System Of Equations & Inequalities-Scientific Notation

A combination of two or more inequalities with single or more variables is known as a system of inequalities. When a situation necessitates various solutions and several constraints on those solutions, inequalities are utilised. 

Scientific notation, often known as exponential notation, is a convenient technique to handle enormously large quantities like the Earth’s mass and minimal values like the weight of a hydrogen atom.

What is Scientific Notation?

The scientific notation allows us to represent extremely large or minimal numbers by multiplying single-digit integers by ten to the exponential power. If the value is substantial, the exponents become positive; if the integer is tiny, the exponent becomes negative.

Scientific notation represents a number n in the format, a * 10b.

In the above case, a is an integer. It means 1 ≤ |a| < 10. Also, b represents an integer.

Components of Scientific Notation:

The main components of elements of scientific notation are as follows:

• Exponent

• Base

• Coefficient

Rules of Scientific Notation:

We must use the following rule to find the exponent or power of ten:

• The starting point must always be ten.

• The coefficient’s exact value is more than or equal to one, but it must be less than ten.

• The exponent should be a non-zero integer, negative or positive.

• Negative and positive numbers and decimal and whole numbers can be used as coefficients.

• The decimal number carries the remainder of the number’s significant digits.

Using Scientific Notation:

Scientific notation is a simplified technique of representing minimal and vast numbers in which we represent quantities in terms of powers of ten. Position the decimal point to the right of the very first digit in a number to express it in scientific notation. 

Make a decimal number between 1 and 10 out of the digits. Count how many times you shifted the decimal point n positions. Multiply the decimals by 10 to the nth exponent. 

n is positive if the decimal is pushed to the left, as in a vast number. n is negative if you shift the decimal right, like a tiny significant number.

The illustration below shows how many positions we need to shift the decimal point just after the single-digit value.

• If the mentioned integer is a multiple of 10, the decimal point must be shifted to the left, and the exponent of ten will be positive.

For instance, 70000 = 7 * 104 is in scientific notation.

• If the specified value is less than 1, the decimal point must be moved to the right, resulting in a negative power of ten.

For instance, 0.0007 = 7 * 0.0001 = 7 * 10-4 is in scientific notation.

Scientific Notation Examples:

The term 5800000 can be written as 5.8 * 106

The term 0.00000082 can be written as 8.2 * 10-7

The above is how you can represent the scientific notation.

Incorporating Scientific Notation into Applications:

Calculating with big or tiny numbers is significantly easier utilising scientific notation and the principles of exponents than it is with ordinary notation. 

Let’s say we’re asked to compute the number of atoms in one litre of water. Each water molecule has three atoms (one oxygen and two hydrogens). 

A drop of water had about 1.32 * 1021 molecules, while 1 L of water contains approximately 1.22 * 104 molecules. 

As a result, 1 L of water contains roughly 3* (1.32 * 1021) * (1.22 * 104) 4.83 * 1025 atoms. 

The decimal terms are multiplied, and the exponents are added. Imagine having to do the maths without knowing how to use scientific notation!

Make sure to put the answer in appropriate scientific notation when completing calculations with scientific notation. 

Consider the product (5 * 104) * (5 * 106) = 25 * 1010, for instance.

Because 25 is more than 10, the solution is not in correct scientific notation. 

Take 25 and divide it by 10 to get a total of 2.5.

It increases the exponent of the answer by 10.

(25) * 1010 = (2.5 * 10) * 1010 = 2.5 * (10 * 1010) = 2.5 * 1011

Changing to Standard Notation from Scientific Notation:

Simply reverse the method to convert a value in scientific notation to conventional notation. If n is positive, change the decimal n positions to the right; if n is negative, change the decimal n positions to the left, and include zeros if required. 

Remember that if n is positive, the number’s value is more significant than one, and if n is negative, the number’s value is below one.

For example: 

To convert 3.22 * 1011, we need to push the decimal point towards the right to the 11th position.

3.22 * 1011 = 322000000000

To convert -5 * 105, we must add 5 zeros to the right positions of (-5)

-5 * 105 = -500000

Positive & Negative Exponents:

When representing big numbers in scientific notation, we utilise positive powers or exponents for base 10. We use negative exponents for base ten while expressing small values in scientific notation. We may deduce from the preceding that a number larger than one can be represented as a positive exponent expression. In contrast, a value less than one can define a negative exponent expression.

Conclusion:

Scientific notation is predicated by the idea that a number may be conceived as a collection of components multiplied together. Finally, we understand the magnitude of a number written in scientific notation by glancing at the exponent on the 10. The number is more than one if the exponent becomes positive, and the greater the exponent, it becomes negative.