Decimals


The Decimal System


1000

 thousands 
100

 hundreds 
 10 

 tens 
 1 

 ones 

$\frac{1}{10}$

  tenths  

$\frac{1}{100}$

 hundredths 

$\frac{1}{1000}$

 thousandths 

Decimal notation can be used to give an 'alternative' way of expressing a fraction

Example :

   'two - tenths' = $\frac{2}{10}$ = 0.2

   'two - hundredths' = $\frac{2}{100}$ = 0.02

   'two-thousandths' = $\frac{2}{1000}$ = 0.002

The dot is referred to as the 'decimal point'.

Staring off with something you know already, like decimal currency, would be a good starter.

You are probably aware, at least subconciously, that 25 pence (i.e. 0.25 pounds) is equal to a quarter of a pound - so you can already convert between some decimals and fractions.

The BBC have an introductory site - click here

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Transforming Decimals into Fractions

Method

  1. Convert the decimal to a fraction
  2. Reduce to simplest form

Examples

0.2 = $\frac{2}{10}$ = $\frac{1}{5}$


0.054 = $\frac{54}{1000}$ = $\frac{27}{500}$


2.75 = $2 \ \frac{75}{100}$ = $2\ \frac{3}{4}$


5.8 = $5\ \frac{8}{10}$ = $5\ \frac{4}{5}$


You can see the logic -

  • if you have one decimal place, you put the 'decimal part' over 10
  • if you have two decimal places you put the 'decimal part' over 100
  • if you have three decimal places, you put the 'decimal part' over 1000

and so on.

And then reduce to lowest terms, where appropriate.

Quick Quiz     Convert these decimals to fractional form, stating the answers in their lowest terms
  1. 1.   0.25
  2. 2.  0.125
  3. 3.   0.35
  4. 4.   0.025
  5. 5.  0.95
  1. 6.   2.75
  2. 7.  3.375
  3. 8.   12.95
  4. 9.   7.625

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Transforming Fractions into Decimals


Method

A fraction can also be considered as a division

For example :

\[ \frac{2}{3} \]

can be considered as either

two-thirds
or as
2 divided by 3


This division will produce the required decimal form of the fraction.

The process of transforming fractions into decimals is quite straightforward. The line separating the numerator from the denominator can also be considered as denoting division - so divide the numerator by the denominator (that's all there is to it, although, in general, you are going to get a large number of decimal places - so you will have to decide how many decimal places you want in the answer)

Examples


3/4 = 3÷ 4 = 0.75

41/8 = 41÷ 8 = 5.125

2/3 = 2 ÷ 3 = 0.666.........   (recurring)

1/7 = 1 ÷ 7 = 0.142857142857..  (could say 0.143 to 3 dec. places, for example)

It is a mathematical fact that this division will always produce a recurring decimal.

Sometimes, this recurring decimal will just be of the form

0.250000000......

where the 'zeros' recur, which will conventionally just be written as

0.25

Sometimes, a non-zero number recurs

\[ \frac{1}{3} = 0.33333..........\]

And sometimes a group of numbers recur

\[ \frac{7}{11} = 0.63636363....... \]

This shows why a number like p cannot be represented as a fraction - because when represented as a fraction it does not recur.

Common conversions worth memorizing

\[\frac{1}{2} =0.5 \]
\[\frac{1}{4} =0.25 \]
\[ \frac{1}{8}= 0.125 \]

and multiples of these, e.g :   $\frac{3}{8} = 0.375$

Quick Quiz     Convert these fractions to decimal form
  1. \[ 1. \ \ \frac{1}{4} \]
  2. \[ 2. \ \ \frac{1}{8} \]
  3. \[ 3. \ \ \frac{13}{43} \]
  4. \[ 4. \ \ \frac{8}{23} \]
  5. \[ 5. \ \ \frac{4}{5} \]
  1. \[ 6. \ \ \frac{22}{7} \]
  2. \[ 7. \ \ \frac{7}{8} \]
  3. \[ 8. \ \ \frac{8}{13} \]
  4. \[ 9. \ \ \frac{39}{23} \]

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Multiplying and Dividing Decimals by Powers of 10

  • Multiplying a decimal by a multiple of 10 shifts the decimal point a number of places to the right .

    • The number of places is equal to the index of 10
    • or alternatively (if you are not knowledgeable about indices), the number of places is equal to the number of zeroes in the multiplier

      • a multiplier of 10 > 1 place
      • a multiplier of 100 > 2 places
      • a multiplier of 1000 > 3 places
      • etc. etc.

