6 big and little numbers

6.1 scientific notation

Understanding exactly how your calculator displays and also handles very large and very tiny numbers is vital if you room to translate the results of calculations correctly. This section focuses on a way of representing numbers recognized as scientific notation.

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Before you start placed your calculator right into the float mode, so it will display up to about 10 digits and return come the home screen ready to do some calculations.

What answer would you intend if friend square 20 million? How plenty of zeros would certainly the answer have? If you perform this on paper, friend should finish up through 4 followed by 14 zeros. However your calculator might not have the ability to display 14 figures, if it is collection to 10 figures. For this reason input this calculation into your calculator and also see what it does:

20000000

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You will certainly probably gain something choose 4e14. The notation e14 have to be read as 10 to the strength 14 (or 10 to the 14) which method fourteen 10s multiplied together. That is composed 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 also called the exponent and method the variety of 10s which room multiplied together. The letter E in 4e14 is offered to mean exponent.

Now rather of 20 million squared, what would certainly 90 million squared be? What would certainly you expect? shot it on your calculator and also see.

Since 20 million squared to be 4 × 1014, you might expect the 90 million squared would certainly 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 truth the very same number as 81 × 1014.

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

However, once a number is written making use of a scientific notation the convention is to usage a number between 1 and also 10 prior to the exponent. So here, rather of 81 × 1014, the number 8.1 is supplied with one extra strength of 10.

Another method of interpreting 8.1 × 1015 is to imagine the the decimal allude in the number 8.1 need to be moved 15 places to the right, pour it until it is full in as many zeros as 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 renowned equation states that E = mc2, the energy E (in joules) connected with a fixed of m kilograms is same to m multiply by c squared, wherein c is the speed of irradiate (in meters every second). Using the value of c provided in your answer to component (a), job-related out the value of E in scientific notation because that a mass of 1 kilogram.

(c) The distance 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 can handle? for example, enter any type of number and also keep squaring. How far can girlfriend 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 manage numbers v exponents of 100 or more. An alert that the calculator produce a article saying ERR:OVERFLOW.


Small numbers an extremely close come zero can additionally be expressed in clinical notation. Because that example, get in a an easy starting worth such together 4.3 and then consistently divide by 10. This deserve to be excellent by pressing

4 . 3

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1 0

And so on.

Watch closely how the numbers go: 4.3, .43, .043, .0043, etc. Climate at some point the screen will jump right into scientific notation. Various calculators use various methods the displaying clinical notation. Watch in your calculator handbook to check how your calculator screens scientific notation and compare it through that offered by the TI-84, which provides .00043 as 4.3e−4.

The e−4 part way that the decimal suggest in the number 4.3 have to be moved four places come the left, which calls for the insertion of some additional zeros.


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What wake up to the an outcome if friend keep splitting by 10? You must see the exponent adjust from −4 to −5 come −6, and so on. Together the negative exponent it s okay larger, the corresponding decimal number it s okay smaller.

As with big numbers, the notation e−4 have to be review as ‘10 to the power −4’ (or simply ‘10 to the minus 4’).

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Do not problem about negative powers the 10 because that the moment. Just think of the notation as a method of creating down tiny numbers whereby the an unfavorable sign reminds you that the decimal suggest must be moved to the left. Below is one more example.