3.3 Scientific Notation

Scientific notation is an efficient, shorthand way of expressing very large or very small numbers. Scientific notation gives us an instant idea of how large or small a number is without having to count all those zeros to determine place value. Scientific notation gets its name because it is primarily used in the sciences, but it can be used in any field.

Scientific notation is composed of a product of two parts; a decimal number part and a power of 10 part. For example, [latex]2.36\times10^3[/latex]. The number part must be a decimal that is greater than or equal to 1 but less than 10. Any number may be expressed in scientific notation. Scientific notation gives us an instant idea of how large or small a number is without having to count all those zeros to determine place value. For example, [latex]2.36\times10^3[/latex] immediately tells us that we are dealing with a number in the thousands since the exponent is 3.

scientific notation

A number is written in scientific notation when it is written as:

[latex]a\times10^n[/latex]

where [latex]0< a\leq1[/latex] and [latex]n[/latex] is an integer.

Scientific Notation

Writing decimal numbers in scientific notation using place value

We write a decimal number using place values. For scientific notation, we write a number based on the place value of its first non-zero digit. For example, the number 9560 has the digit 9 in the thousands place. We may write 9560 as 9.560 thousands, or, since [latex]1000=10^3[/latex], we write [latex]9.560\times10^3[/latex].

Walmart’s net income was about 13.7 billion dollars in 2022[1], which is $13,700,000,000. The place value of the digit 1 is the ten-billions place. We may write the number as 1.37 ten-billions, or, since ten-billions [latex]= 10^{10}[/latex], we write [latex]1.37\times10^{10}[/latex].

Examples

Write the following numbers in scientific notation.

  1. The average home value in Los Angeles in 2022 was about $951,000[2].

Solution

The 9 in $951,000 is in the hundred thousands place, which is [latex]10^5[/latex]. So, [latex]\$951,000=\$9.51\times10^5[/latex] in scientific notation.

  1. The population of the Salt Lake City metro area at the end of 2022 was about 1,258,000 people[3].

Solution

1,258,000 people is 1.258 million people. Million [latex]=10^6[/latex],  so [latex]1,258,000[/latex] people [latex]=1.258\times10^6[/latex] people.

  1. The length of a Queen Fire Ant is about 0.006 meter.[4]

Solution

The first non-zero digit (4) is in the thousandth place. One-thousandth [latex]=10^{-3}[/latex], so [latex]0.004[/latex] meters [latex]=4\times10^{-3}[/latex] meters.

  1. The size of the smallest mite is about 0.0001 meter.[5]

Solution

The first non-zero digit is in the ten-thousandth place. One ten-thousandth [latex]=\times10^{-4}[/latex], so in scientific notation [latex]0.0001[/latex] meters [latex]=1\times10^{-4}[/latex] meters

TRY IT

  1. [latex]3.2 \times 10^5[/latex]
  2. [latex]2 \times 10^{-3}[/latex]
  3. [latex]5.87 \times 10^8[/latex]
  4. [latex]3.007 \times 10^{-12}[/latex]
Show/Hide Answer
  1. [latex]320,000[/latex]
  2. [latex]0.002[/latex]
  3. [latex]587,000,000[/latex]
  4. [latex]0.000000000003007[/latex]

Writing numbers in scientific notation by moving the decimal point

Instead of using the place value of the first non-zero digit in the number to determine the power of 10 in scientific notation, we can count the number of places and direction we must move the decimal point to get from a number between 1 and 10 for the decimal part back to the original number. For example, to write scientific notation for the number 9561, we put the decimal point after the first non-zero digit (9) and use 9.561 as the decimal part. To get back to 9651 from 9.560, we need to move the decimal point of the number 9.561 three places to the right, which is equivalent to multiplying by 1000 or [latex]10^3[/latex]. Therefore, the number 9561 is written as [latex]9.561\times10^3[/latex] in scientific notation.

For the number 0.000007, we need to move the (unwritten) decimal point of the number 7 to the left 6 places to turn 7 into 0.000007. This is equivalent to dividing 7 by [latex]10^6[/latex] or multiplying 7 by [latex]10^{-6}[/latex]. Therefore, the number 0.000007 may be written as [latex]7\times10^{-6}[/latex].

Examples

Write the following numbers in scientific notation.

  1. As of January 2023, the population of the United States is about 336,000,000 people.

Solution

The decimal part is 3.36 (we don’t write all the zeros!) and the decimal point would need to be moved 8 places to the right to get back to 336,000,000. Therefore, 8 is the power on the 10. In scientific notation, 336,000,000 people is written as [latex]3.36\times10^8[/latex] people.

  1. The population of the world in 2023 is around 8 billion people.

Solution

8 billion as a decimal is 8,000,000,000. The decimal part is 8, whose (unwritten) decimal point would need to be moved 9 places to the right to get back to 8,000,000,000. Therefore 8 billion people is written in scientific notation as [latex]8\times10^9[/latex] people.

