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Irrational Numbers

Define and identify numbers that are irrational.

Learning Objectives

After completing this section, you should be able to:

  1. Define and identify numbers that are irrational.
  2. Simplify irrational numbers and express in lowest terms.
  3. Add and subtract irrational numbers.
  4. Multiply and divide irrational numbers.
  5. Rationalize fractions with irrational denominators.

Defining and Identifying Numbers That Are Irrational

We defined rational numbers in the last section as numbers that could be expressed as a fraction of two integers. Irrational numbers are numbers that cannot be expressed as a fraction of two integers. Recall that rational numbers could be identified as those whose decimal representations either terminated (ended) or had a repeating pattern at some point. So irrational numbers must be those whose decimal representations do not terminate or become a repeating pattern.

One collection of irrational numbers is square roots of numbers that aren’t perfect squares. \(x\) is the square root of the number \(a\), denoted \(\sqrt{a}\), if \({x}^{2}=a\). The number \(a\) is the perfect square of the integer \(n\) if \(a={n}^{2}\). The rational number \(\frac{a}{b}\) is a perfect square if both \(a\) and \(b\) are perfect squares.

One method of determining if an integer is a perfect square is to examine its prime factorization. If, in that factorization, all the prime factors are raised to even powers, the integer is a perfect square. Another method is to attempt to factor the integer into an integer squared. It is possible that you recognize the number as a perfect square (such as 4 or 9). Or, if you have a calculator at hand, use the calculator to determine if the square root of the integer is an integer.

Another collection of irrational numbers is based on the special number, pi, denoted by the Greek letter \(\pi\), which is the ratio of the circumference of the diameter of the circle ().

Any multiple or power of \(\pi\) is an irrational number.

Any number expressed as a rational number times an irrational number is an irrational number also. When an irrational number takes that form, we call the rational number the rational part, and the irrational number the irrational part. It should be noted that a rational number plus, minus, multiplied by, or divided by any irrational number is an irrational number.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Simplifying Square Roots and Expressing Them in Lowest Terms

To simplify a square root means that we rewrite the square root as a rational number times the square root of a number that has no perfect square factors. The act of changing a square root into such a form is simplifying the square root.

The number inside the square root symbol is referred to as the radicand. So in the expression \(\sqrt{a}\) the number \(a\) is referred to as the radicand.

Before discussing how to simplify a square root, we need to introduce a rule about square roots. The square root of a product of numbers equals the product of the square roots of those number. Written symbolically, \(\sqrt{a\times b}=\sqrt{a}\times \sqrt{b}\).

Using this formula, we can factor an integer inside a square root into a perfect square times another integer. Then the square root can be applied to the perfect square, leaving an integer times the square root of another integer. If the number remaining under the square root has no perfect square factors, then we’ve simplified the irrational number into lowest terms. To simplify the irrational number into lowest terms when \(n\) is an integer:

Step 1: Determine the largest perfect square factor of \(n\), which we denote \({a}^{2}\).

Step 2: Factor \(n\) into \({a}^{2}\times b\).

Step 3: Apply \(\sqrt{{a}^{2}\times b}=\sqrt{{a}^{2}}\times \sqrt{b}\).

Simplifying a Square Root

Try it.

Simplify the irrational number \(\sqrt{180}\) and express in lowest terms. Identify the rational and irrational parts.

Solution

Begin by finding the largest perfect square that is a factor of 180. We can do this by writing out the factor pairs of 180: \[\begin{array}{l}1\times 180\ 2\times 90\ 3\times 60\ 4\times 45\ 5\times 36\ 6\times 30\ 9\times 20\ 10\times 18\ 12\times 15\end{array}\]

Looking at the list of factors, the perfect squares are 4, 9, and 36. The largest is 36, so we factor the into \(36\times 5={6}^{2}\times 5\). In the formula, \(a=6\) and \(b=5\). Apply \(\sqrt{{a}^{2}\times b}=\sqrt{{a}^{2}}\times \sqrt{b}\).