    Example

    \[ 1.275 \times 10 =12.75\]
    \[1.275 \times 100=127.5\]
    \[1.275 \times 1000=1275\]

  • Dividing a decimal by a multiple of 10 shifts the decimal point a number of places to the left.

    The logic is directly analogous to that for multiplication by 10

    • The number of places is equal to the index of 10, although ignoring the minus sign.
    • or alternatively (if you are not knowledgeable about indices), the number of places is equal to the number of zeroes in the divisor

      • a divisor of 10 > 1 place
      • a divisor of 100 > 2 places
      • a divisor of 1000 > 3 places
      • etc. etc.

    Example

    1.275 ÷ 10 = 0.1275
    1.275 ÷ 100 = 0.01275
    1.275 ÷ 1000 = 0.001275

Quick Quiz     Carry out these calculations
  1. \[1. \ \ 0.23 \times 100 \]
  2. \[ 2. \ \ 9.76 \times 10 \]
  3. \[ 3. \ \ 0.0034 \times 1000 \]
  4. \[ 4. \ \ 0.24 \div 10 \]
  5. \[ 5. \ \ 23.456 \div 100 \]
  1. \[6. \ \ 78.345 \times 1000 \]
  2. \[ 7. \ \ 6.456 \times 1\ 000\ 000 \]
  3. \[ 8.\ \ 9.34 \div 10\ 000 \]
  4. \[ 9.\ \ 0.009 \times 100 \]

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Multiplying Decimals


  Prior knowledge required :   Long multiplication (in Basic Arithmetic)

Method

  1. Multiply the two numbers, ignoring the decimal points
  2. Ensure that the number of digits to the right of the decimal point in the result is the sum of the number of digits to the right of the decimal points from the two original numbers.


Examples


\[ 6 \times 0.2 =1.2\] \[ 0.6 \times 0.2= 0.12 \]
\[6 \times 0.02=0.12\] \[0.06 \times 0.2=0.012\]
\[6 \times 0.002 =0.012\] \[0.006 \times 0.2=0.0012\]
\[6 \times 0.0002=0.0012\]

With respect to the last example in the second column, there are a total of four decimal places originally : (three in 0.006, and one in 0.2), so

  • Step 1 : Multiply 6 × 2 = 12
  • Step 2 : Adjust this 12, by shifting the decimal point by four places to the left $\rightarrow$ 0.0012

Quick Quiz     Without using a calculator, carry out these calculations (these are a bit more complex than the examples above - you will need knowledge of how to do long multiplication)
  1. \[1. \ \ 0.23 \times 1.78 \]
  2. \[ 2. \ \ 9.76 \times 1.2 \]
  3. \[3. \ \ 0.0034 \times 9.45 \]
  4. \[ 4. \ \ 0.24 \times 3.5 \]
  5. \[5. \ \ 23.456 \times 5.6 \]
  1. \[ 6. \ \ 78.345 \times 4.56 \]
  2. \[ 7. \ \ 6.456 \times 0.456 \]
  3. \[8. \ \ 9.34 \times 1.203 \]
  4. \[ 9. \ \ 0.009 \times 3.4 \]

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Dividing Decimals

Prior knowledge required : Long Division (in Basic Arithmetic)

  • To divide a decimal by an integer

    This is very similar to division of an integer by an integer. Use the same method, but in this case, ensure that the position of the decimal point is retained

    Example

             63.2
      ----------
    12 ) 758.4

    The decimal point in the answer is placed above the decimal point in the dividend.

  • To divide a decimal by a decimal

    1. Multiply both decimals by the smallest power of 10 which will make the divisor (second decimal ) into an integer.
    2. Perform the division as in the the previous example )

    Examples

    6 ÷ 0.2 = 60 ÷ 2 = 30
    6 ÷ 0.02 = 600 ÷ 2 = 300
    6 ÷ 0.002 = 6000 ÷ 2 = 3000
    60 ÷ 0.02 = 6000 ÷ 2 = 3000
    0.006 ÷ 0.02 = 0.6 ÷ 2 = 0.3

    Another Example

    5.642 ÷ 0.13 = 564.2 ÷ 13

    and then solve as already described


Note particularly that it is only necessary to make the divisor into an integer. It is not necessary that both numbers should be integers.

