Quantitative Aptitude Notes

Introduction to Standard form, Definition, Importance and Examples

What is Standard Form in Maths?

Introduction

Imagine you are working in the quantum or nuclear industry and you have to write a lot of incredibly large and very small numbers repeatedly. Would you rather write these long and short numbers one at a time, or would you prefer a method for writing them in short form?

We frequently need to express extremely big or very small numbers in research and engineering. The mass of the sun, the mass of electrons and charge of electrons etc., for example, is a crucial statistics in physics; they have numerical values of about twenty to thirty digits.

Rene Descartes invented the approach that modern mathematicians use to express numbers, whereas Archimedes introduced the notion of a standard form. The objective of standard form, according to Arithmetic, is that it allows scientists to manage extremely big and small numerical values without dealing with extreme place values. Counting a lot of 0s while typing or reading is far more mistake-prone than wrongly formatting an exponent, and it is far more difficult to identify when an error happens.


Definition of Standard Form in Maths

“Scientific notation form is a method of describing the absolute value of very big or very small integers as one decimal number greater than one and less than ten multiplied by the matching integer power of ten.”

Or

According to BBC,

“Standard form, or standard notation, is a system of writing numbers which can be particularly useful for working with very large or very small numbers. It is based on using powers of 10 to express how big or small a number is.”

If a number N is written in scientific notation then,

|N| = n x 10^p

Where 1 < n < 10 and p is an integer.

standard form in maths

Why to use Standard Form?

Extremely small and extremely big numbers are frequently encountered in physics. These figures can be stated more effectively using standard form.

Significant digits or significant figures are a separate and crucial notion that has nothing to do with “easy” or “efficiency” – they refer to the “precision” of a value or what many people mistake for “accuracy,” but the term accuracy really refers to something else that is standard form.

Following are the benefits of using standard form in dealing with extreme numerical values.

  • Standard form allows us to make more precise record without messing with extreme data values.
  • Obviously the use of standard form provides a repetition free flow in handling large and small numbers. For example the mass of sun is equal to 13 million earth sized planets.
    This high value contains approximately 31 numbers that cannot be repeated severally in a calculation that’s why we write mass of sun as 1.988×1030kg.
  • Because the numerical quantities are reduced, the computation time is reduced.

Applying mathematical operations for standard form

Let N1 be the first and N2 be the second number then mathematical operations can be applied as:

 

Addition 

|N| = |N1| + |N2|

|N| = n x 10p + m x 10q  

|N| = n + m x 10p+q

 

Subtraction

|N| = |N1| – |N2|

|N| = n x 10p – m x 10q  

|N| = n – m x 10p-q

 

Multiplication

|N| = |N1| x |N2|

|N| = n x 10p x m x 10q  

|N| = (n x m) x 10p+q

 

Division

|N| = |N1| / |N2|

|N| = n x 10p / m x 10q  

|N| = (n / m) x 10p-q

 

Kinds of Standard Form

 The standard form can be divided into several kinds, including as

  1. Notation for engineers
  2. E-notation

 

Notation for engineers

“Engineering notation resembles scientific notation except that the exponent, n, must be a multiple of three, such as 0, 3, 6, 9, 12, -3, -6, and so on.”

52548872 is equivalent to 52.54 106.

E-notation

“In E-notation, the “10” in scientific notation is replaced with only “E.”

When showing the exponent is problematic, this method is used.

It’s written as nEk.

Where n represents the basis and k represents “x 10,” k comes after the E. Below is an example of a comparison between scientific notation and E-notation.

For example 8.8E9 is equal to 8.8×109.


Method to calculate Standard Form numerical

Example 1:

0.000324nm is the distance between two ionic bonds. In standard form, convert this distance.

Solution: Manual method 

Step 1: Place the decimal point on the first non-zero digit. 

00003.24nm

Now the opening zeros are meaningless. 

Step2: Now count the number of digits after decimal point. Write the counted number of digits in power of 10.

3.24×104nm

As a result, we have calculated standard form.

A standard notation calculator can also be used to answer the problem above. Using a scientific manner, it allows us to address problems quickly and effectively. To use a standard notation calculator, follow the steps below.

Solution: By using calculator

 Step 1: Place the given data values into the calculator and press the convert button.

 

Congratulations your standard form has been calculated.

Example 2:

The Sun and Mars are separated by 141,700,000 miles (228,000,000 km). Write their distance in standard form.

Solution: Manual method 

Step 1: Shift the decimal point to the first non-zero digit

1.41700000 miles               and                     2.28000000 km

Step2: Count how many digits there are following the decimal point. Write the counted number of digits in powers of ten. 

1.41 × 108 miles               and                     2.28 × 108 km

As a result, the standard form is computed.


Summary

Archimedes created the notion of standard form, whereas Rene Descartes modified the method advised by Archimedes. Standard form has the benefit of allowing scientists to manage extremely big and small numbers without having to deal with place value extremes.

Using the standard form, we can create more exact records without having to deal with extreme data values. The usage of standard form ensures a smooth flow while dealing with large and tiny numbers.

Since this is a calculation so this can be done by either using manual method or using standard form calculator. For your convenience and precision you can try standard form calculator.


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