Decimal Fractions: Notes and Important Formulas

Definition

Decimal fractions are a fundamental aspect of mathematics, representing numbers that are less than one but more than zero. They are expressed with a decimal point, where the denominator is typically a power of 10. 

Let's explore decimal fractions in detail.

Understanding Decimal Fractions

Decimal fractions are numbers that include a decimal point to indicate values smaller than a whole unit. 

For example, numbers like 0.1 (one tenth), 0.25 (twenty-five hundredths), 0.008 (eight thousandths), and 0.333 (three hundred thirty-three thousandths) are all decimal fractions. 

The digits to the right of the decimal point represent parts of a whole, such as tenths, hundredths, thousandths, and so forth.

Decimal Fractions:

Decimal fractions are numbers less than one that are expressed using a decimal point. They are fractions where the denominator is a power of 10. Here are some examples:

Conversion of Decimal to Vulgar Fraction:

To convert a decimal into a vulgar fraction:

1. Put 1 in the denominator under the decimal point and add zeros equal to the number of decimal places.
   
2. Annexing Zeros and Removing Decimal Signs:
   • Annexing zeros to the right of a decimal fraction doesn't change its value. For example, $0.8=0.80=0.8000.8\; =\; 0.80\; =\; 0.800$0.8=0.80=0.800.
   • When the numerator and denominator have the same number of decimal places, the decimal point can be removed.
     

Operations on Decimal Fractions

Addition and Subtraction: Align decimals and perform as usual.

• Example: $5.9632+0.073=6.03625.9632\; +\; 0.073\; =\; 6.0362$5.9632+0.073=6.0362
  

Multiplication: Multiply without the decimal point, then place the decimal in the product according to the total decimal places.

• Example: $0.2\times 0.02\times 0.002=0.0000080.2\; \backslash times\; 0.02\; \backslash times\; 0.002\; =\; 0.000008$0.2×0.02×0.002=0.000008

Division: Divide as normal and place the decimal in the quotient corresponding to the decimal places in the dividend.

• Example: $0.0204÷17=0.00120.0204\; \backslash div\; 17\; =\; 0.0012$0.0204÷17=0.0012

Recurring Decimals: If figures or sets of figures repeat indefinitely, it's a recurring decimal.

• Example: $\mathrm{0.333...}=0.3‾0.333...\; =\; 0.\backslash overline\{3\}$0.333...=0.3, $\mathrm{3.142857142857...}=3.1428573.142857142857...\; =\; 3.142857$3.142857142857...=3.142857

Pure Recurring Decimal: All figures after the decimal point repeat.
Mixed Recurring Decimal: Some figures repeat while others don't.

• Example: $\mathrm{0.173333...}=0.1730.173333...\; =\; 0.173$0.173333...=0.173