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Scientific Notation

Write numbers in standard or scientific notation.

Learning Objectives

After completing this section, you should be able to:

  1. Write numbers in standard or scientific notation.
  2. Convert numbers between standard and scientific notation.
  3. Add and subtract numbers in scientific notation.
  4. Multiply and divide numbers in scientific notation.
  5. Use scientific notation in computing real-world applications.

Writing Numbers in Standard or Scientific Notation Form

When we say that a number is in scientific notation, we are specifying the form in which that number is written. That form begins with an integer with an absolute value between 1 and 9, then perhaps followed the decimal point and then some more digits. This is then multiplied by 10 raised to some power. When the number only has one non-zero digit, the scientific notation form is the digit multiplied by 10 raised to an exponent. When the number has more than one non-zero digit, the scientific notation form is a single digit, followed by a decimal, which is then followed by the remaining digits, which is then multiplied by 10 to a power.

The following numbers are written in scientific notation:

\(1.45\times {10}^{3}\)

\(-8.345\times {10}^{-4}\)

\(3\times {10}^{2}\)

\(3.14159\times {10}^{0}\)

The following numbers are not written in scientific notation:

Identifying Numbers in Scientific Notation

Try it.

Which of the following numbers are in scientific notation? If the number is not in scientific notation, explain why it is not.

  1. \(-9.67\times {10}^{20}\)
  2. \(145\times {10}^{-8}\)
  3. \(1.45\)
Solution
  1. The number \(-9.67\times {10}^{20}\) is in scientific notation because the absolute value of −9.67 is at least 1 and less than 10.
  2. The number \(145\times {10}^{-8}\) is not in scientific notation because 145 is not at least 1 and less than 10.
  3. The number \(1.45\) is not in scientific notation form. Even though it is at least 1 but less than 10, it is not multiplied by 10 raised to a power.
  • Step 1: Count the number of zeros between the decimal and the first non-zero digit. Label this \(n\).
  • Step 2: Starting with the first non-zero digit of the number, write the digits. If the number was negative, include the negative sign.
  • Step 3: If there is more than one digit, place the decimal after the first digit from Step 2.
  • Step 4: Multiply the number from Step 3 by \({10}^{n+1}\).
  • Step 1: Count the number of digits that are to the left of the decimal point. Label this \(n\).
  • Step 2: Write the digits of the number without the decimal place, if one was present. If the number was negative, include the negative sign.
  • Step 3: If there is more than one digit, place the decimal point after the first digit.
  • Step 4: Multiply the number from Step 3 by \({10}^{n-1}\).
  • If you move the decimal to the left by \(k\) digits, you increase the exponent by \(k\).
  • If you move the decimal to the right by \(k\) digits, you decrease the exponent by \(k\) digits.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Converting Numbers from Scientific Notation to Standard Form

In the previous section, converting a number from standard form to scientific notation was explored. Now, we explore converting from scientific notation back into standard form. Doing so involves moving the decimal according to the power of the 10. The decimal is moved a number of steps equal to the exponent of the 10. As demonstrated previously, when the exponent of the 10 is negative, the decimal is moved to the left and when the exponent of the 10 is positive, the decimal is moved to the right.

Converting from Scientific Notation to Standard Form

Try it.

Convert the following into standard form:

  1. \(2.78\times {10}^{9}\)
  2. \(9.04\times {10}^{-8}\)
Solution
  1. Since the exponent is positive, the decimal moves nine places to the right, so \(2.78\times {10}^{9}\) is \(2,780,000,000\).
  2. Since the exponent is negative, the decimal moves eight places to the left, so \(9.04\times {10}^{-8}\) is \(0.0000000904\).

