6.1 scientific notation
Understanding how your calculator displays and handles very large and very tiny numbers is essential if you space to translate the results of calculations correctly. This section focuses on a method of representing numbers well-known as scientific notation.
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Before friend start put your calculator right into the float mode, so the will display screen up to about 10 digits and also return to the home screen ready to execute some calculations.
What answer would certainly you intend if you square 20 million? How numerous zeros would certainly the prize have? If you perform this ~ above paper, girlfriend should finish up with 4 complied with by 14 zeros. Yet your calculator can not have the ability to display 14 figures, if it is set to 10 figures. So input this calculation right into your calculator and see what that does:
20000000
You will probably get something favor 4e14. The notation e14 must be review as 10 to the power 14 (or 10 come the 14) which means fourteen 10s multiplied together. It is created as 1014. For this reason the calculator answer of 4e14 is 4 × 1014, i beg your pardon is quick for 400000000000000.
The number 14, the is the strength of 10, is likewise called the exponent and method the variety of 10s which are multiplied together. The letter E in 4e14 is offered to mean exponent.
Now instead of 20 million squared, what would 90 million squared be? What would you expect? try it on her calculator and also see.
Since 20 million squared to be 4 × 1014, you might expect that 90 million squared would be 81 × 1014.
However, the calculator reflects a various answer. Shot it and also see.
The calculator"s answer of 8.1e15 or 8.1 × 1015 is in fact the exact same number together 81 × 1014.
Both this numbers stand for 8 100 000 000 000 000.
However, once a number is written using a clinical notation the convention is to use a number in between 1 and also 10 prior to the exponent. So here, instead of 81 × 1014, the number 8.1 is provided with one extra power of 10.
Another means of interpreting 8.1 × 1015 is come imagine the the decimal suggest in the number 8.1 have to be moved 15 locations to the right, pour it until it is full in as numerous zeros as necessary.
(a) The speed of irradiate measured in meters per second, is around 300 million (300 000 000) meters every second. What is this number in clinical notation?
(b) Einstein"s renowned equation says that E = mc2, the power E (in joules) connected with a fixed of m kilograms is same to m multiply by c squared, whereby c is the speed of light (in meters every second). Utilizing the worth of c provided in your answer to component (a), job-related out the worth of E in scientific notation for a fixed of 1 kilogram.
(c) The street from the earth to the sun is about 149 million kilometres. What is this number in clinical notation?
(d) What is the biggest number her calculator have the right to handle? for example, enter any number and keep squaring. How much can friend go before the calculator displays an error message?
Answer
(a) 300 000 000 = 3 × 108
(b)
(c)
(d) The TI-84 calculator cannot manage numbers with exponents of 100 or more. Notification that the calculator produces a blog post saying ERR:OVERFLOW.
Small numbers really close to zero can also be expressed in clinical notation. Because that example, get in a straightforward starting worth such together 4.3 and also then continuously divide through 10. This can be excellent by pressing
4 . 3
And so on.
Watch closely how the numbers go: 4.3, .43, .043, .0043, etc. Then at some suggest the display will jump right into scientific notation. Different calculators use different methods the displaying scientific notation. Watch in your calculator handbook to check how your calculator displays scientific notation and also compare it with that used by the TI-84, which gives .00043 together 4.3e−4.
The e−4 part way that the decimal point in the number 4.3 should be moved four places come the left, which requires the insertion the some additional zeros.
What wake up to the result if friend keep splitting by 10? You should see the exponent readjust from −4 to −5 to −6, and so on. As the negative exponent it s okay larger, the equivalent decimal number gets smaller.
As with large numbers, the notation e−4 need to be review as ‘10 to the strength −4’ (or just ‘10 to the minus 4’).
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Do not issue about negative powers of 10 because that the moment. Simply think of the notation as a method of composing down small numbers where the negative sign reminds you that the decimal allude must be moved to the left. Right here is another example.