integer as fraction where the number itself is the numerator, and the denominator is 1.

For example, 5=5/1. Therefore, all positive integers are rational numbers.

b. -7 = -7/1, so -7 is rational. Rational Equations:

Ex: Show that 8 and -11 are rational Numbers.

8= 8/1 so therefore it's rational

-11= -11/1 therefore it's rational as well

Ex:Why is 4/0 not a rational number?

4/0 can not be written in t he form p/q because q does not = 0 there this not a rational equation. Ex: Show that the following are rational numbers:

a. 0.47

b.0.135

-------------------------------------------------------------

a. 0.47=47/100, so 0.47 is rational

b. 0.135=135/1000=27/200 so 0.135 is rational *All terminating decimal numbers are rational. Irrational Numbers:

*All real numbers are either rational or irrational.

Irrational Numbers cannot be written in the form p/q eher p and q are intergers, q does not equal 0. The set of irrational numbers id denoted by Q'.

Numbers such as are all irrational. Their decimal expansions neither terminate nor recur. Think about about it Equations:

Prove that p 2 + p 3 is irrational.

Show that the following are rational numbers:

a. 0.444444......

b. 0.212121......

c. 0.325325325....... Answer to "Think about it Equations"

Solution: Set x = p 2 + p 3. Squaring, we ¯nd x 2 = 5 + 2 p 6 and (x 2 ¡ 5) 2 = 24. Thus x is a root of equation x 4 ¡ 10x 2 + 1 = 0. However, according to the Rational Root Theorem, the only possible rational roots of this equation are x = 1; ¡1. Neither 1 nor ¡1 satis¯es the equation, so all its roots are irrational. In particular, p 2 + p 3 is irrational.

a. 0.444444....= 4/9

b. 0.212121....=21/99

c. 0.325325325....= 325/999 SPECIAL NUMBER SETS By: Jael Harris

and

Vivienne Eng THE END!! Take Notes and Pay Attention!!

Full transcriptZ: Is the set of all Integers 0,+- 1, +-2, +-3

Q: Is the set of all Rational Numbers or numbers that can be written in the form p/q, where p and q are integers and q doesn't = 0.

R: Is the set of all Real Numbers, which are all numbers which can be placed on a number line. Z+: Is the set of all Positive Integers 1,2,3,4,5,......

Q+: Is the set of all Positive Rational Numbers

R+: Is the set of all Positive Real Numbers. *All terminating and recurring decimal numbers can be shown to be rational Ex: Explain why>

a. Any positive integer is also a rational number

b. -7 is rational number

-----------------------------------------------------------

a. We can write any positive integer as fraction where the number itself is the numerator, and the denominator is 1.

For example, 5=5/1. Therefore, all positive integers are rational numbers.

b. -7 = -7/1, so -7 is rational. Rational Equations:

Ex: Show that 8 and -11 are rational Numbers.

8= 8/1 so therefore it's rational

-11= -11/1 therefore it's rational as well

Ex:Why is 4/0 not a rational number?

4/0 can not be written in t he form p/q because q does not = 0 there this not a rational equation. Ex: Show that the following are rational numbers:

a. 0.47

b.0.135

-------------------------------------------------------------

a. 0.47=47/100, so 0.47 is rational

b. 0.135=135/1000=27/200 so 0.135 is rational *All terminating decimal numbers are rational. Irrational Numbers:

*All real numbers are either rational or irrational.

Irrational Numbers cannot be written in the form p/q eher p and q are intergers, q does not equal 0. The set of irrational numbers id denoted by Q'.

Numbers such as are all irrational. Their decimal expansions neither terminate nor recur. Think about about it Equations:

Prove that p 2 + p 3 is irrational.

Show that the following are rational numbers:

a. 0.444444......

b. 0.212121......

c. 0.325325325....... Answer to "Think about it Equations"

Solution: Set x = p 2 + p 3. Squaring, we ¯nd x 2 = 5 + 2 p 6 and (x 2 ¡ 5) 2 = 24. Thus x is a root of equation x 4 ¡ 10x 2 + 1 = 0. However, according to the Rational Root Theorem, the only possible rational roots of this equation are x = 1; ¡1. Neither 1 nor ¡1 satis¯es the equation, so all its roots are irrational. In particular, p 2 + p 3 is irrational.

a. 0.444444....= 4/9

b. 0.212121....=21/99

c. 0.325325325....= 325/999 SPECIAL NUMBER SETS By: Jael Harris

and

Vivienne Eng THE END!! Take Notes and Pay Attention!!