Adding and Subtracting Numbers in Scientific Notation

To add or subtract numbers in scientific notation, the numbers first need to have the same exponent for the 10s. It is possible to add the following since the powers of 10 match: \(4.5\times {10}^{4}+3.15\times {10}^{4}=7.65\times {10}^{4}\)

Notice that the number parts were added, but the exponent part remained the same. This is due to the distributive property of the real numbers. The \({10}^{4}\) is factored from the two terms, as shown: \(4.5\times {10}^{4}+3.15\times {10}^{4}=(4.5+3.15)\times {10}^{4}=7.65\times {10}^{4}\)

Numbers in scientific notation can be added or subtracted directly using a calculator. Simply enter the values in scientific form and set your calculator to display scientific notation.

Adding and Subtracting Numbers in Scientific Notation with the Same Powers of 10

Try it.

Calculate the following:

  1. \(3.8\times {10}^{-3}+1.006\times {10}^{-3}\)
  2. \(9.61\times {10}^{8}-3.85\times {10}^{8}\)
Solution
  1. Since the powers of 10 match, we use the distributive property of real numbers to factor 10−3 from the numbers. We then add the number parts separately to get 4.806. \(3.8\times {10}^{-3}+1.006\times {10}^{-3}=(3.8+1.006)\times {10}^{-3}=4.806\times {10}^{-3}\)
  2. Since the powers of 10 match, we use the distributive property of real numbers to factor 108 from the numbers. We then subtract the number parts separately to get 5.76. \[9.61\times {10}^{8}-3.85\times {10}^{8}=(9.61-3.85)\times {10}^{8}=5.76\times {10}^{8}\]

Adding and subtracting in scientific notation is straightforward when the exponents are the same. There are two issues that can arise. The first issue is what to do if after adding or subtracting the result is not in scientific notation.

The second issue that might be encountered when adding or subtracting is that the powers of 10 do not match. In that case, one of the numbers must be changed so that the powers of 10 match. It is easiest to make the smaller power of 10 larger to match the other power of 10.

For example, to perform the following, \(4.5\times {10}^{5}+3.9\times {10}^{3}\), we’d change the \(3.9\times {10}^{3}\) so that the power of 10 is 5. To do so, we need to increase the power of 10 and move the decimal in the number part two places to the left. That would alter \(3.9\times {10}^{3}\) into \(0.039\times {10}^{5}\). We would use \(0.039\times {10}^{5}\) in the addition problem, so that the exponents match, allowing the addition to occur. \(4.5\times {10}^{5}+3.9\times {10}^{3}=4.5\times {10}^{5}+0.039\times {10}^{5}=(4.5+0.039)\times {10}^{5}=4.539\times {10}^{5}\)

The steps to take when the exponents of the 10s are not equal are:

Condensed — the full section is in OpenStax Contemporary Mathematics.

Multiplying and Dividing Numbers in Scientific Notation

Multiplying and dividing numbers in scientific notation is somewhat easier than adding or subtracting, because the exponents of the 10s do not have to match. However, it is much more likely that the result will not be in scientific notation, and so that will have to be adjusted at the end. Generally, we multiply or divide the number parts of the two values, and then apply exponent rules to the 10 raised to the powers.

To multiply two numbers in scientific notation:

Step 1: Multiply the number parts.

Step 2: Add the exponents of the 10s.

Step 3: The result is the answer from Step 1 times 10 raised to the answer from Step 2.

Step 4: If the number is not in scientific notation, adjust it appropriately.

Multiplying Numbers in Scientific Notation

Try it.

Calculate the following:

  1. \((4.3\times {10}^{3})\times (1.8\times {10}^{7})\)

  2. \((5\times {10}^{-1}{}^{3})\times (7.3\times {10}^{6})\)

Solution

  1. Step 1: Multiply the number parts to get \(4.3\times 1.8=7.74\).

    Step 2: Add the exponents of the 10s to get \(3+7=10\).

    Step 3: The result is then \(7.74\times {10}^{10}\).

    Step 4: This number is already in scientific notation, so no additional adjustment is necessary, \((4.3\times {10}^{3})\times (1.8\times {10}^{7})=7.74\times {10}^{10}\).

  2. Step 1: Multiply the number parts to get \(5\times 7.3=36.5\).

    Step 2: Add the exponents of the 10s to get \(-13+6=-7\).

