Decimals
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A Decimal is a number that consists of a Whole Part and a Fractional Part, represented as a ratio of two integers. It is a way to express a non-integer value using Fractions.
Definition
A Decimal can be defined as a Rational Number in the form of:
a/b
where ‘a’ is the integer part and ‘b’ is the Denominator, which is a positive integer that divides ‘a’. The Fractional Part of the Decimal is between 0 and 1, but it does not include the integer.
History
The concept of decimals dates back to ancient civilizations, where they used sexagesimal (base-60) number system. The use of decimals became more widespread during the Middle Ages with the introduction of Arabic numerals in the 9th century.
Operations on Decimals
Decimals can be performed using various mathematical operations such as Addition, subtraction, multiplication, and division. Some properties that hold for decimals include:
- Closure: The sum and product of two or more decimals are also decimals.
- Commutative Law: The order in which we add or multiply numbers does not change the result.
- Associative Law: The operation is associative, meaning (a+b)+c = a+(b+c).
- Distributive Law: a(b+c) = ab+ac.
Arithmetic Operations
Addition
The Addition of two decimals involves adding their whole parts and fractional parts separately. For example:
- 0.5 + 0.3 = 0.8 (Whole Part: 0, Fractional Part: 0.8)
- 0.25 + 0.75 = 1.00
Subtraction
Subtracting two decimals involves subtracting their whole parts and fractional parts separately. For example:
- 0.5 - 0.3 = 0.2
- 0.25 - 0.75 = -0.50
Multiplication
Multiplying two decimals involves multiplying their whole parts and fractional parts separately. For example:
- 0.5 × 0.3 = 0.15
- 0.25 × 0.75 = 0.1875
Division
Dividing one Decimal by another involves dividing the Whole Part of the second number by the Numerator and multiplying the Fractional Part by the Denominator. For example:
- 10 ÷ 2 = 5 (integer part: 5, Fractional Part: 1⁄2)
- 3 ÷ 4 = 0.75
Comparison of Decimals
Decimals can be compared using various methods such as:
- Equal Signs: If two decimals are equal, then their whole parts and fractional parts are equal.
- Greater-than and Less-than Signs: If one Decimal is greater than another, then the latter’s Fractional Part must be larger.
Conversion to Fractions
To convert a Decimal to a Fraction, we can use the following methods:
Method 1: Using Place Value
We can express a Decimal as a Fraction using its Place Value. For example:
- 0.5 = 5⁄10 (Whole Part: 5, Fractional Part: 1⁄2)
- 0.3 = 3⁄10 (Whole Part: 3, Fractional Part: 1⁄10)
Method 2: Using Exponents
We can express a Decimal as a Fraction using exponents. For example:
- 0.5 = 10^(-1) (integer Exponent: -1)
- 0.3 = 10^(-1) / 10^(0) (integer Exponent: -1, Fractional Part: 1⁄10)
Real-World Applications
Decimals have numerous real-world applications in various fields such as:
Finance
Decimals are used to represent prices and amounts in finance. For example, the cost of a product is often expressed as $12.50.
Science
Decimals are used to express scientific quantities such as temperatures (°C or °F), pressures (Pa or atm), and speeds (m/s). For example:
- 273.15 K = 0°C
- 101325 Pa = 1 atm
Navigation
Decimals are used in navigation systems such as GPS to determine coordinates and distances.
Terminology
Some common terminology related to decimals includes:
- Decimal point: The dot or Decimal mark that separates the Whole Part from the Fractional Part.
- Fractional Part: The portion of a Decimal that is not represented by an integer.
- Decim: A prefix used to denote a unit of 1⁄10, commonly used in Scientific Notation.
Misconceptions
Some common misconceptions related to decimals include:
- Misconception 1: Decimals are always written with the Decimal point in the Numerator. While it is true that most decimals are represented with the Decimal point in the Numerator, there are exceptions where the Decimal point is in the Denominator.
- Misconception 2: Decimals can never be simplified further. In fact, decimals can often be expressed using simpler Fractions or more precise approximations.
Resources
For further learning and practice, here are some resources:
Online Courses
- “Decimals” by Coursera (University of Michigan)
- “Fractions and Decimals” by edX (Harvard University)