\(\sqrt{{6}^{2}\times 5}=\sqrt{{6}^{2}}\times \sqrt{5}\)

The simplified form of \(\sqrt{180}\) is \(6\sqrt{5}\). In this example, the 6 is the rational part, and the \(\sqrt{5}\) is the irrational part.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Adding and Subtracting Irrational Numbers

Just like any other number we’ve worked with, irrational numbers can be added or subtracted. When working with a calculator, enter the operation and a decimal representation will be given. However, there are times when two irrational numbers may be added or subtracted without the calculator. This can happen only when the irrational parts of the irrational numbers are the same.

To add or subtract two irrational numbers that have the same irrational part, add or subtract the rational parts of the numbers, and then multiply that by the common irrational part.

Subtracting Irrational Numbers with Similar Irrational Parts

Try it.

If possible, subtract the following irrational numbers without using a calculator. If this is not possible, state why.

\[3\sqrt{7}-8\sqrt{7}\]

Solution

Since these two irrational numbers have the same irrational part, \(\sqrt{7}\), we can subtract without using a calculator. The rational part of the first number is 3. The rational part of the second number is 8. Using the formula yields \(3\sqrt{7}-8\sqrt{7}=(3-8)\times \sqrt{7}=-5\sqrt{7}\).

Adding Irrational Numbers with Similar Irrational Parts

Try it.

If possible, add the following irrational numbers without using a calculator. If this is not possible, state why.

\(35\pi +17\pi\)

Solution

Since these two irrational numbers have the same irrational part, \(\pi\), the addition can be performed without using a calculator. The rational part of the first number is 35. The rational part of the second number is 17. Using the formula yields \(35\pi +17\pi =(35+17)\times \pi =52\pi\).

Subtracting Irrational Numbers with Different Irrational Parts

Try it.

If possible, subtract the following irrational numbers without using a calculator. If this is not possible, state why.

\(19\sqrt{3}-5.6\sqrt{7}\)

Solution

The two numbers being subtracted do not have the same irrational part, so the operation cannot be performed.

Multiplying and Dividing Irrational Numbers

Just like any other number that we’ve worked with, irrational numbers can be multiplied or divided. When working with a calculator, enter the operation and a decimal representation will be given. Sometimes, though, you may want to retain the form of the irrational number as a rational part times an irrational part. The process is similar to adding and subtracting irrational numbers when they are in this form. We do not need the irrational parts to match. Even though they need not match, they do need to be similar, such as both irrational parts are square roots, or both irrational parts are multiples of pi. Also, if the irrational parts are square roots, we may need to reduce the resulting square root to lowest terms.

When multiplying two square roots, use the following formula. It is the same formula presented during the discussion of simplifying square roots.

When dividing two square roots, use the following formula.

To multiply or divide irrational numbers with similar irrational parts, do the following:

Step 1: Multiply or divide the rational parts.

Step 2: If necessary, reduce the result of Step 1 to lowest terms. This becomes the rational part of the answer.

Step 3: Multiply or divide the irrational parts.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Rationalizing Fractions with Irrational Denominators

Fractions often represent that some amount is being equally divided into some number of parts. But to conceptualize a fraction in that manner, the denominator needs to be an integer. An irrational number in the denominator interferes with that interpretation of a fraction. Fractions that have denominators that are just the square root of an integer can be altered into fractions with integer denominators using a process called rationalizing the denominator. The process relies on the following property of square roots: \(\sqrt{a}\times \sqrt{a}=a\) and the following property of fractions: \(\frac{a}{b}=\frac{ac}{bc}\) for any non-zero number \(c\).