    Step 3: The result then is \(36.5\times {10}^{-7}\).

    Step 4:Since the number is not in scientific notation, it must be adjusted. To put \(36.5\times {10}^{-7}\) into scientific notation, the decimal moves one to the left, so the exponent would be increased by 1, giving \(3.65\times {10}^{-6}\).

    \((5\times {10}^{-13})\times (7.3\times {10}^{6})=3.65\times {10}^{-6}\)

Dividing Numbers in Scientific Notation

To divide two numbers that are in scientific notation:

Step 1: Divide the number parts.

Step 2: Subtract the exponent of the denominator from the exponent of the numerator.

Step 3: The answer is the result from Step 1 times 10 raised to the result from Step 2.

Step 4: If the number is not in scientific notation, adjust it appropriately.

Dividing Numbers in Scientific Notation

Try it.

Calculate the following:

  1. \((8.4\times {10}^{31})\ /\ (2.1\times {10}^{7})\)
  2. \((4.14\times {10}^{-13})\ /\ (8.28\times {10}^{9})\)
Solution

  1. \((8.4\times {10}^{31})\ /\ (2.1\times {10}^{7})\)

    Step 1: Divide the number parts to get \(8.4\div 2.1=4\).

    Step 2: Subtract the exponent of the denominator from the exponent of the numerator to get \(37-7=24\).

    Step 3: The result is then \(4\times {10}^{24}\).

    Step 4: This number is already in scientific notation, so no adjustment is necessary. \((8.4\times {10}^{31})\ /\ (2.1\times {10}^{7})=4\times {10}^{24}\)

  2. \((4.14\times {10}^{-13})\ /\ (8.28\times {10}^{9})\)

    Step 1: Divide the number parts to get \(4.14\div 8.28=0.5\).

    Step 2: Subtract the exponent of the denominator from the exponent of the numerator to get \(-13-9=-22\).

    Step 3: The result then is \(0.5\times {10}^{-22}\).

    Step 4: Since this number is not in scientific notation, it must be adjusted. To put \(0.5\times {10}^{-22}\) into scientific notation, the decimal needs to move one to the right, so the exponent is decreased by 1, giving \(5\times {10}^{-23}\). \((4.14\times {10}^{-13})\ /\ (8.28\times {10}^{9})=5\times {10}^{-23}\)

Using Scientific Notation in Computing Real-World Applications

As noted at the start of this section, scientific notation is useful when the standard representation of a number is awkward or impractical, which occurs when the numbers being used are extremely large or extremely small. For example, Venus is 67,667,000 miles from the sun. In scientific notation, this is \(6.7667\times {10}^{7}\). Planetary and galaxy distances is one set of numbers that is easier to express using scientific notation.

Calculating Distances

Try it.

How much farther from the sun is Earth compared to Venus if Venus is \(6.7667\times {10}^{7}\) miles from the sun and Earth is \(9.1692\times {10}^{7}\) miles from the sun?

Solution

To determine how much farther Earth is compared to Venus, we’d subtract the distances.
\(9.1692\times {10}^{7}-6.7667\times {10}^{7}=2.4025\times {10}^{7}\).

So, Earth is \(2.4025\times {10}^{7}\) miles farther from the sun than Venus.

Calculating Probability

Try it.

The probability of winning the Mega Millions lottery is published as \(3.304693\times {10}^{-9}\). The probability of being hit by lightning is approximated to be \(2\times {10}^{-6}\). How many times more likely are you to be hit by lightning than win the Mega Millions?

Solution

To find out how many times more likely you are to be hit by lightning, divide the probability of being hit by lightning by the probability of winning the Mega Millions. \[(2\times {10}^{-6})\ /\ (3.304693\times {10}^{-9})=6.609386\times {10}^{3}\]

Step 1: Divide the number parts to get = \(0.6052\) (rounded to the fourth digit).

Step 2: Subtract the exponent of the denominator from the exponent of the numerator to get \(-6-(-9)=3\).

Step 3: The result then is \(0.6052\times {10}^{3}\).

