maths.free › Algebra › 1. Foundations › The Real Numbers

The Real Numbers

Simplify expressions with square roots

Simplify Expressions with Square Roots

Remember that when a number n is multiplied by itself, we write ${n}^{2}$ and read it “n squared.” The result is called the square of n. For example,

\[\begin{array}{llll}{8}^{2} & & & \text{read}\ \text{‘}8\ \text{squared’} \\ 64 & & & 64\ \text{is called the}\ \text{square}\ \text{of}\ 8.\end{array}\]

Similarly, 121 is the square of 11, because ${11}^{2}$ is 121.

Complete the following table to show the squares of the counting numbers 1 through 15.

The numbers in the second row are called perfect square numbers. It will be helpful to learn to recognize the perfect square numbers.

The squares of the counting numbers are positive numbers. What about the squares of negative numbers? We know that when the signs of two numbers are the same, their product is positive. So the square of any negative number is also positive.

\[\begin{array}{llllllllll}{(-3)}^{2}=9 & & & {(-8)}^{2}=64 & & & {(-11)}^{2}=121 & & & {(-15)}^{2}=225\end{array}\]

Did you notice that these squares are the same as the squares of the positive numbers?

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because ${10}^{2}=100,$ we say 100 is the square of 10. We also say that 10 is a square root of 100. A number whose square is $m$ is called a square root of m.

Example

Try it.

Simplify: ⓐ $\sqrt{25}$ ⓑ $\sqrt{121}.$

Solution

ⓐ Since ${5}^{2}=25$	$\begin{array}{l}\sqrt{25} \\ 5\end{array}$
ⓑ Since ${11}^{2}=121$	$\begin{array}{l}\sqrt{121} \\ 11\end{array}$

Example

Try it.

Simplify: ⓐ $\text{-}\sqrt{9}$ ⓑ $\text{-}\sqrt{144}.$

Solution

ⓐ The negative is in front of the radical sign.	$\begin{array}{l}-\sqrt{9} \\ -3\end{array}$
ⓑ The negative is in front of the radical sign.	$\begin{array}{l}-\sqrt{144} \\ -12\end{array}$

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

We have already described numbers as counting numbers, whole numbers, and integers. What is the difference between these types of numbers?

\[\begin{array}{llllll}\text{Counting numbers} & & & & & 1,2,3,4,\text{\ldots } \\ \text{Whole numbers} & & & & & 0,1,2,3,4,\text{\ldots } \\ \text{Integers} & & & & & \text{\ldots }-3,-2,-1,0,1,2,3,\text{\ldots }\end{array}\]

What type of numbers would we get if we started with all the integers and then included all the fractions? The numbers we would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.

All signed fractions, such as $\frac{4}{5},-\ \frac{7}{8},\frac{13}{4},-\ \frac{20}{3}$ are rational numbers. Each numerator and each denominator is an integer.

Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. Each integer can be written as a ratio of integers in many ways. For example, 3 is equivalent to $\frac{3}{1},\frac{6}{2},\frac{9}{3},\frac{12}{4},\frac{15}{5}\text{\ldots }$

An easy way to write an integer as a ratio of integers is to write it as a fraction with denominator one.

\[\begin{array}{lllllll}3=\frac{3}{1} & & & -8=-\ \frac{8}{1} & & & 0=\frac{0}{1}\end{array}\]

Since any integer can be written as the ratio of two integers, all integers are rational numbers! Remember that the counting numbers and the whole numbers are also integers, and so they, too, are rational.

Example

Try it.

Write as the ratio of two integers: ⓐ $-27$ ⓑ 7.31.

Solution

ⓐ Write it as a fraction with denominator 1.	$\begin{array}{l}-27 \\ \frac{-27}{1}\end{array}$
ⓑ Write it as a mixed number. Remember, 7 is the whole number and the decimal part, 0.31, indicates hundredths. Convert to an improper fraction.	$\begin{array}{l}7.31 \\ 7\frac{31}{100} \\ \frac{731}{100}\end{array}$

So we see that $-27$ and 7.31 are both rational numbers, since they can be written as the ratio of two integers.

\[\begin{array}{lllllllllllllllllllll}\text{Integer} & & & & & -2 & & & -1 & & & 0 & & & 1 & & & 2 & & & 3 \\ \text{Decimal form} & & & & & -2.0 & & & -1.0 & & & 0.0 & & & 1.0 & & & 2.0 & & & 3.0\end{array}\]\[\text{These decimal numbers stop.}\]\[\pi =3.141592654...\]

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Locate Fractions on the Number Line

The last time we looked at the number line, it only had positive and negative integers on it. We now want to include fractions and decimals on it.

