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:

• 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. 

• 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. 

• The digits 0 through 9 are used in various combinations to represent different values in each place value.

• The whole number, or the digits to the left of the decimal point, can be expressed as multiples of a positive power of 10

• 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 understand better.

Question:  Write 45.12 in 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 and decimal notation are two methods of writing numbers. The main differences between the two are explained in the table below:
 

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 × 10^-4 = 2.3 × 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 × 10^-3 = 5.2 × 1103 = 5.2 × 11000 = 5.2 × 0.001 = 0.0052

Decimal notation is used in our day-to-day lives. Below are some of its real-life applications:

• Decimal notation is used in finance to represent fractions of whole units, such as cents or paise. E.g., while dealing with interest rates and taxes, we will have to use decimals.

• While recording precise measurements like length, weight, and temperature, we use decimal notations. E.g., the temperature in New York City is 24.8°C.

• We use decimal notation in educational institutions like schools and colleges to record test scores of students. E.g., Amy scored 68.78% in her final assessment.

• In technology and computing, decimal values are essential for accurately describing speed and specifications. For example, 2.5 GB file size or a 3.6 GHz processor speed.

• Decimal notation plays a crucial role in the fuel and energy sector. At gas stations, the prices are often displayed as decimals. E.g., petrol at ₹106.48 per liter.