Step 4: Since this number is not in scientific notation, it must be adjusted. To put \(0.6052\times {10}^{3}\) into scientific notation, the decimal needs to move one place to the right, so the exponent is decreased by 1, giving \(6.052\times {10}^{2}\).

You are \(6.052\times {10}^{2}\), or 605.2, times more likely to be hit by lightning than you are to win the Mega Millions.

Condensed — the full section is in OpenStax Contemporary Mathematics.

What Numbers Could Be Considered “Too Big” or “Too Small”?

One wonders when the numbers we represent become too large or small for consideration. Perhaps the following examples put limits on what is meaningful. The number of particles in the known universe has been estimated at \(4\times {10}^{80}\) particles. The smallest distance that has been measured is \(1\times {10}^{-18}\ m\), though the theoretical smallest measurable value is \(1\times {10}^{-35}\ m\). The distance across the universe is \(4.4\times {10}^{26}\ m\). Considering what those numbers represent, the extreme largest and extreme smallest, they might be numbers that constrain what we should reasonably be expected to deal with.

Key Concepts

  • Some numbers are so large or so small that writing the number out is clumsy and make it difficult to determine the true size of the number. Scientific notation makes the number more readable and make the relative size of the number immediately apparent.
  • A number written in scientific notation is a number at least 1 and smaller than 10 multiplied by 10 raised to an exponent. Converting between scientific notation and standard notation involves correctly applying multiplication and division by powers of 10, which in practice equates to understanding how moving the decimal point of a number impacts the exponent of 10.
  • Adding and subtracting numbers in base 10 requires the exponent of 10 in each number be the same. Once the numbers are converted to have the same exponent with the ten, then the numbers are added or subtracted as indicated, with the power of 10 remaining the same. If the result is not in scientific notation (for instance, the number has exceeded 10), then then number must be converted into scientific notation.
  • Multiplying and dividing numbers in scientific notation is done by multiplying or dividing the number parts, then multiplying or dividing the 10 raised to the power parts, then multiplying those two results. If the new number is not in scientific notation, then the result must be converted into scientific notation.

Videos

  • Converting from Standard Form to Scientific Notation Form
  • Converting from Scientific Notation Form to Standard Form
  • Multiplying Numbers in Scientific Notation
  • Dividing Numbers in Scientific Notation
  • Application of Scientific Notation

Practice (15)

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

  1. Which of the following numbers are in scientific notation? If the number is not in scientific notation, explain why it is not.

    1. \(-9.67\times {10}^{20}\)
    2. \(145\times {10}^{-8}\)
    3. \(1.45\)
    كشفت الإجابة
    1. The number \(-9.67\times {10}^{20}\) is in scientific notation because the absolute value of −9.67 is at least 1 and less than 10.
    2. The number \(145\times {10}^{-8}\) is not in scientific notation because 145 is not at least 1 and less than 10.
    3. The number \(1.45\) is not in scientific notation form. Even though it is at least 1 but less than 10, it is not multiplied by 10 raised to a power.

  2. Write the following numbers in scientific notation form:

    1. 428.9
    2. −0.00000981
    3. 8
    كشفت الإجابة
    1. Since the absolute value of 428.9 is 10 or larger, so we use the process from Case 3, above.

      Step 1: There are three digits to the left of the decimal point, so \(n=3\).

      Step 2: Write the digits of the number without the decimal place, which is 4289.

      Step 3: Since there is more than one digit, place the decimal point after the first digit. We now have 4.289.

      Step 4: Since \(n=3\), we multiply 4.289 by 10 raised to the second power, \(4.289\times {10}^{2}\).

      The scientific notation form of 428.9 is \(4.289\times {10}^{2}\).

    2. Since the absolute value of −0.00000981 is less than 1, we use the process from Case 2.

      Step 1: The number of zeros between the decimal and the first non-zero digit is 5, so \(n=5\).

      Step 2: We write the non-zero digits, including the negative sign, yielding −981.

      Step 3: The decimal gets placed to the right of the first digit, resulting in −9.81.