Let’s start with fractions and locate $\frac{1}{5},-\ \frac{4}{5},3,\frac{7}{4},-\ \frac{9}{2},-5,\ \text{and}\ \frac{8}{3}$ on the number line.

We’ll start with the whole numbers $3$ and $-5.$ because they are the easiest to plot. See .

The proper fractions listed are $\frac{1}{5}\ \text{and}\ -\ \frac{4}{5}.$ We know the proper fraction $\frac{1}{5}$ has value less than one and so would be located between $\text{0 and 1.}$ The denominator is 5, so we divide the unit from 0 to 1 into 5 equal parts $\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5}.$ We plot $\frac{1}{5}.$ See .

Similarly, $-\ \frac{4}{5}$ is between 0 and $-1.$ After dividing the unit into 5 equal parts we plot $-\ \frac{4}{5}.$ See .

Finally, look at the improper fractions $\frac{7}{4},-\ \frac{9}{2},\frac{8}{3}.$ These are fractions in which the numerator is greater than the denominator. Locating these points may be easier if you change each of them to a mixed number. See .

\[\begin{array}{lllllll}\frac{7}{4}=1\frac{3}{4} & & & -\ \frac{9}{2}=-4\frac{1}{2} & & & \frac{8}{3}=2\frac{2}{3}\end{array}\]

shows the number line with all the points plotted.

Example

Try it.

Locate and label the following on a number line: $4,\frac{3}{4},-\ \frac{1}{4},-3,\frac{6}{5},-\ \frac{5}{2},\ \text{and}\ \frac{7}{3}.$

Solution

Locate and plot the integers, $4,-3.$

Locate the proper fraction $\frac{3}{4}$ first. The fraction $\frac{3}{4}$ is between 0 and 1. Divide the distance between 0 and 1 into four equal parts then, we plot $\frac{3}{4}.$ Similarly plot $-\ \frac{1}{4}.$

Now locate the improper fractions $\frac{6}{5},-\ \frac{5}{2},\frac{7}{3}.$ It is easier to plot them if we convert them to mixed numbers and then plot them as described above: $\frac{6}{5}=1\frac{1}{5},-\ \frac{5}{2}=-2\frac{1}{2},\frac{7}{3}=2\frac{1}{3}.$

a < b “a is less than b” when a is to the left of b on the number line
a > b “a is greater than b” when a is to the right of b on the number line

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Locate Decimals on the Number Line

Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.

Example

Try it.

Locate 0.4 on the number line.

Solution

A proper fraction has value less than one. The decimal number 0.4 is equivalent to $\frac{4}{10},$ a proper fraction, so 0.4 is located between 0 and 1. On a number line, divide the interval between 0 and 1 into 10 equal parts. Now label the parts 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. We write 0 as 0.0 and 1 and 1.0, so that the numbers are consistently in tenths. Finally, mark 0.4 on the number line. See .

Example

Try it.

Locate $-0.74$ on the number line.

Solution

The decimal $-0.74$ is equivalent to $-\ \frac{74}{100},$ so it is located between 0 and $-1.$ On a number line, mark off and label the hundredths in the interval between 0 and $-1.$ See .

Which is larger, 0.04 or 0.40? If you think of this as money, you know that $0.40 (forty cents) is greater than $0.04 (four cents). So,

$0.40>0.04$

Again, we can use the number line to order numbers.

a < b “a is less than b” when a is to the left of b on the number line
a > b “a is greater than b” when a is to the right of b on the number line

Where are 0.04 and 0.40 located on the number line? See .

We see that 0.40 is to the right of 0.04 on the number line. This is another way to demonstrate that 0.40 > 0.04.

How does 0.31 compare to 0.308? This doesn’t translate into money to make it easy to compare. But if we convert 0.31 and 0.308 into fractions, we can tell which is larger.