Using these two properties, when a fraction has a square root in the denominator, we can eliminate that square root. Multiply the numerator and denominator by that square root from the denominator, \(\frac{a}{\sqrt{b}}=\frac{a\sqrt{b}}{\sqrt{b}\times \sqrt{b}}\). Then apply \(\sqrt{a}\times \sqrt{a}=a\) to the denominator, yielding \(\frac{a\sqrt{b}}{\sqrt{b}\times \sqrt{b}}=\frac{a\sqrt{b}}{b}\). Notice that there is no longer a square root in the denominator, which allows for interpreting the fraction as dividing a whole into equal parts.

Rationalizing the Denominator

Try it.

Rationalize the denominator of the following:

  1. \(\frac{5}{\sqrt{7}}\)
  2. \(\frac{3\sqrt{6}}{2\sqrt{10}}\)
Solution
  1. The square root in the denominator is \(\sqrt{7}\). In order to rationalize the denominator of \(\frac{5}{\sqrt{7}}\), we need to multiply the numerator and denominator by \(\sqrt{7}\) and simplify.
    \(\frac{5}{\sqrt{7}}=\frac{5\sqrt{7}}{\sqrt{7}\times \sqrt{7}}=\frac{5\sqrt{7}}{7}\)
    The square root is in simplified form, so the final answer is \(\frac{5\sqrt{7}}{7}\).
  2. The square root in the denominator is \(\sqrt{10}\).
    Step 1: In order to rationalize the denominator of \(\frac{3\sqrt{6}}{2\sqrt{10}}\), we need to multiply the numerator and denominator by \(\sqrt{10}\) and simplify.
    \(\frac{3\sqrt{6}}{2\sqrt{10}}=\frac{3\sqrt{6}\times \sqrt{10}}{2\sqrt{10}\times \sqrt{10}}=\frac{3\sqrt{60}}{2\times 10}=\frac{3\sqrt{60}}{20}\)

    Step 2: The 60 under the square root can be factored into the following factor pairs:

    \(1\times 60\ 2\times 30\ 3\times 20\ 4\times 15\ 5\times 12\ 6\times 10\)

    Step 3: The largest square factor of 60 is 4, so we simplify the \(\sqrt{60}\) in the numerator into \(2\sqrt{15}\). We also cancel any common factors.

    \(\frac{3\sqrt{60}}{20}=\frac{3\times 2\sqrt{15}}{20}=\frac{6\sqrt{15}}{20}=\frac{3\sqrt{15}}{10}\)

    This is completely simplified.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Key Concepts

  • Irrational numbers are numbers that cannot be written as an integer divided by another integer. One example is pi, denoted \(\pi\). Another collection of irrational numbers are natural numbers that are not perfect squares.
  • Some irrational numbers can be written as a rational part multiplied by an irrational part. If two irrational numbers have the same irrational parts, they can be added or subtracted.
  • When irrational numbers are similar, on can multiply and divide the numbers without a calculator.
  • Since \(\sqrt{a\times b}=\sqrt{a}\times \sqrt{b}\) , and \(\sqrt{a}\div \sqrt{b}=\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}\), products and quotients of square roots can be determined.
  • Because \(\sqrt{{a}^{2}}=a\) and \(\sqrt{a\times b}=\sqrt{a}\times \sqrt{b}\), it is possible to simplify square root expressions so the radicand contains no perfect square factors.
  • When a fraction has an irrational number as its denominator, it is possible to convert the denominator into a rational number using its conjugate. Doing so involves multiplying the numerator and denominator by the conjugate of the denominator, and then applying the difference of squares formula.
  • With a single square root term
  • Using conjugate numbers for two term denominators

Formulas

  • \[\sqrt{a\times b}=\sqrt{a}\times \sqrt{b}\]
  • \[a\times x\pm b\times x=(a\pm b)\times x\]
  • \[\sqrt{a}\div \sqrt{b}=\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}\]
  • \[{a}^{2}-{b}^{2}=(a-b)(a+b)\]

