6 large and small numbers

6.1 clinical notation

Understanding how your calculator displays and also handles very big and very small numbers is necessary if you space to translate the results of calculations correctly. This section focuses on a means of representing numbers known as scientific notation.

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Before girlfriend start placed your calculator right into the float mode, so that will display up to about 10 digits and also return to the home display screen ready to execute some calculations.

What answer would you suppose if friend square 20 million? How countless zeros would certainly the answer have? If you perform this top top paper, you should end up with 4 adhered to by 14 zeros. Yet your calculator could not have the ability to display 14 figures, if that is set to 10 figures. For this reason input this calculation right into your calculator and see what it does:

20000000

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You will certainly probably gain something choose 4e14. The notation e14 should be check out as 10 come the power 14 (or 10 to the 14) which way fourteen 10s multiplied together. The is created as 1014. Therefore the calculator answer of 4e14 is 4 × 1014, i beg your pardon is brief for 400000000000000.

The number 14, the is the strength of 10, is additionally called the exponent and method the number of 10s which space multiplied together. The letter E in 4e14 is used 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 see.

Since 20 million squared to be 4 × 1014, you can expect the 90 million squared would be 81 × 1014.

However, the calculator reflects a different answer. Try it and see.

The calculator"s answer of 8.1e15 or 8.1 × 1015 is in reality the very same number as 81 × 1014.

Both these numbers stand for 8 100 000 000 000 000.

However, once a number is written utilizing a clinical notation the convention is to use a number in between 1 and also 10 before the exponent. For this reason here, instead of 81 × 1014, the number 8.1 is offered with an extra power of 10.

Another way of interpreting 8.1 × 1015 is come imagine the the decimal allude in the number 8.1 should be relocated 15 areas to the right, filling in as countless zeros together necessary.


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(a) The rate of irradiate measured in meters per second, is around 300 million (300 000 000) meters per second. What is this number in scientific notation?

(b) Einstein"s famed equation states that E = mc2, the power E (in joules) linked with a mass of m kilograms is equal to m multiplied by c squared, wherein c is the rate of irradiate (in meters every second). Using the value of c offered in her answer to part (a), job-related out the worth of E in scientific notation for a fixed of 1 kilogram.

(c) The distance from the planet to the sunlight is around 149 million kilometres. What is this number in clinical notation?

(d) What is the biggest number her calculator deserve to handle? for example, enter any kind of number and keep squaring. How much can you go prior to the calculator display screens an error message?


Answer

(a) 300 000 000 = 3 × 108

(b)

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(c)

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(d) The TI-84 calculator cannot take care of numbers with exponents that 100 or more. Notification that the calculator produce a post saying ERR:OVERFLOW.


Small numbers really close come zero can also be to express in clinical notation. Because that example, go into a an easy starting worth such together 4.3 and also then continuously divide by 10. This can be excellent by pressing

4 . 3

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1 0

And so on.

Watch carefully how the number go: 4.3, .43, .043, .0043, etc. Climate at some point the screen will jump right into scientific notation. Various calculators use various methods of displaying scientific notation. Look in your calculator handbook to examine how your calculator screens scientific notation and also compare it v that used by the TI-84, which provides .00043 as 4.3e−4.

The e−4 part means that the decimal suggest in the number 4.3 should be moved 4 places come the left, which needs the insertion that some extr zeros.


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What wake up to the result if friend keep separating by 10? You need to see the exponent adjust from −4 come −5 to −6, and so on. Together the an unfavorable exponent it s okay larger, the equivalent decimal number gets smaller.

As with large numbers, the notation e−4 need to be check out as ‘10 come the strength −4’ (or just ‘10 to the minus 4’).

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Do not worry about an unfavorable powers that 10 for the moment. Just think the the notation together a means of writing down tiny numbers wherein the negative sign reminds you the the decimal point must be moved to the left. Here is another example.