	0.31	0.308
Convert to fractions.	$\frac{31}{100}$	$\frac{308}{1000}$
We need a common denominator to compare them.
	$\frac{310}{1000}$	$\frac{308}{1000}$

\[\frac{31}{100}=\frac{310}{1000}\ \text{and}\ 0.31=0.310\]

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Key Concepts

Square Root Notation
$\sqrt{m}$ is read ‘the square root of m.’ If $m={n}^{2},$ then $\sqrt{m}=n,$ for $n\ge 0.$
Order Decimals
1. Write the numbers one under the other, lining up the decimal points.
2. Check to see if both numbers have the same number of digits. If not, write zeros at the end of the one with fewer digits to make them match.
3. Compare the numbers as if they were whole numbers.
4. Order the numbers using the appropriate inequality sign.

The Real Numbers

Simplify Expressions with Square Roots

In the following exercises, simplify.

Try it.

$\sqrt{36}$

Solution

Try it.

$\sqrt{4}$

Try it.

$\sqrt{64}$

Solution

Try it.

$\sqrt{169}$

Try it.

$\sqrt{9}$

Solution

Try it.

$\sqrt{16}$

Try it.

$\sqrt{100}$

Solution

Try it.

$\sqrt{144}$

Try it.

$\text{-}\sqrt{4}$

Solution

$-2$

Try it.

$\text{-}\sqrt{100}$

Try it.

$\text{-}\sqrt{1}$

Solution

$-1$

Try it.

$\text{-}\sqrt{121}$

Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

In the following exercises, write as the ratio of two integers.

Try it.

ⓐ 5 ⓑ 3.19

Solution

ⓐ $\frac{5}{1}$ ⓑ $\frac{319}{100}$

Try it.

ⓐ 8 ⓑ 1.61

Try it.

ⓐ $\text{-}12\$ ⓑ 9.279

Solution

ⓐ $\frac{-12}{1}$ ⓑ $\frac{9279}{1000}$

Try it.

ⓐ $\text{-}16$ ⓑ 4.399

In the following exercises, list the ⓐ rational numbers, ⓑ irrational numbers

Try it.

$0.75,0.22\overset{\text{-}}{3},1.39174\text{\ldots }$

Solution

ⓐ $0.75,0.22\overset{\text{-}}{3}$ ⓑ $1.39174\text{\ldots }$

Try it.

$0.36,0.94729\text{\ldots },2.52\overset{\text{-}}{8}$

Try it.

$0.4\overset{\text{-}}{5},1.919293\text{\ldots },3.59$

Solution

ⓐ $0.4\overset{\text{-}}{5},3.59$ ⓑ $1.919293\text{\ldots }$

Try it.

$0.1\overset{\text{-}}{3},0.42982\text{\ldots },1.875$

In the following exercises, identify whether each number is rational or irrational.

Try it.

ⓐ $\sqrt{25}$ ⓑ $\sqrt{30}$

Solution

ⓐ rational ⓑ irrational

Try it.

ⓐ $\sqrt{44}$ ⓑ $\sqrt{49}$

Try it.

ⓐ $\sqrt{164}$ ⓑ $\sqrt{169}$

Solution

ⓐ irrational ⓑ rational

Try it.

ⓐ $\sqrt{225}$ ⓑ $\sqrt{216}$

In the following exercises, identify whether each number is a real number or not a real number.

Try it.

ⓐ $\text{-}\sqrt{81}$ ⓑ $\sqrt{-121}$

Solution

ⓐ real number ⓑ not a real number

Try it.

ⓐ $\text{-}\sqrt{64}$ ⓑ $\sqrt{-9}$

Try it.

ⓐ $\sqrt{-36}$ ⓑ $\text{-}\sqrt{144}$

Solution

ⓐ not a real number ⓑ real number

Try it.

ⓐ $\sqrt{-49}$ ⓑ $\text{-}\sqrt{144}$

In the following exercises, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers.

Try it.

$-8,0,1.95286\text{\ldots },\frac{12}{5},\sqrt{36},9$

Solution

ⓐ $0,\sqrt{36},9$ ⓑ $-8,0,\sqrt{36},9$ ⓒ $-8,0,\frac{12}{5},\sqrt{36},9$ ⓓ $1.95286\text{\ldots }$ ⓔ $-8,0,1.95286\text{\ldots },\frac{12}{5},\sqrt{36},9$

Try it.

