Decimal Notation

Decimal notation is a way of writing numbers, both whole numbers and fractions, using a base-ten system. It relies on ten unique digits (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) and a decimal point to indicate the separation between the whole number part and the fractional part.

The key aspects of decimal notation are:

1. The base ten system relies on powers of 10, with each position to the left of the decimal point representing a positive power of 10 and each position to the right representing a negative power of 10. 
   
    
2. The decimal point is an important mathematical symbol. Its main function is to separate the whole number and make it distinct from the fractional part of the number. 
   
    
3. The digits 0 through 9 are used in various combinations to represent different values in each place value.
   
    
4. The whole number, or the digits to the left of the decimal point, can be expressed as multiples of a positive power of 10.
   
    
5. Digits to the right of the decimal point are written as multiples of negative powers of 10. Decimal numbers follow a base-10 system, which means each place value is ten times smaller than its predecessor.

Let’s consider an example in the form of a question to better understand.

Question:  Write 45.12 in a fraction and with base 10.

Answer: 

 In fraction is written as 4512/100

  With the base 10, it is written as 4 × 10¹ + 5 × 10⁰ + 1 × 10^-1 + 2 × 10^-2.
 

Scientific notation is a way of writing very large or very small numbers in the form

a×10ⁿ
 

where 1 ≤ a < 10 and n is an integer. This notation makes it easy to write very large or minimal numbers in a compact form. 

Example:

• 3,200,000 → 3.2 × 10⁶ (decimal moved 6 places to the left).

• 0.000045 → 4.5×10⁻⁵ (decimal moved 5 places to the right).

The table below illustrates the comparison between decimal notation and scientific notation.

Converting a decimal to scientific notation requires rewriting the number using powers of 10. The steps are as follows:
 

Step 1: Move the decimal point to get a number between 1 and 10

Step 2: Count the number of places moved (n)

Step 3: Express the number as a product with a power of 10 (10n if greater than 1, 10-n if between 0 and 1).

Step 4: Double-check the result

Let’s consider an example to understand this better 

Write 0.00023 in scientific notation.

The step-by-step process for this would be as follows:

Step 1: Move the decimal point to identify a number between 1 and 10
0.00023 → 2.3 (moving the decimal 4 places to the right)

Step 2: Count the number of places moved (n)
n = 4

Step 3: Express the number as a product with a power of 10 (10n if greater than 1, 10-n if between 0 and 1).
2.3 × 10^-4

Step 4: Double-check the result
\(2.3 \times 10^{-4} = 2.3 \times 0.0001 = 0.00023 \)

For converting scientific to decimal notion, write the number in its standard form. Steps used for converting scientific to decimal notation are as follows: 
 

Step 1: Identify the exponent (n) of 10

Step 2: Shift the decimal point ‘n’ places to the right if the exponent is positive, or to the left if it is negative, adding zeros as necessary

Step 3: Cross-check the result.

For example: 
Write 5.2 × 10^-3 in decimal form.

The step-by-step process for this would be as follows:

Step 1:Identify the exponent (n) of 10
n = -3

Step 2: We should shift the decimal point 3 places towards the left because n is -3, and negative values demand a shift towards the left.
5.2 = 0.0052

Step 3: Verify the result.

\(5.2 \times 10^{-3} = 5.2 \times 10^3 = 5.2 \times 10{,}000 = 5.2 \times 0.001 = 0.0052 \)

Decimal notation is a system for representing numbers using a base-10 structure, where values are expressed as whole units and fractions separated by a decimal point. It allows precise representation, calculation, and communication of numerical information. Below are some of its real-life applications:

1. Finance and Economics: Decimal notation is used to represent the fractional units of currency, such as dollars and cents. It is crucial for accurately calculating interest rates, taxes, and financial transactions. For example: A bank account balance of $1,245.75 or a tax rate of 7.25% requires precise decimal representation.
   
    
2. Scientific Measurements: In science and engineering, decimals are used for precise measurements of length, mass, volume, and temperature. Accuracy is essential for experiments, research, and technical work. Example: The average body temperature of a patient might be recorded as 98.6°F, or the distance between two points in a physics experiment could be 12.47 meters.
   
    
3. Education and Assessment: Decimal notation is widely used in schools and universities to record the test scores, grades, and GPA calculations. For instance: A student may score 92.75% on a standardized test, or a GPA might be 3.67 on a 4.0 scale.
   
    
4. Technology and Computing: Decimal values are essential in computer science, electronics, and IT for describing specifications, processing speeds, and data sizes. Example: A laptop may have 16.5 GB of RAM, or a processor might operate at 3.45 GHz.
   
    
5. Energy, Transportation, and Commerce: In industries like fuel, transportation, and retail, decimals are used for pricing, measurements, and efficiency calculations. For example: Gasoline may be priced at $3.589 per gallon, or an airline ticket may cost $425.75.

Decimals are numbers that include a fractional part, and understanding them is important in mathematics. Two key skills when working with decimals are rounding and comparing.

• Rounding decimals 

• Comparing decimals 

The table below illustrates the comparison between rounding decimals and comparing decimals.