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Welcome to our Math lesson on Very Big and Very Small Numbers, this is the fifth lesson of our suite of math lessons covering the topic of Standard Form, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
Very Big and Very Small Numbers
Standard numbers are not suitable to use when expressing normal values if we don't have any strong reason for this. For example, it is not necessary to write 38 as 3.8 × 101 or 0.41 as 4.1 × 10-1 unless this is explicitly demanded.
We normally use the standard form for convenience - to express either very large or very small numbers. For example, instead of saying, "the green light has a frequency of about 545 trillion hertz", we say, "the frequency of green light is about 5.45 × 1014 hertz". This saves us the time and effort to write numbers with many zeroes at the end.
The same thing can be said for very small numbers as well. Thus, instead of saying "the diameter of a hydrogen atom is 0.106 nm" (nm means 'nanometre', which is equal to one billionth of one metre), we say "the diameter of a hydrogen atom is 1.02 × 10-10 m". This saves us a lot of previous time in units conversions and other annoying actions deriving from the use of such numbers.
The general rules applied when converting very small or very large ordinary numbers into standard form are as follows:
- If a whole number A = abcdef…n is very big, the decimal point shifts n - 1 positions due left and the index of ten also becomes n - 1, where n is the number of digits of the original number. Hence, the number becomes A = a.bcdef…n × 10n-1. For example, in the number A = 327,416, we have n = 6. Thus, we obtain for the standard form of this number: A = 3.27416 × 105.
- If a big number contains a decimal part (if it is not whole therefore), the above rule is applied by starting from the decimal point. However, the digits after the decimal point are all written when the number is converted into the standard form. For example, if A = 30,681.27 is converted to standard form (n = 5), it becomes A = 3.068127 × 104.
- If a very small decimal number B = 0.0000abcd needs to be converted into standard form, we shift the decimal point from its actual position to after the first non-zero digit (here after a). As for the index, we count how many positions the decimal point has shifted and this number is written as a negative index. In our case, we have a shift by five positions due right, so B = a.bcd × 10-5.
Example 5
The wavelength of yellow light is 580 nm. Given that the speed of light is 300,000 km/s, calculate the frequency of yellow light in hertz. The equation of light waves is
speed = wavelength × frequency
where speed is measured in metres per second.
Solution 5
First, we write all values in standard form. Thus, since nanometre is one billionth of metre or 10-9 m and 1 km = 1000 m = 103 m, we have
Wavelength = 580 nm = 580 × 10-9 m = 5.80 × 10-7 m
Speed = 300,000 km/s = 300,000 × 1000 m/s = 300,000,000 m/s = 3.0 × 108 m/s
Thus, since
speed = wavelength × frequency
frequency = speed/wavelength
= 3.0 × 108 m/s/5.80 × 10-7 m
≈0.517 × 108-(-7) Hz
= 0.517 × 1015 Hz
= 5.17 × 1014 Hz
More Standard Form Lessons and Learning Resources
Powers and Roots Learning Material| Tutorial ID | Math Tutorial Title | Tutorial | Video
Tutorial | Revision
Notes | Revision
Questions |
|---|
| 7.3 | Standard Form | | | | |
| Lesson ID | Math Lesson Title | Lesson | Video
Lesson |
|---|
| 7.3.1 | The Meaning of Standard Form. | | |
| 7.3.2 | Writing Decimals in Standard and Decomposed Form | | |
| 7.3.3 | Addition and Subtraction with Numbers in Standard and Decomposed Form | | |
| 7.3.4 | Multiplication and Division of Numbers in Standard Form | | |
| 7.3.5 | Very Big and Very Small Numbers | | |
| 7.3.6 | Powers of Numbers Written in the Standard Form | | |
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