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 notation
Similarly 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!
Examples 2 × 103 = 2,000 4.1 × 10–5 = 0.000041 2.9 × 1014 = 290,000,000,000,000
Write 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 × 10–2
Write the following numbers in scientific notation.
Write the following numbers so that they have the exponent of 10 shown. For example, 1.2 x 1011 = 0.012 x 1013 = 120. x 109
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.
509 + 71 ------ 580
In order to add or subtract two numbers in scientific notation both numbers must be expressed using the same power of ten. Otherwise, the digits with the same place value will not be lined up and you will end up with a wrong answer. For example, it is clear that when adding 509 and 71 you should line up the 7 under the 0 and the 1 under the nine:
(5.09 × 102) + (7.1 × 101) (5.09 × 102) = (7.1 × 101) --------------- and this is wrong! ==> 12.19 × 103
However, when the same two numbers are written in scientific notation our usual instinct to line numbers up on the decimal point is betrayed (see at right).
This result is clearly incorrect because the answer we know is correct (580) is written as 5.80 × 102 in scientific notation. The problem is that we did not line up the numbers based on their place values. Instead we lined them up on the decimal points. When using scientific notation the decimal point alone is not enough of a guide to the value of the digits in the number. The exponent communicates the correct place value of each digit based on the following rule: the place value of the digit to the left of the decimal point is given by the exponent.
We normally think about place value based on names: ones, tens, hundreds or tenths, hundredths, thousandths. Place value can be usefully abbreviated by using the powers of ten that are used when writing a number using scientific notation. See the table at right. The '5' in '509' is in the hundreds place or, as we may now call it, the 102 place. So when we write it in scientific notation with the decimal to the right of the 5 it becomes 5.09 × 102. We could write it in other ways, too. For example, if the decimal were placed after the '0', which is in the tens place, then it becomes 50.9 × 101. The number has the same value but we have just re-written it with a new power of ten. The power of ten we use just depends on the place value of the digit just to the left of the decimal.
This discussion suggests a way in which we might set up the arithmetic for adding and subtracting numbers expressed in scientific notation. Take a look at the following example.
5.4 × 1012 + 2.3 × 1011Write the digits in a chart with each digit in the column that correctly gives its place value (see the table at left).
Now that we have lined up the digits correctly and added the two numbers we have only to re-write the answer to give each digit its proper place value. By putting the decimal to the right of the '5' we can look at the top of the chart to see that the correct power of ten to give its place value is 12. So the answer is 5.63 × 1012. Of course, another way to do the problem is to adjust one number to have the same power of ten as the other number. Then it is simply a matter of lining up the decimals to add or subtract. Worked out in this way the problem looks like this:
(5.4 x 1012) + (2.3 x 1011) = (5.4 x 1012) + (0.23 x 1012) = 5.63 x 1012
Addition and Subtraction Example (3 x 104) + (2 x 103) = (3 x 104) + (0.2 x 104) = 3.2 x 104 = 32,000
Add or subtract. Express all answers in scientific notation.
Do as many of the following problems as you need to do to get good at the arithmetic. They have all been written to be done with no help at all from a calculator. As an additional exercise you might try checking your mental arithmetic using the calculator’s scientific notation function. But be careful! Some of the problems will not be possible on the calculator because the numbers are so large the calculator can’t handle them!
Do these problems as homework.