$-9,-3\frac{4}{9},\text{-}\sqrt{9},0.40\overset{\text{-}}{9},\frac{11}{6},7$

Try it.

$\text{-}\sqrt{100},-7,-\ \frac{8}{3},-1,0.77,3\frac{1}{4}$

Solution

ⓐ none ⓑ $\text{-}\sqrt{100},-7,-1$ ⓒ $\text{-}\sqrt{100},-7,-\ \frac{8}{3},-1,0.77,3\frac{1}{4}$ ⓓ none ⓔ $\text{-}\sqrt{100},-7,-\ \frac{8}{3},-1,0.77,3\frac{1}{4}$

Try it.

$-6,-\ \frac{5}{2},0,0.\overset{\text{———}}{714285},2\frac{1}{5},\sqrt{14}$

Locate Fractions on the Number Line

In the following exercises, locate the numbers on a number line.

Try it.

$\frac{3}{4},\frac{8}{5},\frac{10}{3}$

Solution

Try it.

$\frac{1}{4},\frac{9}{5},\frac{11}{3}$

Try it.

$\frac{3}{10},\frac{7}{2},\frac{11}{6},4$

Solution

Try it.

$\frac{7}{10},\frac{5}{2},\frac{13}{8},3$

Try it.

$\frac{2}{5},-\ \frac{2}{5}$

Solution

Try it.

$\frac{3}{4},-\ \frac{3}{4}$

Try it.

$\frac{3}{4},-\ \frac{3}{4},1\frac{2}{3},-1\frac{2}{3},\frac{5}{2},-\ \frac{5}{2}$

Solution

Try it.

$\frac{2}{5},-\ \frac{2}{5},1\frac{3}{4},-1\frac{3}{4},\frac{8}{3},-\ \frac{8}{3}$

In the following exercises, order each of the pairs of numbers, using < or >.

Try it.

$-1___-\ \frac{1}{4}$

Solution

Try it.

$-1___-\ \frac{1}{3}$

Try it.

$-2\frac{1}{2}___-3$

Solution

Try it.

$-1\frac{3}{4}___-2$

Try it.

$-\ \frac{5}{12}___-\ \frac{7}{12}$

Solution

Try it.

$-\ \frac{9}{10}___-\ \frac{3}{10}$

Try it.

$-3___-\ \frac{13}{5}$

Solution

Try it.

$-4___-\ \frac{23}{6}$

Locate Decimals on the Number Line In the following exercises, locate the number on the number line.

Try it.

0.8

Solution

Try it.

$-0.9$

Try it.

$-1.6$

Solution

Try it.

3.1

In the following exercises, order each pair of numbers, using < or >.

Try it.

$0.37___0.63$

Solution

Try it.

$0.86___0.69$

Try it.

$0.91___0.901$

Solution

Try it.

$0.415___0.41$

Try it.

$-0.5___-0.3$

Solution

Try it.

$-0.1___-0.4$

Try it.

$-0.62___-0.619$

Solution

Try it.

$-7.31___-7.3$

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Simplify: ⓐ $\sqrt{25}$ ⓑ $\sqrt{121}.$

Bonisa impendulo

ⓐ
Since ${5}^{2}=25$ $\begin{array}{l}\sqrt{25} \\ 5\end{array}$
ⓑ
Since ${11}^{2}=121$ $\begin{array}{l}\sqrt{121} \\ 11\end{array}$
Simplify: ⓐ $\sqrt{36}$ ⓑ $\sqrt{169}.$

Bonisa impendulo

ⓐ 6 ⓑ 13
Simplify: ⓐ $\sqrt{16}$ ⓑ $\sqrt{196}.$

Bonisa impendulo

ⓐ 4 ⓑ 14

Simplify: ⓐ $\text{-}\sqrt{9}$ ⓑ $\text{-}\sqrt{144}.$

Bonisa impendulo

ⓐ The negative is in front of the radical sign.	$\begin{array}{l}-\sqrt{9} \\ -3\end{array}$
ⓑ The negative is in front of the radical sign.	$\begin{array}{l}-\sqrt{144} \\ -12\end{array}$

Simplify: ⓐ $\text{-}\sqrt{4}$ ⓑ $\text{-}\sqrt{225}.$

Bonisa impendulo

ⓐ $-2$ ⓑ $-15$
Simplify: ⓐ $\text{-}\sqrt{81}$ ⓑ $\text{-}\sqrt{100}.$

Bonisa impendulo

ⓐ $-9$ ⓑ $-10$

Write as the ratio of two integers: ⓐ $-27$ ⓑ 7.31.

