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Number System

Laws of Exponents

The exponents can be numbers or constants; they can also be variables. Exponents are generally positive real numbers, but they can also be negative numbers.

 

First Law
am × an = a m+n
Example:
22 × 23 = 22+3 = 25 = 32

Second Law
\(\frac{a^m}{a^n}\) = am-n
Example:
\(\frac{2^5}{2^3}\) = 25-3 = 22 = 4

Third Law
(ab)m = am × bm
Example:
(2 × 3)2 =22 × 32 = 4 × 9 = 36

Fourth Law
(am)n = am×n
Example:
(22)3 = 22×3 = 26 = 64


 

Natural Numbers
Tallying numbers 1, 2, 3, 4, 5, ....... etc. are called Natural numbers. Set of characteristic numbers is by and large meant by N.


Whole Numbers
All the regular numbers together with zero are called Whole numbers. The numbers 0, 1, 2, 3, 4, 5, ....... and so on are called Whole numbers. Set of Whole numbers is by and large meant by W. Each Natural number is a Whole number.

Integers
Every single normal number, zero and negatives of the regular numbers are called Integers, i.e. ......– 5, – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5,...........etc. are whole numbers. Set of Integers is for the most part signified by I or Z. Each Whole number is an Integer.

Rational Numbers
Numbers that can be composed as pqpq, where p and q are whole numbers and q ≠ 0 are called Rational numbers. The accumulation of Rational numbers is meant by Q. Between any two balanced numbers there exists boundlessly numerous rational numbers.

Irrational Numbers
Numbers which can't be communicated as pqpq, where p and q are whole numbers and q ≠ 0.

The arrangement of silly numbers is meant by Q−Q. 2\(\sqrt{2}\), 3\(\sqrt{3}\), 7\(\sqrt{7}\) are the cases of silly numbers.
The prochord of the length of periphery of a circle to the length of its distance across is constantly steady. It is an unreasonable number and indicated by π. Decimal construction of π is non-ending and non-rehashing. Estimation of π = 3.14159265......... Rough estimation of π is 227227, yet not equivalent to the correct value.

Pythagoras Theorem
In a privilege calculated triangle, the square of the hypotenuse is equivalent to the whole of the squares of the other two sides.

OB2 = OA2 + AB2
OB2 = 12 + 12
OB2 = 2
OB = \(\sqrt{2}\)

Irrational numbers can be represented to on the number line utilizing Pythagoras hypothesis.

Real Numbers
Real Numbers include:
  • Whole Numbers(like 0, 1, 2, 3, 4, etc)
  • Rational Numbers(like \(\frac{3}{4}\), 0.125, 0.333..., 1.1, etc )
  • Irrational Numbers(like π, \(\sqrt{2}\), etc )

Real Numbers can also be positive, negative or zero.

So ... what is not a Real Number?
  • Imaginary Numberslike \(\sqrt{−1}\) (the square root of minus 1) are not Real Numbers
  • Infinityis not a Real Number.

 

Mathematicians likewise play with some exceptional numbers that that aren't Real Numbers.

 

The Real Number Line

The Real Number Line resembles a geometric line.
A point is picked on hold to be the "starting point". Focuses to the privilege are sure, and focuses to one side are negative.

A separation is been "1", then entire numbers are separated: {1,2,3,...}, and furthermore in the negative bearing: {...,−3,−2,−1}

Any point on hold is a Real Number:

  • The numbers could be entire (like 7)
  • or reasonable (like \(\frac{20}{9}\))
  • or unreasonable (like π)

Be that as it may, we won't discover Infinity, or an Imaginary Number.

Why are they called "Real" Numbers?
Since they are not Imaginary Numbers.
The Real Numbers had no name before Imaginary Numbers were considered. They got called "Genuine" on the grounds that they were not Imaginary. That is the genuine answer!

Real does not mean they are in the real world
They are not called "Genuine" in light of the fact that they proof the benefit of something genuine.

In mathematics we like our numbers immaculate, when we compose 0.5 we mean precisely half.

Be that as it may, in this present reality half may not be correct (take a stab at slicing an apple precisely down the median).

 


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