Primary Mathematics/Decimals

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Decimals[edit]

Decimals are a means of representing a partial number without using fractions. You will see them when dealing with currency, and may find them in other situations as well. They will otherwise appear to be normal numbers.

Addition and Subtraction[edit]

When adding or subtracting decimal numbers, the decimal points need to be aligned. If this causes missing digits on the right, then treat them as zeros.

 16.42
+ 2.3
------
 18.72

Multiplication[edit]

When multiplying numbers, the right-most digits are aligned, with the decimal point being placed within the normal location for the number. Multiply the two numbers as if they are normal numbers to get the initial result. Next, count the number of digits to the right of the decimal points of the two numbers; the total of the two counts indicates the number of digits to the right of the decimal point.

  8.32 -- 2 digits after decimal
 * 3.2 -- 1 digit after decimal
------
 1 664
24 96
------
36.624 -- 2 + 1 = 3 digits after decimal

Division[edit]

Normally, the decimal point in the quotient is in the same location as the dividend. If a decimal point appears in the divisor, the quotient's point is moved to the right by the same number of digits in the quotient.

Fractions to Decimals[edit]

Converting a fraction to a decimal simply involves long division. However, some numbers may repeat infinitely as they can't be described in decimal form.

If you need to add digits to continue the decimal, simply add zeros to the quotient.

      2.375
    ___
24 / 57
     48
     --
      9.0
      7.2
      ---
      1.80
      1.68
      ----
        120
        120
        ---
          0

In the example above, you get an exact value.

      2.157...
    ___
19 / 41
     38
    ---
      3.0
      1.9
      ----
      1.10
       .95
      ----
        150 
        133
        ---
         17

In the example above, the number continues on without end. In general, you will want to stop after only a few digits or until you find the repeating pattern. When you find the repeating pattern, a simple notation is to draw a single line above the digits that continuously appear.

Converting to fractions[edit]

Finite-length decimals are trivially converted into fractions. The numberator contains the number without the decimal point, and the denominator starts with 1, and has an additional '0' appended for each digit to the right of the decimal point.

Infinite-length decimals are more complex, but are possible. The procedure for converting them requires a knowledge of Algebra, but will be summarised in an example. First, write x as the repeating decimal:

       x =      0.71428571428571428571428571428571...

Find the length of the repetition, and both sides by 10 until they line up again:

1000000x = 714285.71428571428571428571428571428571...

Subtract the first equation from the second, which should look like this:

 999999x = 714285

Divide both sides by the left number:

       x = 714285/999999

Reduce to lowest terms. In this case, both sides are divisible by 142857:

       x = 5/7

Significant digits[edit]

In practice, decimal numbers (such as those obtained from measurements), may have a degree of error. Thus, it might not make sense to always keep full precision when subtracting 0.001 from 1 million.

When adding or subtracting, a result is normally kept to the same precision as the least precise number. When multiplying or dividing, a result contains the same number of significants as the one with the least number of significant digits.

When dealing with measurements, there is usually only need for at most three significant digits. Handheld calculators, having no direct concept of significant digits, will simply display them all and will have to be rounded manually.

More information on significant digits will be found in a later chapter.


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