Bonisa impendulo

ⓐ Write it as a fraction with denominator 1.	$\begin{array}{l}-27 \\ \frac{-27}{1}\end{array}$
ⓑ Write it as a mixed number. Remember, 7 is the whole number and the decimal part, 0.31, indicates hundredths. Convert to an improper fraction.	$\begin{array}{l}7.31 \\ 7\frac{31}{100} \\ \frac{731}{100}\end{array}$

So we see that $-27$ and 7.31 are both rational numbers, since they can be written as the ratio of two integers.

Write as the ratio of two integers: ⓐ $-24$ ⓑ 3.57.

Bonisa impendulo

ⓐ $\frac{-24}{1}$ ⓑ $\frac{357}{100}$
Write as the ratio of two integers: ⓐ $-19$ ⓑ 8.41.

Bonisa impendulo

ⓐ $\frac{-19}{1}$ ⓑ $\frac{841}{100}$

Given the numbers $0.58\overset{\text{-}}{3},0.47,3.605551275...$ list the ⓐ rational numbers ⓑ irrational numbers.

Bonisa impendulo

ⓐ Look for decimals that repeat or stop.	The 3 repeats in $0.58\overset{\text{-}}{3}$. The decimal 0.47 stops after the 7. So $0.58\overset{\text{-}}{3}$ and 0.47 are rational.
ⓑ Look for decimals that neither stop nor repeat.	$3.605551275\text{\ldots }$ has no repeating block of digits and it does not stop. So $3.605551275\text{\ldots }$ is irrational.

For the given numbers list the ⓐ rational numbers ⓑ irrational numbers: $0.29,0.81\overset{\text{-}}{6},2.515115111\text{\ldots }.$

Bonisa impendulo

ⓐ $0.29,0.81\overset{\text{-}}{6}$ ⓑ $2.515115111\text{\ldots }$
For the given numbers list the ⓐ rational numbers ⓑ irrational numbers: $2.6\overset{\text{-}}{3},0.125,0.418302\text{\ldots }$

Bonisa impendulo

ⓐ $2.6\overset{\text{-}}{3},0.125$ ⓑ $0.418302\text{\ldots }$
For each number given, identify whether it is rational or irrational: ⓐ $\sqrt{36}$ ⓑ $\sqrt{44}.$
Bonisa impendulo
1. ⓐ Recognize that 36 is a perfect square, since ${6}^{2}=36.$ So $\sqrt{36}=6,$ therefore $\sqrt{36}$ is rational.
2. ⓑ Remember that ${6}^{2}=36$ and ${7}^{2}=49,$ so 44 is not a perfect square. Therefore, the decimal form of $\sqrt{44}$ will never repeat and never stop, so $\sqrt{44}$ is irrational.
For each number given, identify whether it is rational or irrational: ⓐ $\sqrt{81}$ ⓑ $\sqrt{17}.$

Bonisa impendulo

ⓐ rational ⓑ irrational
For each number given, identify whether it is rational or irrational: ⓐ $\sqrt{116}$ ⓑ $\sqrt{121}.$

Bonisa impendulo

ⓐ irrational ⓑ rational
For each number given, identify whether it is a real number or not a real number: ⓐ $\sqrt{-169}$ ⓑ $\text{-}\sqrt{64}.$
Bonisa impendulo
1. ⓐ There is no real number whose square is $-169.$ Therefore, $\sqrt{-169}$ is not a real number.
2. ⓑ Since the negative is in front of the radical, $\text{-}\sqrt{64}$ is $-8,$ Since $-8$ is a real number, $\text{-}\sqrt{64}$ is a real number.
For each number given, identify whether it is a real number or not a real number: ⓐ $\sqrt{-196}$ ⓑ $\text{-}\sqrt{81}.$

Bonisa impendulo

ⓐ not a real number ⓑ real number
For each number given, identify whether it is a real number or not a real number: ⓐ $\text{-}\sqrt{49}$ ⓑ $\sqrt{-121}.$