      Step 4: Since \(n=5\), we multiply −9.81 by 10 raised to the fourth power, \(-9.81\times {10}^{-6}\).

      The scientific notation form of −0.00000981 is \(9.81\times {10}^{-6}\).

    3. Since 8 is a single-digit integer, apply Case 1. The scientific notation form of 8 is \(8\times {10}^{1}\).

  3. Change \(456.142\times {10}^{5}\) by moving the decimal two places to the left.

    كشفت الإجابة

    Since we are moving the decimal to the left by two places, we increase the exponent of 10 by 2, so that the exponent is now 7. This gives us \(456.142\times {10}^{5}=4.56142\times {10}^{7}\).

  4. Change \(12.3\times {10}^{2}\) by moving the decimal five places to the right.

    كشفت الإجابة

    Since we are moving the decimal to the right by five places, we decrease the exponent of 10 by 5, so that exponent is now −3. This give us \(12.3\times {10}^{2}=1230000.0\times {10}^{-3}\).

  5. Convert the following into standard form:

    1. \(2.78\times {10}^{9}\)
    2. \(9.04\times {10}^{-8}\)
    كشفت الإجابة
    1. Since the exponent is positive, the decimal moves nine places to the right, so \(2.78\times {10}^{9}\) is \(2,780,000,000\).
    2. Since the exponent is negative, the decimal moves eight places to the left, so \(9.04\times {10}^{-8}\) is \(0.0000000904\).

  6. Calculate the following:

    1. \(3.8\times {10}^{-3}+1.006\times {10}^{-3}\)
    2. \(9.61\times {10}^{8}-3.85\times {10}^{8}\)
    كشفت الإجابة
    1. Since the powers of 10 match, we use the distributive property of real numbers to factor 10−3 from the numbers. We then add the number parts separately to get 4.806. \(3.8\times {10}^{-3}+1.006\times {10}^{-3}=(3.8+1.006)\times {10}^{-3}=4.806\times {10}^{-3}\)
    2. Since the powers of 10 match, we use the distributive property of real numbers to factor 108 from the numbers. We then subtract the number parts separately to get 5.76. \[9.61\times {10}^{8}-3.85\times {10}^{8}=(9.61-3.85)\times {10}^{8}=5.76\times {10}^{8}\]

  7. Calculate the following:

    1. \(7.03\times {10}^{13}+8.5\times {10}^{13}\)
    2. \(4.3\times {10}^{21}-4.613\times {10}^{21}\)
    كشفت الإجابة

    1. Since the powers of 10 match, we add the number parts and multiply that by \({10}^{13}\): \(7.03\times {10}^{13}+8.5\times {10}^{13}=(7.03+8.5)\times {10}^{13}=15.53\times {10}^{13}\).

      However, \(15.53\times {10}^{13}\) is not in scientific notation because the absolute value of 15.53 is more than 10. To put this number in scientific notation, the decimal needs to move one to the left. To balance that move, the power of 10 must be increased by 1. So, the answer in scientific notation is \(1.553\times {10}^{14}\).

    2. Since the powers of 10 match, we add the number parts: \(4.3\times {10}^{21}-4.613\times {10}^{21}=(4.3-4.613)\times {10}^{21}=-0.313\times {10}^{21}\)

      However, \(-0.313\times {10}^{21}\) is not in scientific notation because it is less than 1. To put it in scientific notation, the decimal needs to move one to the right. To balance that move, the power of 10 must be decreased by 1. So, the answer in scientific notation is \(-3.13\times {10}^{20}\).

  8. Calculate the following:

    \[6.1\times {10}^{4}+4.8\times {10}^{5}\]

    كشفت الإجابة

    Step 1: The lower exponent is 4. To make this equal to the larger exponent, we increased it by 1.

    Step 2: Since the smaller exponent was increased by 1, move the decimal one to the left, so the addition become \(6.1\times {10}^{4}+4.8\times {10}^{5}=0.61\times {10}^{5}+4.8\times {10}^{5}\).