Videos

  • The Philosophy of the Pythagoreans
  • Using Desmos to Find the Square Root of a Number
  • Simplifying Square Roots
  • Rationalizing the Denominator

Practice (12)

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

  1. Determine which of the following are perfect squares.

    1. 45
    2. 81
    3. \(\frac{9}{28}\)
    4. \(\frac{144}{400}\)
    Хариулт
    1. The prime factorization of 45 is \(45={3}^{2}\times 5\). Since the 5 is not raised to an even power, 45 is not a perfect square.
    2. The prime factorization of 81 is \({3}^{4}\). All the prime factors are raised to even powers, so 81 is a perfect square.
    3. We must determine if both the numerator and denominator of \(\frac{9}{28}\) are perfect squares for the rational number to be a perfect square. The numerator is 9, and as mentioned above, 9 is a perfect square (it is 3 squared). Now we check the prime factorization of the denominator, 28, which is \(28={2}^{2}\times 7\). Since 7 is not raised to an even power, 28 is not a perfect square. Since the denominator is not a perfect square, \(\frac{9}{28}\) is not a perfect square.
    4. We must determine if both the numerator and denominator of \(\frac{144}{400}\) are perfect squares for the rational number to be a perfect square. The numerator is 144. The prime factorization of 144 is \(144={2}^{4}\times {3}^{2}\). Since all the prime factors of 144 are raised to even powers, 144 is a perfect square. Now we check the prime factorization of the denominator, 400, which is \(400={2}^{4}\times {5}^{2}\). Since all the prime factors of 400 are raised to even powers, 400 is a perfect square. Since the numerator and denominator of \(\frac{144}{400}\) are perfect squares, \(\frac{144}{400}\) is a perfect square.

  2. Identify which of the following numbers are irrational.

    1. \(\sqrt{35}\)
    2. \(0.\overset{\bar}{15}\)
    3. \(\sqrt{121}\)
    4. \(4\pi\)
    Хариулт
    1. 35 can be factored as \(5\times 7\), showing that 35 is not the square of an integer or a rational number. This mean its square root is an irrational number.
    2. Since \(0.\overset{\bar}{15}\) is a decimal with a repeating pattern, it is rational, so it is not an irrational number.
    3. \(121={11}^{2}\). Since 121 is the square of an integer, its square root is a rational number.
    4. Since \(4\pi\) is a multiple of pi, it is irrational. In this case, the rational part of the number is 4, while the irrational part is \(\pi\).

  3. Simplify the irrational number \(\sqrt{180}\) and express in lowest terms. Identify the rational and irrational parts.

    Хариулт

    Begin by finding the largest perfect square that is a factor of 180. We can do this by writing out the factor pairs of 180: \[\begin{array}{l}1\times 180\ 2\times 90\ 3\times 60\ 4\times 45\ 5\times 36\ 6\times 30\ 9\times 20\ 10\times 18\ 12\times 15\end{array}\]

    Looking at the list of factors, the perfect squares are 4, 9, and 36. The largest is 36, so we factor the into \(36\times 5={6}^{2}\times 5\). In the formula, \(a=6\) and \(b=5\). Apply \(\sqrt{{a}^{2}\times b}=\sqrt{{a}^{2}}\times \sqrt{b}\).

    \(\sqrt{{6}^{2}\times 5}=\sqrt{{6}^{2}}\times \sqrt{5}\)

    The simplified form of \(\sqrt{180}\) is \(6\sqrt{5}\). In this example, the 6 is the rational part, and the \(\sqrt{5}\) is the irrational part.

  4. Simplify the irrational number \(\sqrt{330}\) and express in lowest terms. Identify the rational and irrational parts.