Bonisa impendulo

ⓐ real number ⓑ not a real number
Given the numbers $-7,\frac{14}{5},8,\sqrt{5},5.9,\text{-}\sqrt{64},$ list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers.
Bonisa impendulo
1. ⓐ Remember, the whole numbers are 0, 1, 2, 3, … and 8 is the only whole number given.
2. ⓑ The integers are the whole numbers, their opposites, and 0. So the whole number 8 is an integer, and $-7$ is the opposite of a whole number so it is an integer, too. Also, notice that 64 is the square of 8 so $\text{-}\sqrt{64}=-8.$ So the integers are $-7,8,\text{-}\sqrt{64}.$
3. ⓒ Since all integers are rational, then $-7,8,\text{-}\sqrt{64}$ are rational. Rational numbers also include fractions and decimals that repeat or stop, so $\frac{14}{5}\ \text{and}\ 5.9$ are rational. So the list of rational numbers is $-7,\frac{14}{5},8,5.9,-\sqrt{64}.$
4. ⓓ Remember that 5 is not a perfect square, so $\sqrt{5}$ is irrational.
5. ⓔ All the numbers listed are real numbers.
For the given numbers, list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers: $-3,\text{-}\sqrt{2},0.\overset{\text{-}}{3},\frac{9}{5},4,\sqrt{49}.$

Bonisa impendulo

ⓐ $4,\sqrt{49}$ ⓑ $-3,4,\sqrt{49}$ ⓒ $-3,0.\overset{\text{-}}{3},\frac{9}{5},4,\sqrt{49}$ ⓓ $\text{-}\sqrt{2}$ ⓔ $-3,\text{-}\sqrt{2},0.\overset{\text{-}}{3},\frac{9}{5},4,\sqrt{49}$
For the given numbers, list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers: $\text{-}\sqrt{25},-\ \frac{3}{8},-1,6,\sqrt{121},2.041975\text{\ldots }$

Bonisa impendulo

ⓐ $6,\sqrt{121}$ ⓑ $\text{-}\sqrt{25},-1,6,\sqrt{121}$ ⓒ $\text{-}\sqrt{25},-\ \frac{3}{8},-1,6,\sqrt{121}$ ⓓ $2.041975\text{\ldots }$ ⓔ $\text{-}\sqrt{25},-\ \frac{3}{8},-1,6,\sqrt{121},2.041975\text{\ldots }$
Locate and label the following on a number line: $4,\frac{3}{4},-\ \frac{1}{4},-3,\frac{6}{5},-\ \frac{5}{2},\ \text{and}\ \frac{7}{3}.$

Bonisa impendulo

Locate and plot the integers, $4,-3.$
Locate the proper fraction $\frac{3}{4}$ first. The fraction $\frac{3}{4}$ is between 0 and 1. Divide the distance between 0 and 1 into four equal parts then, we plot $\frac{3}{4}.$ Similarly plot $-\ \frac{1}{4}.$
Now locate the improper fractions $\frac{6}{5},-\ \frac{5}{2},\frac{7}{3}.$ It is easier to plot them if we convert them to mixed numbers and then plot them as described above: $\frac{6}{5}=1\frac{1}{5},-\ \frac{5}{2}=-2\frac{1}{2},\frac{7}{3}=2\frac{1}{3}.$
Locate and label the following on a number line: $-1,\frac{1}{3},\frac{6}{5},-\ \frac{7}{4},\frac{9}{2},5,-\ \frac{8}{3}.$

Bonisa impendulo
Locate and label the following on a number line: $-2,\frac{2}{3},\frac{7}{5},-\ \frac{7}{4},\frac{7}{2},3,-\ \frac{7}{3}.$

Bonisa impendulo

Order each of the following pairs of numbers, using < or >. It may be helpful to refer .