  1. The distance from Earth to Mars is about 33.9 million miles.

Solution

33.9 million as a decimal is 33,900,000. Therefore, the decimal part of scientific notation is 3.39 and the decimal point would need to be moved 7 places to the right to get back to 33,900,000. In scientific notation, 33.9 million miles [latex]=3.39\times10^7[/latex] miles.

  1. A single red blood cell is about 0.0000078 meters in diameter.[6]

Solution

For the number 0.0000078, the first non-zero digit (7) is in the place value millionths ([latex]10^{-6}[/latex]). Therefore, in scientific notation [latex]0.0000078[/latex] meters [latex]=7.8\times10^{-6}[/latex].

TRY IT

Convert the following decimal numbers into scientific notation.

  1. [latex]3,543,000,000,000[/latex]
  2. [latex]0.00000234[/latex]
  3. [latex]234,500,000,000,000[/latex]
  4. [latex]0.000007097[/latex]
  5. A bacteria is about one millionth of a meter in diameter.
  6. A hydrogen atom is about 0.0000000000529 meters.[7]
Show/Hide Answer
  1. [latex]3.543\times10^{12}[/latex]
  2. [latex]2.34\times10^{-6}[/latex]
  3. [latex]2.345\times10^{14}[/latex]
  4. [latex]7.097\times10^{-6}[/latex]
  5. [latex]1\times10^{-6}[/latex] meters
  6. [latex]5.29\times10^{-11}[/latex] meters

Of course, if we can convert decimal numbers into scientific notation, we can convert numbers written in scientific notation into decimal numbers. The exponent on the 10 in scientific notation will tell us how many places to move the decimal point in the number part of the scientific notation. A positive exponent will require us to move the decimal point to the right, while a negative exponent will require us to move the decimal point to the left.

For example, [latex]4.887\times10^{5}[/latex] becomes 488,700 as a decimal number. The decimal point that was after the 4 in the scientific notation number moves 5 places to the right. This requires us to add two zeros onto 4887.

Example

Homo-sapiens have lived on Earth for around [latex]3\times10^5[/latex] years.[8] Express the age of homo-sapiens in scientific notation.

Solution

To convert [latex]3\times10^5[/latex] to a decimal number the exponent tells us to move the decimal point in 3. five places to the right:  3. becomes 300,000.

So, [latex]3\times10^5=300,000[/latex].

 

TRY IT

Convert the following numbers written in scientific notation to decimal numbers.

  1. [latex]3.543\times10^{12}[/latex]
  2. [latex]2.34\times10^{-6}[/latex]
  3. [latex]2.345\times10^{14}[/latex]
  4. [latex]7.097\times10^{-6}[/latex]
  5. The amount of water surface area on the Earth is about [latex]1.4\times10^8[/latex] square miles. Write this as a decimal number.
  6. The age of the Earth is estimated to be [latex]4.543\times10^9[/latex] years old [9]. Express the age of the Earth as a decimal.
  7. As of January 27, 2023, the US National Debt is about $31.5 x 1013 (Note that this is not written in scientific notation because the number part, 31.5, should be between 1 and 10).
  8. Amazon’s net income in 2021 was [latex]3.336\times10^{10}[/latex] dollars[10].
Show/Hide Answer
  1. [latex]3,543,000,000,000[/latex]
  2. [latex]0.00000234[/latex]
  3. [latex]234,500,000,000,000[/latex]
  4. [latex]0.000007097[/latex]
  5. [latex]140,000,000[/latex] square miles
  6. [latex]4,543,000,000[/latex] years
  7. [latex]\$31.5\times10^{13}=(\$3.15\times10)\times10^{13}=\$3.15\times10^{14}[/latex]
  8. $33,360,000,000
  1. https://www.macrotrends.net/stocks/charts/WMT/walmart/net-income ↵
  2. https://www.zillow.com/home-values/12447/los-angeles-ca/ ↵
  3. https://www.statista.com/statistics/815769/salt-lake-city-metro-area-population/ ↵
  4. https://pestsamurai.com/ant-size-chart-and-comparison/ ↵
  5. https://www.orkincanada.ca/pests/mites/ ↵
  6. https://www.britannica.com/science/blood-biochemistry/Red-blood-cells-erythrocytes ↵
  7. https://education.jlab.org/qa/how-much-of-an-atom-is-empty-space.html ↵
  8. https://www.forbes.com/sites/startswithabang/2019/05/15/what-was-it-like-when-the-first-humans-arose-on-earth/?sh=2aba07f36997 ↵
  9. https://education.nationalgeographic.org/resource/resource-library-age-earth ↵
  10. https://www.statista.com/statistics/266288/annual-et-income-of-amazoncom/ ↵