Quick Quiz     Without using a calculator, carry out these calculations (these are more difficult than the examples. You will need to know how to do long division.
  1. \[ 1. \ \ 0.23 \div 1.78 \]
  2. \[ 2. \ \ 9.76 \div 1.2 \]
  3. \[ 3. \ \ 0.0034 \div 9.45 \]
  4. \[ 4. \ \ 0.24 \div 3.5 \]
  5. \[ 5. \ \ 23.456 \div 5.6 \]
  1. \[6.\ \ 78.345 \div 4.56 \]
  2. \[7. \ \ 6.456 \div 0.456 \]
  3. \[ 8. \ \ 9.34 \div 1.203 \]
  4. \[9. \ \ 0.009 \div 3.4 \]

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Addition and Subtraction of Decimals

Method

  1. Write down the two (or more) numbers directly beneath each other, lining up the decimal points.
  2. Add or subtract as normal

Examples

2.4 + 0.0002 gives 2.4000
0.0002
-------
2.4002
1.8 - 0.643 gives


1.800
0.643
-------
1.157

Quick Quiz     Carry out these calculations
  1. \[1. \ \ 0.23 + 1.78 \]
  2. \[2. \ \ 9.76 + 1.2 \]
  3. \[ 3. \ \ 0.0034 + 9.45 \]
  4. \[ 4. \ \ 0.24 + 3.5\]
  5. \[ 5. \ \ 23.456 + 5.6 \]
  1. \[ 6. \ \ 78.345 + 4.56\]
  2. \[ 7. \ \ 6.456 + 0.456 \]
  3. \[ 8. \ \ 9.34 + 1.203\]
  4. \[ 9. \ \ 0.009 + 3.4 \]

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Rounding to n Decimal Places

Method

  1. To round to the nth. decimal place, consider the (n+1)th. digit
    • For values < 5 leave the nth. digit unchanged
    • For values > 5 add one to the nth. digit

  2. Ignore all subsequent digits

Examples

\[134.0351 \ \mbox{rounded to 3 decimal places (d.p.)} = 134.035\]
\[134.0351\ \mbox{rounded to 2 decimal places (d.p.)} = 134.04\]
\[ 134.0351\ \mbox{rounded to 1 decimal places (d.p.)} = 134.0\]

Quick Quiz     Carry out these roundings, in the manner requested
  1. \[ 1. \ \ 1.234\ \mbox{to 1 d.p.} \]
  2. \[ 2. \ \ 9.76456\ \mbox{to 2 d.p.} \]
  3. \[ 3. \ \ 0.0034\ \mbox{to 3 d.p.} \]
  4. \[ 4. \ \ 0.2463\ \mbox{to 3 d.p.} \]
  5. \[5. \ \ 23.456\ \mbox{to 2 d.p.} \]
  1. \[6. \ \ 78.345\ \mbox{to 2 d.p.} \]
  2. \[ 7. \ \ 6.456456\ \mbox{to 5 d.p.} \]
  3. \[8. \ \ 9.34203\ \mbox{to 1 d.p.} \]
  4. \[ 9. \ \ 0.009342\ \mbox{to 4 d.p.} \]

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Rounding to 'n' Significant Figures


The first significant figure is the first non-zero digit

Method

  1. To round to the nth. significant figure, consider the (n + 1)th. non-zero digit

    • For values < 5 leave the nth. digit unchanged
    • For values > 5 add one to the nth digit

  2. All subsequent digits become zero

Example

1374.0351 rounded to 1 significant figure (s.f.) = 1000
1374.0351 rounded to 2 significant figures (s.f.) = 1400
1374.0351 rounded to 3 significant figures (s.f.) = 1370
1374.0351 rounded to 4 significant figures (s.f.) = 1374

Quick Quiz     Carry out these roundings, in the manner requested
  1. \[ 1. \ \ 1.234\ \mbox{to 1 sig figs.} \]
  2. \[2. \ \ 9.76456\ \mbox{to 2 sig figs.} \]
  3. \[ 3. \ \ 0.0034\ \mbox{to 3 sig figs.} \]
  4. \[ 4. \ \ 0.2463\ \mbox{to 3 sig figs.} \]
  5. \[ 5. \ \ 23.456\ \mbox{to 2 sig figs.} \]
  1. \[6. \ \ 78.345\ \mbox{to 2 sig figs.} \]
  2. \[ 7. \ \ 6.456456\ \mbox{to 5 sig figs.} \]
  3. \[ 8. \ \ 9.34203\ \mbox{to 1 sig figs.} \]
  4. \[ 9. \ \ 0.009342\ \mbox{to 4 sig figs.} \]

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