Bonisa impendulo

ⓐ $-\ \frac{2}{3}$ is to the right of $-1$ on the number line.	$\begin{array}{l}-\ \frac{2}{3}___-1 \\ -\ \frac{2}{3}>-1\end{array}$
ⓑ $-3\frac{1}{2}$ is to the right of $-3$ on the number line.	$\begin{array}{l}-3\frac{1}{2}___-3 \\ -3\frac{1}{2}<-3\end{array}$
ⓒ $-\ \frac{3}{4}$ is to the right of $-\ \frac{1}{4}$ on the number line.	$\begin{array}{l}-\ \frac{3}{4}___-\ \frac{1}{4} \\ -\ \frac{3}{4}<-\ \frac{1}{4}\end{array}$
ⓓ $-2$ is to the right of $-\ \frac{8}{3}$ on the number line.	$\begin{array}{l}-2___-\ \frac{8}{3} \\ -2>-\ \frac{8}{3}\end{array}$

Order each of the following pairs of numbers, using < or >:
ⓐ $-\ \frac{1}{3}___-1$ ⓑ $-1\frac{1}{2}___-2$ ⓒ $-\ \frac{2}{3}___-\ \frac{1}{3}$ ⓓ $-3___-\ \frac{7}{3}.$

Bonisa impendulo

ⓐ > ⓑ > ⓒ < ⓓ <
Order each of the following pairs of numbers, using < or >:
ⓐ $-1___-\ \frac{2}{3}$ ⓑ $-2\frac{1}{4}___-2$ ⓒ $-\ \frac{3}{5}___-\ \frac{4}{5}$ ⓓ $-4___-\ \frac{10}{3}.$

Bonisa impendulo

ⓐ < ⓑ < ⓒ > ⓓ <
Locate 0.4 on the number line.

Bonisa impendulo

A proper fraction has value less than one. The decimal number 0.4 is equivalent to $\frac{4}{10},$ a proper fraction, so 0.4 is located between 0 and 1. On a number line, divide the interval between 0 and 1 into 10 equal parts. Now label the parts 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. We write 0 as 0.0 and 1 and 1.0, so that the numbers are consistently in tenths. Finally, mark 0.4 on the number line. See .
Locate on the number line: 0.6.

Bonisa impendulo
Locate on the number line: 0.9.

Bonisa impendulo
Locate $-0.74$ on the number line.

Bonisa impendulo

The decimal $-0.74$ is equivalent to $-\ \frac{74}{100},$ so it is located between 0 and $-1.$ On a number line, mark off and label the hundredths in the interval between 0 and $-1.$ See .
Locate on the number line: $-0.6.$

Bonisa impendulo
Locate on the number line: $-0.7.$

Bonisa impendulo

Order $0.64___0.6$ using $<$ or $>.$

Bonisa impendulo

Write the numbers one under the other, lining up the decimal points.	$\begin{array}{l}0.64 \\ 0.6\end{array}$
Add a zero to 0.6 to make it a decimal with 2 decimal places. Now they are both hundredths.	$\begin{array}{l}0.64 \\ 0.60\end{array}$
64 is greater than 60.	$64>60$
64 hundredths is greater than 60 hundredths.	$0.64>0.60$
	$0.64>0.6$

Order each of the following pairs of numbers, using $<\ \text{or}\ >\text{:}\ 0.42___0.4.$

Bonisa impendulo

>
Order each of the following pairs of numbers, using $<\ \text{or}\ >\text{:}\ 0.18___0.1.$

Bonisa impendulo

>

Order $0.83___0.803$ using $<$ or $>.$

Bonisa impendulo

	$0.83___0.803$
Write the numbers one under the other, lining up the decimals.	$\begin{array}{l}0.83 \\ 0.803\end{array}$
They do not have the same number of digits. Write one zero at the end of 0.83.	$\begin{array}{l}0.830 \\ 0.803\end{array}$
Since $830>803$, 830 thousandths is greater than 803 thousandths.	$0.830>0.803$
	$0.83>0.803$

Order the following pair of numbers, using $<\ \text{or}\ >\text{:}\ 0.76___0.706.$

Bonisa impendulo

>
Order the following pair of numbers, using $<\ \text{or}\ >\text{:}\ 0.305___0.35.$

Bonisa impendulo

<

Use $<$ or $>$ to order $-0.1___-0.8.$

Bonisa impendulo

	$-0.1___-0.8$
Write the numbers one under the other, lining up the decimal points. They have the same number of digits.	$\begin{array}{l}-0.1 \\ -0.8\end{array}$
Since $-1>-8$, −1 tenth is greater than −8 tenths.	$-0.1>-0.8$

Symbols used here

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

\pi

pi
Ratio of a circle's circumference to its diameter, 3.14159…

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\neq

not equal
The two sides are different.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: The Real Numbers

Simplify expressions with square roots
Identify integers, rational numbers, irrational numbers, and real numbers
Locate fractions on the number line
Locate decimals on the number line
Write the numbers one under the other, lining up the decimal points.
Check to see if both numbers have the same number of digits. If not, write zeros at the end of the one with fewer digits to make them match.
Compare the numbers as if they were whole numbers.
Order the numbers using the appropriate inequality sign.