    Хариулт

    Begin by finding the largest perfect square that is a factor of 330. We can do this by writing out the factor pairs of 330: \[\begin{array}{l}1\times 330\ 2\times 165\ 3\times 110\ 5\times 66\ 6\times 55\ 10\times 33\ 11\times 30\ 15\times 22\end{array}\]

    Looking at the list of factors, there are no perfect squares other than 1, which means \(\sqrt{330}\) is already expressed in lowest terms. In this case, 1 is the rational part, and \(\sqrt{330}\) is the irrational part. Though we could write this as \(1\sqrt{330}\), but the product of 1 and any other number is just the number.

  5. Simplify the irrational number \(\sqrt{2,548}\) and express in lowest terms. Identify the rational and irrational parts.

    Хариулт

    Begin by finding the largest perfect square that is a factor of 2,548. We can do this by writing out the factor pairs of 2,548: \[\begin{array}{l}1\times 2548\ 2\times 1274\ 4\times 637\ 7\times 364\ 13\times 196\ 14\times 182\ 26\times 98\ 28\times 91\ 49\times 52\end{array}\]

    Looking at the list of factors, the perfect squares are 4, 49, and 196. The largest is 196, so we factor the 2,548 into \(196\times 13={14}^{2}\times 13\). In the formula, \(a=14\) and \(b=5\). Apply \(\sqrt{{a}^{2}\times b}=\sqrt{{a}^{2}}\times \sqrt{b}\).

    \(\sqrt{{14}^{2}\times 13}=\sqrt{{14}^{2}}\times \sqrt{13}\)

    The simplified form of \(\sqrt{2,548}\) is \(14\sqrt{13}\). In this example, 14 is the rational part, and \(\sqrt{13}\) is the irrational part.

  6. If possible, subtract the following irrational numbers without using a calculator. If this is not possible, state why.

    \[3\sqrt{7}-8\sqrt{7}\]

    Хариулт

    Since these two irrational numbers have the same irrational part, \(\sqrt{7}\), we can subtract without using a calculator. The rational part of the first number is 3. The rational part of the second number is 8. Using the formula yields \(3\sqrt{7}-8\sqrt{7}=(3-8)\times \sqrt{7}=-5\sqrt{7}\).

  7. If possible, add the following irrational numbers without using a calculator. If this is not possible, state why.

    \(35\pi +17\pi\)

    Хариулт

    Since these two irrational numbers have the same irrational part, \(\pi\), the addition can be performed without using a calculator. The rational part of the first number is 35. The rational part of the second number is 17. Using the formula yields \(35\pi +17\pi =(35+17)\times \pi =52\pi\).

  8. If possible, subtract the following irrational numbers without using a calculator. If this is not possible, state why.

    \(19\sqrt{3}-5.6\sqrt{7}\)

    Хариулт

    The two numbers being subtracted do not have the same irrational part, so the operation cannot be performed.

  9. Perform the following operations without a calculator. Simplify if possible.

    1. \(3\sqrt{15}\div (8\sqrt{3})\)
    2. \(14.7\sqrt{135}\div (3\sqrt{5})\).
    Хариулт
    1. In this division problem, \(3\sqrt{15}\div (8\sqrt{3})\), notice that the irrational parts of these numbers are similar. They are both square roots, so follow the steps given above.

      Step 1: Divide the rational parts. \(3\div 8=\frac{3}{8}\)

      Step 2: If necessary, reduce the result of Step 1 to lowest terms. The 3 and 8 have no common factors, so \(\frac{3}{8}\) is already in lowest terms.

      Step 3: Divide the irrational parts. \(\sqrt{15}\div \sqrt{3}=\frac{\sqrt{15}}{\sqrt{3}}=\sqrt{\frac{15}{3}}\)

      Step 4: If necessary, reduce the result from Step 3 to lowest terms. The radicand can be reduced, which yields \(\sqrt{5}\).

      Step 5: The result is the product of the rational and irrational parts, which is \(\frac{3}{8}\sqrt{5}\).

    2. In this division problem, \(14.7\sqrt{135}\div (3\sqrt{5})\), notice that the irrational parts of these numbers are similar. They are both square roots, so follow the steps given above.

