Number System means a system of representing numbers on the number line( number line extending from (-∞ (infinity) to +∞). Number line can be defined as an imaginary line on which each and every number can be represented by using symbols. This line can extend to both sides, right side to zero will contain all positive numbers while left side to the zero will contain all the negative numbers.
This line is capable of representing rational as well as irrational numbers.
- Rational Numbers: These are the numbers which can be represented in the form of fractions say a/b where both a and b are integers and b should not be equal to zero.
Example: 5/4, 11/8, 7/4 are all rational numbers.
- Irrational numbers: The numbers which cannot be represented in the form of a/b where a and b are integers and b is not equal to 0 are called Irrational numbers. The numbers which are not rational are Irrational. Irrational numbers were first discovered by Greek mathematician Pythagoras around 400 BC.
Example: √2, √3, √15 are all irrational numbers.
Square root (√)
The square root of a number means that the power of that number is 1/2 or 0.5. √ this symbol is known as radical sign or radix and the number under the radical sign will be called as radicand. It is represented as √2, √3, √4 etc. It can be said that the root of any number A is B if B2 = A. For example square root of 4 is 2 because 22 = 4. The square root of 3 is √3 because (√3)2 = 3
Here should also consider the negative terms as 2 and -2 both are the square roots of 4 as,
2 × 2 = (-2) × (-2) = 4
So, if a number is multiplied by the self then it will produce some specific number and the numbers which are being multiplied are called the square roots of that specific number. As in above example multiplication of 2 with 2 and -2 with -2 produces the same number 4 so both 2 and -2 are the square roots.
By this, it can be concluded that all positive numbers will have two roots one is √a and the other is -√a where a is some positive number. Now lets have some basic formulation of square roots
- √a × √a = a, √3 × √3 = 3
- a√b × a√b = a2 × b, 2√3 × 2√3 = 22 × 3
- √a × b = √a√b, √2 × 3 = √2√3
- √(a/b) = √a/√b, √(2/3) = √2/√3
- a√c + b√c = (a + b)√c, 2√5 + 3√5 = (2 + 3)√5
- a√c – b√c = (a – b)√c, 2√5 – 3√5 = (2 – 3)√5
a/(b + c√n) here will be rationalized the denominator by multiplying by (b – c√n)/(b – c√n). b – c√n is the conjugate of b + c√n.
Note:
Conjugate: The conjugate of any binomial is same as the number but with opposite sign.
For example, the conjugate of A + B will be A – B.
How do you divide a square root addition?
Solution:
The task is to divide a square root addition means we are given an addition of square root numbers or say an irrational number like √2 + √5 and it is needed to find a way to perform division operation on it. Lets understand what is Square root addition is?
Square Root Addition
Add square roots directly as done for natural numbers so for adding square roots there are some rules.
- Only add two square roots if the values under √ are equal.
- Only add numbers that are in front of √ so these numbers are called coefficients.
For example, 6√2 + 4√2 = 10√2
√2 + √3 ≠ √5
Divide a square root addition
The task is to divide a square root addition so for that let’s have a square root addition
a + √b
Now there are two cases:
- If one wants to divide it with a rational number X
- If one wants to divide it with an irrational number √X
Sample Problems
Question 1: Divide the square root addition by a rational number, (4 + √20)/2.
Solution:
Case 1: Dividing a square root addition by a rational number
(a + √b)/X = a/X + √b/X and then simplify
= 4/2 + √20/2
= 2 + (√5 × 2 × 2)/2
= 2 + (2√5)/2
= 2 + √5
Question 2: Divide the square root addition by irrational number, (2 + √7)/(2 + √3)
Solution:
Case 2: Dividing a square root addition by an irrational number, In this case, multiply the numerator and denominator by denominator’s conjugate and then simplify.
Here denominator’s conjugate is 2- √3
So, multiply it with numerator and denominator.
(2 + √7) × (2 – √3)/(2 + √3) × (2 – √3)
(4 – 2√3 + 2√7 – √7√3)/1
1 in denominator is obtained as (a + b)(a – b) = a2 – b2
4 – 2 √3 + 2 √7 – √21
One can simplify it further but the above answer is also complete this is how division on square root addition is performed.