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Mastering Standard Form: A Guide to Definition, Steps, and Examples

Thursday - February 22, 2024

The standard form of numbers plays an important role in the study of mathematics and other disciplines especially in the field of astronomy to write larger calculations. The standard form of numbers is the core dimensional concept behind complex calculations.

The standard form of numbers enables scientists to express long measurements and quantities in a precise and accurate way. The standard form can be used in daily life in working with significant figures as well and it is more common in various disciplines.

In this article, we will elaborate the definition & explanation of standard form; we’ll also discuss how to write numbers in standard form with examples.

Standard Form of Numbers:

Expressing numbers in standard form also known as, scientific notation or exponential form provides a simplified and convenient way to signify extremely large or small values. The typical format for presenting a number in standard form is:

m x 10n where

  • 1 ≤ m < 10 and sometimes we pronounce it as a coefficient.
  • n Є is the exponent of 10 and it can be both positive or negative.

The standard form of the numbers plays an important role in dealing with astronomical numbers and microscopic measurements. The scientists and astronomers always write in standard from to express smaller and larger observations & distances.

Understanding this form becomes necessary to apprehend the distance between celestial bodies such as stars or galaxies that depending on their astronomical and microscopic values are extremely high or small.

Note: The coefficient in standard form is limited to 1 or more digits but less than 10.

How to Write Numbers in Standard Form?

  • The first step in standardizing a number is to locate the coefficient “a” by shifting the decimal point so that it lies to the left of the first non-zero digit.
  • Next, find out the (power) exponent n by calculating the number of places or positions the decimal point moved.
  • The exponent is positive if the original number is higher than or equal to 1, and negative if the original number is less than 1.

Examples of writing numbers in standard form.

Example 1:

What will be the given number 657 000 000 000 000 in standard form or scientific notation?

SOLUTION:

Step 1: The non-zero digits (657) make up the coefficient, which we can determine.

Step 2: The decimal point will be located after the first non-zero number like 6.57

Step 3: Observe the number of digits (numbers) after 6. There are 14 digits to which the decimal point has crossed to come in the standard position. This will be the exponent of 10 i.e. 1014.

Step 4: So, the given number in standard form will be expressed as 6.57 x 1014.

Example 2:

What will be the number 0.000 000 000 000 000 0849 in standard form or scientific number?

SOLUTION:

Step 1: We identify the coefficient i.e. the non-zero digits (849) make up the coefficient, which we can determine.

Step 2: Place the decimal point after the first non-zero digit i.e. 8.49

Step 3: We count the number of digits before 8. There are 17 digits to which the decimal point has crossed to come in standard position from left to right. This will be the exponent of 10 i.e. 10-17.

Step 4: So, the given number standard form will be expressed as 8.49 x 10-17.

Example 3:

What will be the number 0.000 000 000 000 0793 x 105 in standard form or scientific number?

SOLUTION:

Step 1: We identify the coefficient i.e. the non-zero digits (793) make up the coefficient, which we can determine.

Step 2: Place the decimal point after the first non-zero digit i.e. 7.93

Step 3: We count the number of digits before 8. There are 14 digits to which the decimal point has crossed to come in standard position from left to right. This will be the exponent of 10 i.e. 10 ^-14.

Step 4: Now the number can be written as 8.49 x 10-14 x 105

Step 5: Simplify the powers of 10 as 8.49 x 10-14 + 5. So, the given number in standard form will be expressed as 8.49 x 10-9.

Example 4:

What will be the given number (52 00 00 000 x 107) / (26 000 x 103) in standard form or scientific notation?

SOLUTION:

Step 1: The non-zero digits (52) and (26) make up the coefficient, which we can determine.

Step 2: The decimal point will be located after the first non-zero number like 6.5 and 2.6

Step 3: Observe the number of digits (numbers) after 6 and 2. There are 8 digits after the digit 6 and 4 after the digit 2 to which the decimal point has crossed to come in the standard position. This will be the exponent of 10 i.e. 108 and 104. So, the numbers can be written as 5.2 x 108 x 107 and 2.6 x 104 x 103.

Step 4: Simplify the powers of 10 as

(5.2 x 108 x 107) / (2.6 x 104 x 103) = (5.2 / 2.6) x 108 + 7 – (4 + 3)

 = 2 x 1015 – 7 = 2 x 108

So, the given number in standard form will be expressed as 2 x 108.

Final Words

The numbers in standard form are a universal notation that signifies the description of very large or very small numbers. In this article, we covered the concept of the important term the standard form. We explored its basic definition and explained this universal concept. We also described a useful method to write the numbers in the standard form with solved examples.

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