      Step 1: Divide the rational parts. \(14.7\div 3=4.9\)

      Step 2: If necessary, reduce the result of Step 1 to lowest terms. This rational number is expressed as a decimal so will not be reduced.

      Step 3: Divide the irrational parts. \(\sqrt{135}\div \sqrt{5}=\frac{\sqrt{135}}{\sqrt{5}}=\sqrt{\frac{135}{5}}\)

      Step 4: If necessary, reduce the result from Step 3 to lowest terms. The radicand can be reduced, which yields \(\sqrt{\frac{135}{5}}=\sqrt{27}=\sqrt{9\times 3}=3\sqrt{3}\).

      Step 5: The result is the product of the rational and irrational parts, which is \(4.9\times 3\sqrt{3}=14.7\sqrt{3}\).

  10. Perform the following operations without a calculator. Simplify if possible.

    1. \((19\sqrt{3})\times (5.6\sqrt{12})\)
    2. \(13\pi \times 8\pi\)
    Хариулт
    1. In this multiplication problem, \((19\sqrt{3})\times (5.6\sqrt{12})\), notice that the irrational parts of these numbers are similar. They are both square roots. Follow the process above.

      Step 1: Multiply the rational parts. \(19\times 5.6=106.4\)

      Step 2: If necessary, reduce the result of Step 1 to lowest terms. This rational number is expressed as a decimal and will not be reduced.

      Step 3: Multiply the irrational parts. \(\sqrt{3}\times \sqrt{12}=\sqrt{3\times 12}=\sqrt{36}\)

      Step 4: If necessary, reduce the result from Step 3 to lowest terms. The radicand is 36, which is the square of 6. The irrational part reduces to \(\sqrt{36}=6\).

      Step 5: The result is the product of the rational and irrational parts, which is \(106.4\times 6=638.4\).

      Notice that sometimes multiplying or dividing irrational numbers can result in a rational number.

    2. In this multiplication problem, \(13\pi \times 8\pi\), notice that the irrational parts of these numbers are the same, \(\pi\). Follow the process above.

      Step 1: Multiply the rational parts. \(13\times 8=104\)

      Step 2: If necessary, reduce the result of Step 1 to lowest terms. That result is an integer.

      Step 3: Multiply the irrational parts. \(\pi \times \pi ={\pi }^{2}\)

      Step 4: If necessary, reduce the result from Step 3 to lowest terms. This cannot be reduced.

      Step 5: The result is the product of the rational and irrational parts, which is \(104{\pi }^{2}\).

  11. Rationalize the denominator of the following:

    1. \(\frac{5}{\sqrt{7}}\)
    2. \(\frac{3\sqrt{6}}{2\sqrt{10}}\)
    Хариулт
    1. The square root in the denominator is \(\sqrt{7}\). In order to rationalize the denominator of \(\frac{5}{\sqrt{7}}\), we need to multiply the numerator and denominator by \(\sqrt{7}\) and simplify.
      \(\frac{5}{\sqrt{7}}=\frac{5\sqrt{7}}{\sqrt{7}\times \sqrt{7}}=\frac{5\sqrt{7}}{7}\)
      The square root is in simplified form, so the final answer is \(\frac{5\sqrt{7}}{7}\).
    2. The square root in the denominator is \(\sqrt{10}\).
      Step 1: In order to rationalize the denominator of \(\frac{3\sqrt{6}}{2\sqrt{10}}\), we need to multiply the numerator and denominator by \(\sqrt{10}\) and simplify.
      \(\frac{3\sqrt{6}}{2\sqrt{10}}=\frac{3\sqrt{6}\times \sqrt{10}}{2\sqrt{10}\times \sqrt{10}}=\frac{3\sqrt{60}}{2\times 10}=\frac{3\sqrt{60}}{20}\)

      Step 2: The 60 under the square root can be factored into the following factor pairs:

      \(1\times 60\ 2\times 30\ 3\times 20\ 4\times 15\ 5\times 12\ 6\times 10\)

      Step 3: The largest square factor of 60 is 4, so we simplify the \(\sqrt{60}\) in the numerator into \(2\sqrt{15}\). We also cancel any common factors.