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Zama wena

Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Okuningi Algebra

Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value

ⓐ Since \({5}^{2}=25\)	\(\begin{array}{l}\sqrt{25} \\ 5\end{array}\)
ⓑ Since \({11}^{2}=121\)	\(\begin{array}{l}\sqrt{121} \\ 11\end{array}\)

ⓐ Since \({5}^{2}=25\)	\(\begin{array}{l}\sqrt{25} \\ 5\end{array}\)
ⓑ Since \({11}^{2}=121\)	\(\begin{array}{l}\sqrt{121} \\ 11\end{array}\)

ⓐ \(-\ \frac{2}{3}\) is to the right of \(-1\) on the number line.	\(\begin{array}{l}-\ \frac{2}{3}___-1 \\ -\ \frac{2}{3}>-1\end{array}\)
ⓑ \(-3\frac{1}{2}\) is to the right of \(-3\) on the number line.	\(\begin{array}{l}-3\frac{1}{2}___-3 \\ -3\frac{1}{2}<-3\end{array}\)
ⓒ \(-\ \frac{3}{4}\) is to the right of \(-\ \frac{1}{4}\) on the number line.	\(\begin{array}{l}-\ \frac{3}{4}___-\ \frac{1}{4} \\ -\ \frac{3}{4}<-\ \frac{1}{4}\end{array}\)
ⓓ \(-2\) is to the right of \(-\ \frac{8}{3}\) on the number line.	\(\begin{array}{l}-2___-\ \frac{8}{3} \\ -2>-\ \frac{8}{3}\end{array}\)

Write the numbers one under the other, lining up the decimal points.	\(\begin{array}{l}0.64 \\ 0.6\end{array}\)
Add a zero to 0.6 to make it a decimal with 2 decimal places. Now they are both hundredths.	\(\begin{array}{l}0.64 \\ 0.60\end{array}\)
64 is greater than 60.	\(64>60\)
64 hundredths is greater than 60 hundredths.	\(0.64>0.60\)
	\(0.64>0.6\)

	\(0.83___0.803\)
Write the numbers one under the other, lining up the decimals.	\(\begin{array}{l}0.83 \\ 0.803\end{array}\)
They do not have the same number of digits. Write one zero at the end of 0.83.	\(\begin{array}{l}0.830 \\ 0.803\end{array}\)
Since \(830>803\), 830 thousandths is greater than 803 thousandths.	\(0.830>0.803\)
	\(0.83>0.803\)

ⓐ The negative is in front of the radical sign.	\(\begin{array}{l}-\sqrt{9} \\ -3\end{array}\)
ⓑ The negative is in front of the radical sign.	\(\begin{array}{l}-\sqrt{144} \\ -12\end{array}\)

ⓐ Write it as a fraction with denominator 1.	\(\begin{array}{l}-27 \\ \frac{-27}{1}\end{array}\)
ⓑ Write it as a mixed number. Remember, 7 is the whole number and the decimal part, 0.31, indicates hundredths. Convert to an improper fraction.	\(\begin{array}{l}7.31 \\ 7\frac{31}{100} \\ \frac{731}{100}\end{array}\)

ⓐ Look for decimals that repeat or stop.	The 3 repeats in \(0.58\overset{\text{-}}{3}\). The decimal 0.47 stops after the 7. So \(0.58\overset{\text{-}}{3}\) and 0.47 are rational.
ⓑ Look for decimals that neither stop nor repeat.	\(3.605551275\text{\ldots }\) has no repeating block of digits and it does not stop. So \(3.605551275\text{\ldots }\) is irrational.

	\(-0.1___-0.8\)
Write the numbers one under the other, lining up the decimal points. They have the same number of digits.	\(\begin{array}{l}-0.1 \\ -0.8\end{array}\)
Since \(-1>-8\), −1 tenth is greater than −8 tenths.	\(-0.1>-0.8\)