6 Large and tiny numbers

6.1 Scientific notation

Understanding exactly how your calculator screens and handles exceptionally big and exceptionally little numbers is necessary if you are to analyze the outcomes of calculations properly. This area concentrates on a means of representing numbers well-known as clinical notation.

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Before you begin put your calculator right into the float mode, so it will screen up to about 10 digits and also return to the house display ready to do some calculations.

What answer would certainly you expect if you square 20 million? How many kind of zeros would certainly the answer have? If you carry out this on paper, you have to end up through 4 complied with by 14 zeros. However your calculator can not have the ability to screen 14 numbers, if it is set to 10 numbers. So input this calculation right into your calculator and also watch what it does:



You will most likely obtain somepoint choose 4e14. The notation e14 should be read as 10 to the power 14 (or 10 to the 14) which means fourteen 10s multiplied together. It is written as 1014. So the calculator answer of 4e14 is 4 × 1014, which is short for 400000000000000.

The number 14, that is the power of 10, is additionally dubbed the exponent and also suggests the variety of 10s which are 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 your calculator and watch.

Due to the fact that 20 million squared was 4 × 1014, you could suppose that 90 million squared would certainly be 81 × 1014.

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

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

Both these numbers represent 8 100 000 000 000 000.

However before, when a number is created using a clinical notation the convention is to usage a number in between 1 and 10 prior to the exponent. So here, rather of 81 × 1014, the number 8.1 is supplied through an additional power of 10.

Anvarious other way of interpreting 8.1 × 1015 is to imagine that the decimal point in the number 8.1 must be relocated 15 locations to the ideal, filling in as many type of zeros as vital.


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

(b) Einstein"s well known equation states that E = mc2, the energy E (in joules) linked with a mass of m kilograms is equal to m multiplied by c squared, wright here c is the rate of light (in meters per second). Using the worth of c given in your answer to part (a), work out the worth of E in clinical notation for a mass of 1 kilogram.

(c) The distance from the Earth to the Sun is about 149 million kilometres. What is this number in scientific notation?

(d) What is the largest number your calculator can handle? For example, enter any kind of number and store squaring. How much can you go prior to the calculator screens an error message?


(a) 300 000 000 = 3 × 108





(d) The TI-84 calculator cannot manage numbers with exponents of 100 or even more. Notice that the calculator produces a message saying ERR:OVERFLOW.

Small numbers incredibly cshed to zero have the right to also be expressed in scientific notation. For example, enter an easy starting value such as 4.3 and also then continuously divide by 10. This have the right to be done by pressing

4 . 3

1 0

And so on.

Watch closely exactly how the numbers go: 4.3, .43, .043, .0043, etc. Then at some allude the display screen will certainly jump into clinical notation. Different calculators usage various approaches of displaying scientific notation. Look in your calculator handbook to examine how your calculator displays clinical notation and also compare it with that used by the TI-84, which gives .00043 as 4.3e−4.

The e−4 component means that the decimal point in the number 4.3 need to be relocated 4 areas to the left, which requires the insertion of some added zeros.


What happens to the outcome if you keep dividing by 10? You should watch the exponent adjust from −4 to −5 to −6, and so on. As the negative exponent gets larger, the corresponding decimal number gets smaller sized.

Just like big numbers, the notation e−4 have to be check out as ‘10 to the power −4’ (or ssuggest ‘10 to the minus 4’).

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Do not worry around negative powers of 10 for the minute. Just think of the notation as a way of composing dvery own small numbers wright here the negative sign reminds you that the decimal allude should be moved to the left. Here is an additional example.