    Step 3: Now add the numbers, \(0.61\times {10}^{5}+4.8\times {10}^{5}=5.41\times {10}^{5}\)

    Step 4: The result is in scientific notation, so no additional adjustment is necessary.

    \(6.1\times {10}^{4}+4.8\times {10}^{5}=5.41\times {10}^{5}\)

  9. Calculate the following:
    \(7.9\times {10}^{-15}-6.8\times {10}^{-13}\)

    كشفت الإجابة

    Step 1: The lower exponent is −15 and the larger is −13. To make −15 equal to the larger exponent, we increased it by 2.

    Step 2: Since the smaller exponent increased by 2, move the decimal two to the left. The subtraction changes to \(7.9\times {10}^{-15}-6.8\times {10}^{-13}=0.079\times {10}^{-13}-6.8\times {10}^{-13}\).

    Step 3: Subtract the numbers, \(0.079\times {10}^{-13}-6.8\times {10}^{-13}=-6.721\times {10}^{-13}\).

    Step 4: The result is in scientific notation, so no additional adjustment is necessary.
    \(7.9\times {10}^{-15}-6.8\times {10}^{-13}=-6.721\times {10}^{-13}\)

  10. Calculate the following:

    1. \((4.3\times {10}^{3})\times (1.8\times {10}^{7})\)

    2. \((5\times {10}^{-1}{}^{3})\times (7.3\times {10}^{6})\)

    كشفت الإجابة

    1. Step 1: Multiply the number parts to get \(4.3\times 1.8=7.74\).

      Step 2: Add the exponents of the 10s to get \(3+7=10\).

      Step 3: The result is then \(7.74\times {10}^{10}\).

      Step 4: This number is already in scientific notation, so no additional adjustment is necessary, \((4.3\times {10}^{3})\times (1.8\times {10}^{7})=7.74\times {10}^{10}\).

    2. Step 1: Multiply the number parts to get \(5\times 7.3=36.5\).

      Step 2: Add the exponents of the 10s to get \(-13+6=-7\).

      Step 3: The result then is \(36.5\times {10}^{-7}\).

      Step 4:Since the number is not in scientific notation, it must be adjusted. To put \(36.5\times {10}^{-7}\) into scientific notation, the decimal moves one to the left, so the exponent would be increased by 1, giving \(3.65\times {10}^{-6}\).

      \((5\times {10}^{-13})\times (7.3\times {10}^{6})=3.65\times {10}^{-6}\)

  11. Calculate the following:

    1. \((8.4\times {10}^{31})\ /\ (2.1\times {10}^{7})\)
    2. \((4.14\times {10}^{-13})\ /\ (8.28\times {10}^{9})\)
    كشفت الإجابة

    1. \((8.4\times {10}^{31})\ /\ (2.1\times {10}^{7})\)

      Step 1: Divide the number parts to get \(8.4\div 2.1=4\).

      Step 2: Subtract the exponent of the denominator from the exponent of the numerator to get \(37-7=24\).

      Step 3: The result is then \(4\times {10}^{24}\).

      Step 4: This number is already in scientific notation, so no adjustment is necessary. \((8.4\times {10}^{31})\ /\ (2.1\times {10}^{7})=4\times {10}^{24}\)

    2. \((4.14\times {10}^{-13})\ /\ (8.28\times {10}^{9})\)

      Step 1: Divide the number parts to get \(4.14\div 8.28=0.5\).

      Step 2: Subtract the exponent of the denominator from the exponent of the numerator to get \(-13-9=-22\).

      Step 3: The result then is \(0.5\times {10}^{-22}\).

      Step 4: Since this number is not in scientific notation, it must be adjusted. To put \(0.5\times {10}^{-22}\) into scientific notation, the decimal needs to move one to the right, so the exponent is decreased by 1, giving \(5\times {10}^{-23}\). \((4.14\times {10}^{-13})\ /\ (8.28\times {10}^{9})=5\times {10}^{-23}\)

  12. How much farther from the sun is Earth compared to Venus if Venus is \(6.7667\times {10}^{7}\) miles from the sun and Earth is \(9.1692\times {10}^{7}\) miles from the sun?

