Scientific Notation

Introduction

Scientific notation is a method of displaying very large or very minuscule numbers in a simplified format. It is a crucial topic for the class 11th examination. These notes cover all the phases in which scientific notation is discussed, including the rules of scientific notation, some issues that can occur and their solutions, and scientific notation examples. The research incorporates scientific notation in standard form. 

Scientific Notation

As mentioned in the introduction, scientific notation allows us to capture very large or very small numbers by multiplying single-digit integers by 10 to the degree of the relevant exponent. If the quantity is very high, the coefficient is positive; otherwise, it is negative. To understand this more deeply, you can learn about power and expansions.

The following is a general depiction of scientific notation:

a10b where 1 ≤ a < 10

The Rules of Scientific Notation

To find the power or argument of 10, we must follow these rules and steps:

• The starting point should always be ten
• The multiplier must be a non-zero numeric, which might be favourable or unfavourable
• The comparative value of the correlation coefficient should be higher than or equal to one but less than ten
• Pleasant and unpleasant numbers, including whole and decimal values, can be used as coefficients
• The remainder of the number’s integer value is carried by the mantissa
• Let us see after how many spots we need to relocate the punctuation mark after the single-digit value
• If the provided integer is a massive number of 10, the numerical value must be moved to the left, and the strength of ten must be positive
• 6000 = 6103 is an example of scientific notation
• If the specified value is less than one, the decimal point must be moved to the right, resulting in a negative power of ten
• 0.006 = 6×0.001 = 610^-3 is an example of mathematical notation

Scientific Notation Examples

Some scientific notation examples include: 

490000000 = 4.9108 

1230000000 = 1.23109

50500000 =5.05107 

0.000000097 = 9.710^-8 

0.0000212 = 2.1210^-5

Exponents: Positive and Negative

When expressing huge numbers in scientific notation, we utilise positive expansions for base 10. For example, 20000 equals 2104, where 4 is the favourable exponent.

When expressing tiny integers in mathematical notation, we utilise negative radicals for foundation. For example,  0.0002 = 210^-4, where -4 is the negative integer.

We may thus conclude that numbers that are higher than one can be expressed as expressions with positive exponents; whereas quantities that are less than one can be written as expressions with negative exponents.

Issues and Solutions

Question 1: Write the value 0.00000046 in scientific notation.

Solution: Increase the decimal point to the right of 0.00000046 by 7 places.

To generate the number 4.6, the numerical value was shifted 7 places to the right.

Since the numbers are less than ten, the decimal point is pushed to the right. As a result, we use a negative expression here.

The scientific notation is as follows:

4.6 x 0.0000001 = 4.610^-7

Question 2: Write the number 301000000 in scientific notation.

Solution: Shift the decimal 8 places to the left such that it is oriented correctly according to the upper left corner non-zero digits, 3.01000000. Remove all the zeros and multiply the result by ten.

The number is now equal to 3.01.

We use a favourable exponent because the integer is bigger than 10 and the fraction is pushed to the left.

As a result, the scientific notation of the value is 3.01108.

Question 3. Convert 1.36107 from scientific notation to customary notation 

Solution: 1.36107 scientific notation is already given.

7 is the exponent

Because the argument 7 is positive, we must shift the decimal point 7 places to the privilege.

Therefore, 1.3610b = 1.36 x 10000000 = 1.36,000,000

Questions for Practice

Problem 1: Convert the following numbers to scientific notation:

28100000

7890000000

0.00000542

Problem 2: Put the following in standard form:

3.55105

2.8910^-6

9.810^-2

Scientific Notation to Standard Form

To overcome the problem of reading huge numbers, the standard form of a number is established. Any value between 1.0 and 10.0 that can be represented in decimal form,  multiplied by the power of ten, is in the standard form. 1.5102 is an example.

Example

Problem:

In standard form, the number is 72800000000000.

Solution:

72800000000000 is the given number.

7.281013 is the conventional form/scientific notation of the given integer.

Number To Scientific Notation

The appropriate scientific notation format is a10b, where ‘a’ is a numerical or decimal number with the exact amount more than or equal to only about ten, or 1 ≤ |a| < 10. b is the power of ten that is necessary for scientific notation to be mathematically faithful to the real number.

In your quantity, move the punctuation mark until there is just one non-zero digit to the left. an is the resultant decimal number.

• Count the number of times you shifted the decimal point. This is the number b
• If you shift the decimal point to the left, b becomes positive
• If you shift the decimal point to the right, b becomes negative
• If you didn’t have to change the decimal point, then b = 0

Convert 357,096 to scientific notation as an example.

• 3.57096, a = 3.57096 
• We shifted the decimal to the left so that b becomes positive, resulting in b = 5
• In scientific notation, the number 357,096 is 3.57096105

Convert 0.005600 to scientific notation as an example.

• To get a = 5.600, shift the decimal three spaces to the right and eliminate the leading zeros
• We shift the decimal to the right, making b negative. b = -3
• 5.610^-3 is the scientific notation for the number 0.005600

It is worth noting that we don’t delete the following 0s because they are actually to the right of the exponent and are therefore important numbers.

E notation is similar to scientific notation, except that the letter e is used instead of 10.

Conclusion

As we all know, whole numbers can be extended indefinitely. However, such large numbers cannot be written on a sheet of paper. Furthermore, the numbers that appear after the decimal in the millions place need to be expressed in a simpler format. As a result, representing a few integers in their extended form is challenging. We therefore use scientific notation. It is important to learn about numbers in the general form as well. The standard form of a number is designed to alleviate the challenge of reading large numbers.