You will learn the basics of scientific notation in this activity. Scientific notation is a way of writing very large and very small numbers with a minimum of fuss and without having to count a lot of zeros.
In science we are often forced to use numbers that are inconveniently large or small. For example, the number of stars in the Andromeda Galaxy is
Another example is the mass of a single He (helium) atom:
Because these numbers are such a pain to write out, and more importantly, so difficult to do calculations with we use scientific notation. Scientific notation is like an abbreviation for numbers.
The number 200,000,000,000 can be re-written as the product of 2 and 100,000,000,000 (2 × 100,000,000,000). Conveniently, the number 100,000,000,000 can be re-written as a power of ten: 1011. (Count the zeros: there are 11 of them.) To write 200,000,000,000 in scientific notation you write the two factors together like this:
When using a calculator to do calculations with this number you will need one that can handle scientific notation. You can tell whether your calculator can do it by looking for a button that says e, ee, or exp. When you use these buttons you are telling the calculator “times ten to the power of”. Never use any key combination that results in 10^X since this can cause problems when the calculator follows the rules of order of operations. Many calculators (and documents that cannot produce superscriptslike this) use an alternate way to write scientific notation. They would display the number 2 × 1011 as
This is read the same way as 2 × 1011: two times ten to the eleventh power.
The mass of the He atom is written this way using scientific notation:
Here is how to go about changing a number from normal notation to scientific notation.
Write 6,750,000 using scientific notationSimilarly a number like 1.1 millionths of a second (0.000 001 1 s) can be written as 1.1 × 10-6 s. The difference is that you count digits to the left of the new decimal position. When you count to the left the exponent of the base will be negative. Negative exponents indicate that a number is a fraction: that is, that the number is less than one. Be careful not to count the zero in front of the old decimal point!
smaller bigger
Fraction 1/000 1/100 1/10
Decimal notation 0.001 0.01 0.1 1 10 100 1,000
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Scientific notation 10-3 10-2 10-1 100 101 102 103
Examples 2 × 103 = 2,000 4.1 × 10–5 = 0.000041 2.9 × 1014 = 290,000,000,000,000Write the following numbers in decimal form.
Examples 602,200,000,000,000,000,000,000 = 6.022 × 1023 0.00132 = 1.32 × 10–3 0.0193 = 1.93 × 10–2Write the following numbers in scientific notation.
Examples 5 × 104 < 5 × 109 6 × 10–9 > 3 × 10–9Write a greater than sign or less than sign between each of the following pairs of numbers to show which is larger.
Now you should know how to write numbers using scientific notation. If you do not yet understand any of the exercises on the previous page, please ask for help before going on.
Numbers written in scientific notation are easy to multiply and divide We’ll start with some problems using just powers of ten.
Multiplication Examples 102 × 104 = 102 + 4 = 106 1014 × 10–6 = 1014 + (-6) = 108 Division Examples 102 ÷ 104 = 102 – 4 = 10–2 1014 ÷ 10–6 = 1014 – (-6) = 1020 106 —— = 106 – 4 = 102 104
When multiplying, multiply the numbers and add the exponents. When dividing, divide the numbers and subtract the exponents. Finally, once you have an answer in scientific notation change the answer so that the number multiplied by a power of 10 is between 1 and 10:
Multiplication and Division Examples (2 x 102) × (6 x 103) = (6 x 103) ÷ (2 x 102) = 12 x 105 = 1.2 x 106 3 x 101 (2.7 x 109) × (4.2 x 103) = (2.16 x 102) ÷ (1.2 x 106) = 11.34 x 1012 = 1.134 x 1013 1.8 x 10–4 (5.6 x 10–9) × (2.3 x 103) = (4.5 x 102) ÷ (9.0 x 10–5) = 12.88 x 10–6 = 1.288 x 10–5 0.50 x 107 = 5.0 x 106 Multiply or divide as indicated. Simplify all answers so that the number multiplied by a power of 10 is between 1 and 10.
The first thing to note when adding and subtracting numbers in scientific notation is what order of magnitude the numbers are in. Order of magnitude refers to the difference between two exponents of 10. The number 1.2 × 102 is one order of magnitude smaller than the number 2.4 × 103. Likewise, 6.02 × 1023 is 20 orders of magnitude larger than 6.0 × 103. Numbers should only be added together (or subtracted from one another) when they are within 2 or at most 3 orders of magnitude. If you need to add two numbers that differ by more than 3 orders of magnitude (like 104 and 109) just write down the larger number. It is so much larger that the smaller number adds too small an amount to matter. If you must subtract a smaller number from a larger one (say 102 – 10–3) just write the larger number as the answer: the number you are subtracting is too small to matter.
Note that you must make the exponents the same order of magnitude to add or subtract. To make a number have a larger exponent, you must write the number smaller. To make a number have a smaller exponent, you must write the number larger. That just means moving the decimal point:
These numbers all mean the same thing! (1.25 if not using scientific notation) 0.0125 × 102 0.125 × 101 1.25 × 100 12.5 × 10–1 125 × 10–2 1,250 × 10–3 To change 1.25 × 100 so that the exponent is 2 you must multiply by 102 and divide by 102: 0.0125 = 1.25 ÷ 102 102 = 100 × 102 so 1.25 × 100 = 0.0125 × 102 Write the following numbers so that they have the exponent of 10 shown. Also write each number in decimal notation. One is done for you.
Addition and Subtraction Examples
(3 x 104) + (2 x 103) = (3 x 104) + (0.2 x 104)
= 3.2 x 104 = 32,000
(3 x 104) - (2 x 103) = (30 x 103) - (2 x 103)
= 28 x 103 = 2.8 x 104
Change numbers so that both in each problem have the same exponent, then
add or subtract. If numbers differ by more than 2 orders of magnitude,
then write down the larger number without performing any operation.
Do these problems as homework.