Decimal fractions and percentages

Definitions and notation

  • A decimal fraction \(F\) is one that is expressed as the sum \[F=\tfrac{a}{10}+\tfrac{b}{100}+\tfrac{c}{1000}+\cdots\]

    where \(a,b,c,\cdots\) are any of the digits \(0,1,2,3,4,5,6,7,7,9\).

  • This sum can have as many rational numbers as needed, possibly an infinite number of them.
  • The decimal fraction is written as a sequence of those digits following a decimal point, which is conventionally preceded by a zero. For example, \[0\color{red}{.}\color{blue}{3}\color{green}{7}\color{violet}{2}=\tfrac{\color{blue}{3}}{10}+\tfrac{\color{green}{7}}{100}+\tfrac{\color{violet}{2}}{1000}\] Each one of those digits is called a decimal place.
  • A decimal number that is similar to a mixed number has a whole part in addition to the fractional part. The whole part is written before the decimal point. For example, \[5\color{red}{.}\color{blue}{0}\color{green}{2}\color{violet}{4}=5+\tfrac{\color{blue}{0}}{10}+\tfrac{\color{green}{2}}{100}+\tfrac{\color{violet}{4}}{1000}\]
  • A terminating decimal number is one that has a finite number of digits in the notation above.
  • A repeating decimal number is one that has an infinite number of digits, but they repeat themselves in a certain pattern. For example,
    • \(\frac{1}{3}=0.3333333\cdots\), and the \(3\) keeps repeating forever. We express the repeating part with a bar above the repeating decimals, so we write it as \(\frac{1}{3}=0.\overline{3}\).
    • The repeating part may come after a number of fixed decimals. For example, \(\frac{1}{6}=0.16666666\cdots\), and the \(6\) keeps repeating forever; therefore, we write it as \(\frac{1}{6}=0.1\overline{6}\).
    • The repeating part is not necessarily a single digit. For example, \(\frac{1}{7}=0.142857142857142857\cdots\), which we then write as \(\frac{1}{7}=0.\overline{142857}\).
  • A decimal number that does not terminate and is not a repeating decimal represents an irrational number; i.e., a number that cannot be written as a ratio of two integers.
  • A percentage is the number of parts for each one hundred parts; i.e., it is a rational number with the denominator fixed as \(100\). It is denoted by the sign \(\%\), written after a decimal number. For example \(37.1\%\) means \(37.1\) parts for each \(100\) parts or \(\tfrac{37.1}{100}\). A percentage has no units because the numerator and denominator have the same unit of measurement.

Interconversion

Decimal and percentage

  • A decimal number is converted to percentage by multiplying by \(100\).
  • A percentage is converted to a decimal number by dividing by \(100\).

Rational or mixed number to decimal number

  • A rational number is converted into a decimal number by performing long division of the numerator by the denominator. We will always end up with either a terminating decimal or a repeating decimal.
  • If we only need the decimal value to a certain number of digits, we do not need to complete the operation. We only need one or two decimal places beyond what is required, then we can approximate the result. For example, if we need \(\tfrac{12}{13}\) only to 3 decimal places, we find the result to 4 places, which is \[\tfrac{12}{13}=0.9230\dots\] Then we approximate that the desired result, which is \[\tfrac{12}{13}\approx0.923\]
  • To convert a mixed number into decimal, we only need to convert the fractional part, since the whole part is the same in both cases.

Decimal number to rational number

  • If the fractional part of a decimal number is terminating, we simply take the sequence of digits after the decimal point as the numerator, and \(10^n\) as the denominator, where \(n\) is the number of digits in this sequence. For example \(0.325\) has three digits after the decimal point. We can then put this rational number in lowest terms. Therefore, \[0.325=\frac{325}{10^3}=\frac{325}{1000}=\frac{13}{40}\]
  • If the fractional part contains a repeating pattern, there is a process by which we can convert it to a rational number, but it is outside the current scope.

  • Now that we have the whole part and the fractional part, we have a mixed number. We can then convert it to a rational number.

Adding and subtracting decimal numers

  • We align the numbers at the decimal point.
  • We add the digits from right to left as usual, with carryover as appropriate.

Multiplying and dividing decimal numbers

  • We note the number of decimal places in the dividend (call it \(a\)) and that in the divisor (call it \(b\)).
  • We remove the decimal place from the numerator and the denominator, then perform long division of these numbers
  • The decimal place in the result is then shifted to the right by \(b-a\) if \(b\gt a\), or to the left by \(a-b\) if \(a\gt b\). We do not need to shift the decimal place in the result if \(a=b\).

For example, if we are doing \(3.27\div 0.3\), we note that \(a=2\) and \(b=1\). We then perform \(327\div 3=109\). Because \(a\gt b\), we shift the decimal place by \(a-b=1\) place to the left, and the result is \(10.9\).

Raising a decimal number to a power

There is no easy way of doing this. Raising to a power is essentially repeated multiplication, so this is what we do.