# 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:

#### Questions & Answers

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
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
sure. what is your question?
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice