# What is 0 1 0 001

In the “Place value systems” chapter you learned that natural numbers can be represented with the help of place value systems. The number \ (1532 \) can then be read, for example, as \ begin {align *} 1532 = 2 \ times 1 + 3 \ times 10 + 5 \ times 100 + 1 \ times 1000. \ end {align *}
If you want to represent fractions using place value systems, you need decimal numbers. The idea is as follows: just as you assign a digit to tens, hundreds, thousands, etc., you can assign a digit to tenths, hundredths, and thousandths. Numbers in this representation will be Decimal numbers called. The place in the decimal notation that says when the tenths, hundredths, etc. begin is marked with a comma, which is why decimal numbers are also (somewhat casually) spoken of Point numbers. The number \ (103.14 \) means \ begin {align *} 103.14 = 1 \ times 100 + 0 \ times 10 + 3 \ times 1 + 1 \ times \ frac {1} {10} + 4 \ cdot \ frac {1} {100} = 103 \ frac {14} {100} = \ frac {10314} {100}. \ end {align *}
The number \ (0.2022 \) means \ begin {align *} 0.2022 = 0 \ cdot 1 + 2 \ cdot \ frac {1} {10} +0 \ cdot \ frac {1} {100} +2 \ cdot \ frac {1} {1000} +2 \ cdot \ frac {1} {10000} = \ frac {2022} {10000}. \ end {align *}
The part of a decimal number before the decimal point becomes the integer part called the part after the decimal point broken part called. You already know this distinction between a whole and a broken part from dealing with improper fractions! The digits that describe the integer part (i.e. the numbers before the decimal point) are called Pre-decimal places, and those that describe the broken part are called Decimal places.

You already know the division of an object or a unit into tenths, hundredths, etc. from many things in daily life: A meter is in \ (10 ​​\) decimeters, in \ (100 \) centimeters and in \ (1000 \) millimeters divided, so \ (1 dm = \ frac {1} {10} m \), \ (1 cm = \ frac {1} {100} m \) and \ (1mm = \ frac {1} {1000} m \). \ (236 \) centimeters are \ (2.36 \) meters, \ (43 \) millimeters are \ (0.0043 \) meters, and so on. A euro is divided into \ (100 \) cents, so \ (35 \) cents are nothing more than \ (0.35 \) euros.

Division into decimal numbers
Source: Wikipedia

It is often the case that one is only interested in a certain number of decimal places, since the decimal places to the far right have little influence on the size of the decimal number (in the number \ (12.3459 \) the \ (5 \) stands for thousandths and the \ (9 \) for ten thousandths). That is why one is interested in the decimal number with a given number of decimal places that is as close as possible to the given decimal number. This procedure is called Round: If, for example, \ (12.3459 \) should be rounded to the third place after the comma, this means that one looks for the decimal number with \ (3 \) places after the decimal point that differs least from \ (12.3459 \) . This will be the number \ (12.346 \) (\ (12.345 \) it cannot be, because the number \ (9 \) ensures that the number \ (12.3459 \) is closer to \ (12.346 \) ), one writes \ (12.3459 \ approx 12.346 \). You can remember: if you want to round to the \ (n \) th decimal place, consider the \ (n + 1 \) th decimal place: if this is less than \ (5 \), the \ (n \) - th decimal place and you "cut" all further decimal places from the \ (n \) th place; if the \ (n + 1 \) th decimal place is greater than \ (5 \) (as in the example above), then the \ (n \) th decimal place increases by \ (1 \) and then you cut here as well all other decimal places. If the decimal place to which it is rounded is already a zero, this can also be omitted, e.g. \ (13.001 \) is rounded to two places: \ (13.101 \ approx 13.10 = 13.1 \)

The numbers \ (\ frac {1} {10} = 0.1; \ frac {1} {100} = 0.01; \ frac {1} {1000} = 0.0001 \) etc. are "negative powers of ten“(Attention: these numbers are of course not negative in the sense of whole numbers! What is meant here are negative exponents, e.g. \ (0.001 \) can be written as \ (10 ​​^ {- 3} \). The negative exponent is the Number of places after the decimal point. You can find out more about this in the chapter "Power arithmetic laws") Multiplying (positive and negative) powers of ten by decimal numbers is comparatively easy: you move the decimal point by the size of the exponent to the left or to the right; if the exponent is positive, the comma is shifted to the right, otherwise to the left: \ (13.65 \ times 100 = 136.5 \), \ (2530.002 \ times 0.01 = 25.30002 \). When dividing decimal numbers and powers of ten everything happens exactly the other way around: here the decimal point is shifted by the size of the exponent to the left if it is positive and to the right if it is negative: \ (905.5 \ div 100 = 9.055 \ ); \ (30.5042 \ div 0.01 = 30504.2 \). If you can't seem to move the comma, you can think of a series of zeros to the left of the whole or to the right of the broken part, they don't change the number. For example \ begin {align *} 34.7 \ cdot 0.0001 = 000000034.7 \ cdot 0.001 = 0.00347, \ end {align *} \ begin {align *} 34.7 \ div 0.0001 = 34.7000000 \ div 0.0001 = 347 000 \ end {align *} or \ begin {align *} 28 \ cdot 100 = 28.0000000 \ cdot 100 = 2800.00000 = 2800. \ End {align *}
In the case of a terminating decimal number, one can obviously always find a whole number and a negative power of ten (by shifting the decimal point), which when multiplied together result in the given decimal number. The whole number you are looking for can even be easily read from the decimal number. This way it is very easy to convert decimal numbers to fractions; we have for example: \ begin {align *} 0.45 = 45 \ cdot 0.01 = 45 \ cdot \ frac {1} {100} = \ frac {45} {100} = \ frac {9} {20 }, \ end {align *} or
\ begin {align *} 5.55 = 555 \ cdot 0.01 = 555 \ cdot \ frac {1} {100} = \ frac {555} {100} = \ frac {111} {20} \ end {align *}
Once you have understood this, it will be easy for you to convert terminating decimal numbers (those are those that only have a finite number of places after the decimal point; but there are also decimal numbers with an infinite number of places after the decimal point, which you will encounter later) into fractions; the other way around, converting fractions into decimal numbers is a little more difficult. This is explained in the next chapter. But first a few exercises so that you can deepen your knowledge.

#### exercise

Calculate!

\ begin {align *}
& a) && 3.6 \ cdot 10 \
& b) && 3,6 \ div 10 \
& c) && 30.06 \ div 1000 \
& d) && 15.273273 \ cdot 0.001 \
& e) && 0.00043 \ div 0.001
\ end {align *}

### solution

\ begin {align *}
& a) && 3.6 \ cdot 10 = 36 \
& b) && 3.6 \ div 10 = 0.36 \
& c) && 30.06 \ div 1000 = 0.03006 \
& d) && 15.273273 \ cdot 0.001 = 0.015273273 \
& e) && 0.00043 \ div 0.001 = 0.43
\ end {align *}

Round...
\ begin {align *}
& a) && 0.645 \ to \ one \ place \ after \ the \ comma! \
& b) && 0.645 \ to \ two \ places \ after \ the \ comma! \
& c) && 1,004 \ to \ one \ place \ after \ the \ comma!
\ end {align *}

### solution

\ begin {align *}
& a) && 0.645 \ approx 0.6 \
& b) && 0.645 \ approx 0.65 \
& c) && 1.004 \ approx 1.0 = 1 \
\ end {align *}

Write as a fraction and shorten if necessary!

\ begin {align *}
& a) && 0.34 \
& b) && 0.555 \
& c) && 1.25 \
& d) && 5.000001
\ end {align *}

### solution

\ begin {align *}
& a) && 0.34 = \ frac {34} {100} = \ frac {17} {50} \
& b) && 0.555 = \ frac {555} {1000} = \ frac {111} {200} \
& c) && 1,25 = \ frac {125} {100} = \ frac {5} {4} \
& d) && 5,000001 = \ frac {5000001} {1000000}
\ end {align *}