And the radius of an electron is thought to be even smaller again, at about one onethousandth of the radius of a proton!

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1 Guided Inst./Prac.: Scientific Notation M8001 Did you know that the radius of a hydrogen atom is about five one hundred-billionths of a meter? That s zero, point zero, zero, zero, zero, zero, zero, zero, zero, zero, zero, five meters. Within the hydrogen atom are even smaller particles called protons and electrons. The radius of a proton is known to be about one-quadrillionth of a meter that s one fifteen places to the right of the decimal point! And the radius of an electron is thought to be even smaller again, at about one onethousandth of the radius of a proton! As you can imagine, writing out numbers like these all the time could get quite tedious! This is why many scientists and mathematicians use a system to represent very large and very small numbers in a way that makes them easy to read and write this system is called scientific notation. Scientific notation is a way to write a number as a decimal number greater than or equal to one, and less than ten, multiplied by a power of ten. To help us understand how scientific notation is used to write very small numbers, let s first review the powers of ten. We know that ten to the power of zero equals one, ten to the power of one equals ten, ten to the power of two equals one hundred, ten to the power of three equals one thousand, and so on.

2 Okay, we ve reviewed powers of ten for place values to the left of the decimal point, but what about the powers of ten for place values to the right of the decimal point that is, tenths, hundredths, thousandths and so on? Well, let s look at the pattern here. We know that one hundred is written as ten to the power of two, ten is written as ten to the power of one, and one is written as ten to the power of zero. Notice that as we move to the right, each place value is one-tenth of the previous place value. Also, notice that the power of ten decreases by one for each place value we move to the right. Now let s continue this pattern: the value of the next place to the right is found by taking one-tenth of one, which is one-tenth, so this is the tenths place. And decreasing the power of ten by one more gives ten to the power of negative one! The place to the right of that is the hundredths place, so here we can write ten to the power of negative two. And the power of ten, or exponent, continues to decrease by one for each place value we move to the right. Now we know that the place values on the right side of the ones place represent numbers between zero and one. So numbers between zero and one can be expressed using negative powers of ten! One tenth can be written as ten to the power of negative one; one-hundredth can be written as ten to the power of negative two, and so on. How is one ten-thousandth expressed using a power of ten?

3 So what about scientific notation? Remember that it involves writing a number as a decimal number greater than or equal to one, and less than ten times a power of ten. Well we ve seen that one ten-thousandth, for example, can be expressed as ten to the power of negative four but that s just the same as one point zero times ten to the power of negative four. Now we ve written one ten-thousandth in scientific notation! Remember the hydrogen atom we talked about earlier? Let s see how we can use scientific notation to represent its radius, which is about zero point zero, zero, zero, zero, zero, zero, zero, zero, zero, zero five meters, or five one hundred billionths of a meter. To see how we write this number in scientific notation, we can first extend the place value chart out to the one hundred billionths place. We can see from the chart that the value of the one hundred billionths place can be written as ten to the power of negative eleven. So five one hundred billionths can be written as five point zero times ten to the power of negative eleven. The radius of a hydrogen atom, written in scientific notation, is five point zero times ten to the power of negative eleven meters. We don t always have a place value table at our fingertips, but there is a three-step method we can use to make writing numbers in scientific notation easier. The first step is to move the decimal point until the number is greater than or equal to one and less than ten. In this case, we move the decimal point to the right until the number is five point zero. The second step is to identify the direction we move the decimal point. Think of it this way: if we move the decimal point to the right to change a number in standard form to

4 scientific notation, the exponent is negative. And if we move the decimal point to the left to change a number in standard form to scientific notation, the exponent is positive. In this example, we move the decimal to the right, so we know the exponent is negative. The third step is to count how many places we move the decimal point. In this example, the decimal is moved eleven places, so our exponent is negative eleven. These three steps can make it easier when working with very large numbers or very small numbers. The radius of an iron atom has been calculated to be about zero point zero, zero, zero, zero, zero, zero, zero, zero, zero, one four meters. What is this length in scientific notation? We can also convert from scientific notation to standard form. As an example, let s look at one point zero times ten to the power of negative eighteen. We know this number is much less than one, so we ll move the decimal point to the left. Because the number in the exponent is negative eighteen, we should move the decimal point eighteen places to the left and drop the power of ten. So, in standard form, one point zero times ten to the power of negative eighteen can be rewritten as zero point zero, zero, zero, zero, zero, zero, zero, zero, zero, zero, zero, zero, zero, zero, zero, zero, zero, one. That number is called one-quintillionth! We ve been looking at very small numbers so far, but remember that we can also write numbers that aren t quite so small in scientific notation for example, three point four five times ten to the power of negative four. We can use our rule to write them in standard form also here the exponent is negative four, so we move the decimal point four places to the left, to get zero point zero, zero, zero, three, four, five.

5 Try this one. How is one point three five times ten to the power of negative five written in standard notation? In this lesson we talked about how to write numbers between zero and one using scientific notation. To convert a number between zero and one from standard form to scientific notation, we can: Move the decimal point until the number is greater than or equal to one and less than ten. Next, we identify the direction we move the decimal point. If we move the decimal point to the right to change a number in standard form to scientific notation, the exponent is negative. Remember that when using scientific notation, we multiply by a power of ten. So now we count how many places we moved the decimal point to find the exponent of ten. To convert a number between zero and one from scientific notation to standard notation, we can: Move the decimal point the same number of places to the left as the exponent of ten, and then drop the power of ten. Do your best to answer each problem correctly on the first try so you won t lose any points. Most people can see objects with a width as small as zero point zero zero zero one meters from a distance of a meter. How is this width written in scientific notation? What is two point one four times ten to the power of negative four meters written in standard form?

6 A ten liter sample of swimming pool water was found to contain four point one one times ten to the power of negative two grams of chlorine disinfectant. What is this amount written in standard form? What is three point zero two times ten to the power of negative seven meters written in standard form? The common cold virus has a diameter of about twenty-eight billionths of a meter. Express this in scientific notation. A dust particle was found to be one point six nine times ten to the power of negative five inches in diameter. What is one point six nine times ten to the power of negative five inches written in standard form? The width of a plant cell is zero point zero zero zero zero one two seven six meters. How is this width written in scientific notation? The size of a flea egg is approximately zero point zero zero zero five one three meters. Express this in scientific notation. If you're ready to exit, click Done.