Primary Mathematics/Decimals

Decimals

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.In the decimal system during expression of numbers the denominators are presented in the positive integral power of 10. E.g. 56/100=0.56 In the above example the denominator is present as 10^2

Addition and Subtraction

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

When adding and subtracting decimals, make sure you line the problem up vertically. You do almost exactly like you do with normal adding and subtracting decimals. Adding and subtracting decimals is not that hard, because if you have learned how to add and subtract normal two digit and higher, numbers, you will most likely get the hang of adding and subtracting decimals because, like I already said, adding and subtracting decimals is almost exactly like adding and subtracting normal numbers. When you are adding and subtracting decimals, make sure the problem is lined up with the correct place value like this:

14.65

+12.6_

The equation I just wrote is the exact same as this one.   14.65
     +12.60

You would also do that when you subtract. When adding and subtracting decimals make sure the decimals are lined up. Also, put the zero in (other wise it could get confusing) so it acts as a placeholder and you get an accurate answer. When you get an answer and there's a zero at the very end after the numbers behind the decimal point, you can take out that zero because it really doesn't mean anything. In fact, you could add a trillion zeros to the end of the decimal, 1.4 but it would still mean 1.4. Here is an example of what I’m talking about. 13.500000000 is the same as this number 13.5 Here is an example of what not to do. 134.056 this number is not the same as this number. 134.56 When you place the decimal in your answer, it is very simple. It’s the same with adding and subtracting, so that’s not something you need to worry about. Say you did 4.3+5.1, and you got 94. You're not done. You need to place your decimal still! Like I said, don’t worry! It’s simple. Your problem is setup like this: 4.3

                                            +5.1
                                   you got this= 94 all you do is bring the decimal down and your final answer is= 9.4 For subtracting decimals you do the same thing bring the decimal down.

Multiplication

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

If you multiply whole numbers you can multiply decimals too. (if you do not know how to multiply whole numbers learn how to do that first or else most of this will not make sense.) To do so, you don't have to put a decimal and a zero to line up the problem with it's own place value like you do with adding and subtracting decimals. You just line it up like you would with a normal multiplication problem. Here is what that would look like:

     24
    x.3  

You would get this 72 but you need to place your decimal. This is what the answer would look like when it is finished: 7.2 Now lets do a problem where both problems have a decimal.

 7.5

x 6.5

     You would get this: 4875

Then you count how many numbers are behind the decimal in the equation. (2) then you would get this: 48.75 As you can see it's not that hard.Here is a harder problem:

     12.566
    *34.666
 you would get this: 435612956 Then you count how many numbers are behind the decimals in the numbers. There is 6, so you count from the left to the right, 6 numbers. Your new answer is 435.612956 Multiplying decimals and multiplying whole numbers are really similar, just don’t forget to add the decimal or else you’ll get the problem very wrong.

Division

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

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

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

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.