      \(\frac{3\sqrt{60}}{20}=\frac{3\times 2\sqrt{15}}{20}=\frac{6\sqrt{15}}{20}=\frac{3\sqrt{15}}{10}\)

      This is completely simplified.

  12. Rationalize the denominator of \(\frac{4}{6+\sqrt{10}}\).

    Хариулт

    Step 1: We recognize that the denominator is the sum of two numbers where one or both involve square roots. This means the conjugate can be used to remove the square root from the denominator.

    Step 2: To do so, we multiply the numerator and the denominator each by the conjugate of the denominator. Since the denominator is \(6+\sqrt{10}\), the conjugate we will use is \(6-\sqrt{10}\).

    Step 3: The conjugate is multiplied by the numerator and the denominator.

    \(\frac{4}{6+\sqrt{10}}\times \frac{6-\sqrt{10}}{6-\sqrt{10}}\)

    Step 4: Remembering how a number times its conjugate works, this becomes

    \(\frac{4}{6+\sqrt{10}}\times \frac{6-\sqrt{10}}{6-\sqrt{10}}=\frac{4\times (6-\sqrt{10})}{{6}^{2}-{(\sqrt{10})}^{2}}\).

    Step 5: In the numerator, we apply the distributive property. Using it yields
    \(\frac{4\times (6-\sqrt{10})}{{6}^{2}-{(\sqrt{10})}^{2}}=\frac{24-4\sqrt{10}}{36-10}=\frac{24-4\sqrt{10}}{26}\).

    Step 6: Notice that the denominator no longer contains a square root. It has been rationalized. If desired, this can then be written as a rational number minus an irrational number, by recalling that \(\frac{a-b}{c}=\frac{a}{c}-\frac{b}{c}\).

    Applying that to the answer, we have \(\frac{4}{6+\sqrt{10}}=\frac{24-4\sqrt{10}}{26}=\frac{24}{26}-\frac{4\sqrt{10}}{26}\).

    Step 7: With a bit of cancellation, this reduces to \(\frac{4}{6+\sqrt{10}}=\frac{24}{26}-\frac{4\sqrt{10}}{26}=\frac{12}{13}-\frac{2\sqrt{10}}{13}\).

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…
\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\prod_{k=1}^{n} a_k
product
Multiply a_k for k = 1 up to n.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.
a \equiv b \pmod n
congruent modulo n
n divides a − b; a and b have the same remainder.
a \mid b,\ \gcd(a,b)
divides, greatest common divisor
b is a multiple of a; the largest number dividing both.
\varphi(n),\ \pi(x)
Euler's totient, prime-counting function
Count of 1..n coprime to n; number of primes up to x.
\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n
integers modulo n
The remainders 0…n−1 with clock arithmetic.
a \bmod n
remainder
What is left after dividing a by n.

How to: Irrational Numbers

  1. Define and identify numbers that are irrational.
  2. Simplify irrational numbers and express in lowest terms.
  3. Add and subtract irrational numbers.
  4. Multiply and divide irrational numbers.
  5. Rationalize fractions with irrational denominators.
  6. 45
  7. 81
  8. The prime factorization of 45 is

Questions people ask

Why are primes so important?

Every integer factors into primes in exactly one way, so primes are the atoms of multiplication. Cryptography relies on that factoring being easy to state and hard to do.

How do I tell whether a big number is prime?

Trial division up to the square root works for small numbers. For large ones, probabilistic tests (Miller–Rabin) give an answer that is wrong with negligible probability, and deterministic tests (AKS) exist but are slower.

Өөрийнхөөг турш

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

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