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

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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:

#### Questions & Answers

explain and give four Example hyperbolic function
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
please can go further on polynomials quadratic
Abdullahi
hi mam
Mark
I need quadratic equation link to Alpa Beta
find the value of 2x=32
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
use the y -intercept and slope to sketch the graph of the equation y=6x
how do we prove the quadratic formular
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1