1.1 Real numbers: algebra essentials  (Page 3/35)

 Page 3 / 35

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

1. $\frac{7}{77}$
2. $\sqrt{81}$
3. $4.27027002700027\dots$
4. $\frac{91}{13}$
5. $\sqrt{39}$
1. rational and repeating;
2. rational and terminating;
3. irrational;
4. rational and repeating;
5. irrational

Real numbers

Given any number n , we know that n is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers    . As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or –). Zero is considered neither positive nor negative.

The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of 0. Any real number corresponds to a unique position on the number line.The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the real number line    as shown in [link] .

Classifying real numbers

Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?

1. $-\frac{10}{3}$
2. $\sqrt{5}$
3. $-\sqrt{289}$
4. $-6\pi$
5. $0.615384615384\dots$
1. $-\frac{10}{3}\text{\hspace{0.17em}}$ is negative and rational. It lies to the left of 0 on the number line.
2. $\sqrt{5}\text{\hspace{0.17em}}$ is positive and irrational. It lies to the right of 0.
3. $-\sqrt{289}=-\sqrt{{17}^{2}}=-17\text{\hspace{0.17em}}$ is negative and rational. It lies to the left of 0.
4. $-6\pi \text{\hspace{0.17em}}$ is negative and irrational. It lies to the left of 0.
5. $0.615384615384\dots \text{\hspace{0.17em}}$ is a repeating decimal so it is rational and positive. It lies to the right of 0.

Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?

1. $\sqrt{73}$
2. $-11.411411411\dots$
3. $\frac{47}{19}$
4. $-\frac{\sqrt{5}}{2}$
5. $6.210735$
1. positive, irrational; right
2. negative, rational; left
3. positive, rational; right
4. negative, irrational; left
5. positive, rational; right

Sets of numbers as subsets

Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram, such as [link] .

Sets of numbers

The set of natural numbers    includes the numbers used for counting: $\text{\hspace{0.17em}}\left\{1,2,3,...\right\}.$

The set of whole numbers    is the set of natural numbers plus zero: $\text{\hspace{0.17em}}\left\{0,1,2,3,...\right\}.$

The set of integers    adds the negative natural numbers to the set of whole numbers: $\text{\hspace{0.17em}}\left\{...,-3,-2,-1,0,1,2,3,...\right\}.$

The set of rational numbers    includes fractions written as

The set of irrational numbers    is the set of numbers that are not rational, are nonrepeating, and are nonterminating:

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Miranda
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Miranda Drice
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Miranda
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Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
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Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
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Jeffrey
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Jeffrey
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Miranda
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Jeffrey
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Miranda
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Miranda
Jeffrey
Jeffrey
Miranda
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Miranda
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Steve
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Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
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Miranda
what is the solution of the given equation?
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Miranda
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Jeffrey
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
cos(- z)=cos z
Mustafa
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
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Abdel
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Nokwanda
C'est comment
Abdel
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SORIE
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SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function