    كشفت الإجابة

    To determine how much farther Earth is compared to Venus, we’d subtract the distances.
    \(9.1692\times {10}^{7}-6.7667\times {10}^{7}=2.4025\times {10}^{7}\).

    So, Earth is \(2.4025\times {10}^{7}\) miles farther from the sun than Venus.

  13. The probability of winning the Mega Millions lottery is published as \(3.304693\times {10}^{-9}\). The probability of being hit by lightning is approximated to be \(2\times {10}^{-6}\). How many times more likely are you to be hit by lightning than win the Mega Millions?

    كشفت الإجابة

    To find out how many times more likely you are to be hit by lightning, divide the probability of being hit by lightning by the probability of winning the Mega Millions. \[(2\times {10}^{-6})\ /\ (3.304693\times {10}^{-9})=6.609386\times {10}^{3}\]

    Step 1: Divide the number parts to get = \(0.6052\) (rounded to the fourth digit).

    Step 2: Subtract the exponent of the denominator from the exponent of the numerator to get \(-6-(-9)=3\).

    Step 3: The result then is \(0.6052\times {10}^{3}\).

    Step 4: Since this number is not in scientific notation, it must be adjusted. To put \(0.6052\times {10}^{3}\) into scientific notation, the decimal needs to move one place to the right, so the exponent is decreased by 1, giving \(6.052\times {10}^{2}\).

    You are \(6.052\times {10}^{2}\), or 605.2, times more likely to be hit by lightning than you are to win the Mega Millions.

  14. Sometimes it is entertaining to determine the time it takes for something to happen. Fingernails grow about \(8.032\times {10}^{-11}\) km per minute. How many kilometers long would fingernails be after \(6\times {10}^{4}\) minutes?

    كشفت الإجابة

    To find the length of the fingernails after the specified time, we multiply their rate of growth and the time they’ve grown. \((8.032\times {10}^{-11})\times (6\times {10}^{4})=48.192\times {10}^{-7}=4.8192\times {10}^{-6}\)

    So, after \(6\times {10}^{4}\) minutes, the fingernails would be \(4.8192\times {10}^{-6}\) km long. To put this in perspective, \(1\times {10}^{-6}\) km is a millimeter, and \(6\times {10}^{4}\) minutes is about 4.16 days. So, after about 4.16 days, fingernails have grown about 4.8 millimeters.

  15. As mentioned in the opening to this section, it is estimated that we’re producing 2.5 quintillion bytes of data per day. A good estimate is that there are 7.674 billion people on the planet. Convert both of those numbers to scientific notation, and then determine how much data is being generated per person each day.

    كشفت الإجابة

    Written in standard form, 2.5 quintillion is 2,500,000,000,000,000,000. Changing that to scientific notation, move the decimal 18 places, so 2.5 quintillion bytes = \(2.5\times {10}^{18}\) bytes. Writing 7.647 billion in scientific notation would be \(7.647\times {10}^{9}\) because a billion is 1,000,000,000 = \({10}^{9}\). So, to find out how much data is being produced daily per person, we would divide these two numbers. \(\frac{2.5\times {10}^{18}}{7.647\times {10}^{9}}=0.327\times {10}^{9}=3.27\times {10}^{8}\)

    In standard form, that’s 327,000,000 bytes per person, so 327 million bytes of data daily are being produced per person.

Symbols used here

\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: Scientific Notation

  1. Write numbers in standard or scientific notation.
  2. Convert numbers between standard and scientific notation.
  3. Add and subtract numbers in scientific notation.
  4. Multiply and divide numbers in scientific notation.
  5. Use scientific notation in computing real-world applications.
  6. The number
  7. The number
  8. The number

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.

جرّب نفسك

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

أكثر في Number Theory

Prime factorisationPrime numbersGCD and LCMModular arithmeticDivisorsSequencesNumber basesDiophantine equationsFermat's little theorem and Euler's theoremRSA: cryptography from number theory