 |
| Image Credit: Pixabay.com |
➧ It is denoted by ℝ.
➧ Every real number is either rational or irrational.
➧ Every real number is represented by a unique point on the number line. Also, every point on the number line represents a unique real number. This is why we call the number line, the real number line.
➧ In the 1870s two German mathematicians, Cantor and Dedekind, showed that: Corresponding to every real number, there is a point on the real number line, and corresponding to every point on the number line, there exists a unique real number.
Representing Real numbers on Number line
➧ It is easy to represent integers on number line and we know how to do it including fractions and decimals up to few decimal places.➧ How can we represent numbers with more than two digits after decimal point on the number line?
How can we represent non-terminating decimal numbers on the number line?
The process of successive magnification
The process of visualisation of representation of numbers on the number line, through a magnifying glass, is known as the process of successive magnification.It is possible by sufficient successive magnifications to visualise the position (or representation) of a real number with a terminating and non-terminating decimal expansion on the number line.
➢ Let us locate 3.765 on the number line.
↬ Integral part of the number is positive 3, therefor the number lie between 3 and 4. Locate 3 and 4 on the number line.
| ↬ Suppose we divide the portion between 3 & 4 into ten equal divisions & mark each point of division as in figure, then first mark is 3.1, the second mark is 3.2 & so on. ↬ Suppose we are using magnifying glass which will give us the clear view of subdivisions as in figure. ↬ The number at first decimal place is 7, therefore we look at the portion between 3.7 & 3.8. Again suppose this portion is divided into ten equal parts then the first mark is 3.71, the second mark is 3.72 & so on. |  |
↬ Therefore, 3.765 is the 5th mark in these subdivisions.
Operations on Real numbers
➧ Rational numbers satisfy the commutative, associative and distributive laws for addition and multiplication.Rational numbers are ‘closed’ with respect to addition, subtraction, multiplication and division (except by zero).
➧ Irrational numbers also satisfy the commutative, associative and distributive laws for addition and multiplication.
The sum, difference, quotients and products of irrational numbers are not always irrational i.e., not always closed.
(i) The sum or difference of a rational number and an irrational number is irrational.
2+√3 = 2+1.71... = 3.71...
(ii) The product or quotient of a non-zero rational number with an irrational number is irrational.
2×√3 = 2√3
(iii) If we add, subtract, multiply or divide two irrationals, the result may be rational or irrational.
2√3 +√3 = (2+1)√3 = 3√3,
√3-√3 = 0,
√3×√3 = 3,
Finding Roots of any Positive Real number x geometrically.
(Negative numbers cannot have square roots as square of any real number i.e. square number is always positive).
→ Mark the distance 3.5 units from a fixed point A on a given line to obtain a point B such that AB = 3.5 units
→ From B, mark a distance of 1 unit and mark the new point as C. BC = 1 unit
→ Find the mid-point of AC and mark that point as O. Draw a semicircle with centre O and radius OC.
→ Draw a line BD perpendicular to AC passing through B and intersecting the semicircle at D. Then, BD = √3.5.
➢ More generally, to find √x , for any positive real number x, we mark B so that AB = x units, and, as above, mark C so that BC = 1 unit. Then, as we have done for the case x = √3.5, we find BD=√x .
➢ Proof - We can prove this result using the Pythagoras Theorem.
→ The radius of the circle is x +1/2 units.
OC = OD = OA = x +1/2 units.
OB = x- (x +1/2) = x -1/2
→ In right-angled triangle ∆OBD,
BD2 = OD2 – OB2 (by the Pythagoras Theorem)
⇒ BD2 = (x +1/2) - (x -1/2) = 4x/4 = x
⇒ BD = √x
This construction gives us a visual, and geometric way of showing that √x exists for all real numbers x > 0.
The position of √x on the number line.
→ If we want to know the position of x on the number line, then let us treat the line BC as the number line, with B as zero, C as 1, and so on.→ Draw an arc with centre B and radius BD, which intersects the number line in E. Then, E represents √x .
➧ We can extend the idea of square roots to cube roots, fourth roots, and in general nth roots, where n is a positive integer.
→ We can define n√a for a real number a > 0 and a positive integer n, let a > 0 be a real number and n be a positive integer. Then n√a = b, if bn = a and b > 0.
→ The symbol ‘ √’ used in √2 , 3√8, n√a , etc. is called the radical sign.
Identities relating to square roots of Real numbers.
(i) √ab = √a √b(ii) √(a/b) = √a /√b
(iii) (√a + √b) ( √a − √b) = (√a)2 − (√b)2
= a − b
(iv) (a + √b) ( a − √b) = (a)2 − (√b)2
= a2 − b
(v) (√ a +√ b) (√c + √d ) = √a(√c + √d ) + √b(√c + √d )
= √a√c + √a√d + √b√c+ √b√d
(vi) (√ a +√ b)2 = (√a)2 + 2√a√b +(√b)2
= a + 2√ab + b
→ √ab = √a √b
→ √(a/b) = √a /√b
→ (√a + √b) ( √a − √b) = a − b
→ ( a + √b) ( a − √b) = a2 − b
→ (√ a +√ b) (√c + √d ) = √ac + √ad + √bc + √bd
→ (√ a +√ b)2 = a + 2√ab + b
Rationalising the denominator
When an expression in the denominator is irrational i.e., terms with root (number under the radical sign √), the process of converting it to an equivalent expression of rational number is called rationalising the denominator.→ Thus it is the process of eliminating of radicals in the denominator of a algebraic fraction.
→ Rationalise the denominator of 1/√2
1/√2 × √2/√2 = 1×√2/√2×√2 = √2/2
→ Rationalise the denominator of 1/2+√3
1/2+√3 × (2⁻√3)/2⁻√3 = 2⁻√3/4-3 = 2⁻√3/1 = 2⁻√3
Laws of Exponents for Real numbers
➧ We already know the following laws of exponents, (Here a, n and m are integers. a is called the base and m and n are the exponents.)(i) am × an = am+n
(ii) (am)n = amn
(iii) am /an = am−n, m > n
(iv) ambm = (ab)m
(v) a0= 1.
(vi) 1/an = a−n (putting m=0 in (iii))
➧ Let a > 0 be a real number and n a positive integer. Then n√a = b, if bn = a and b > 0.
In the language of exponents, we definen√a = a1/n (bn = a ⇔ b = a1/n)
➧ Let a > 0 be a real number. Let m and n be integers such that m and n have no common factors other than 1, and n > 0. Then,
am/n = ( n√a)m = n√am
➧ We now have the following extended laws of exponents:
Let a > 0 be a real number and p and q be rational numbers. Then, we have
(i) ap . aq = ap+q
(ii) (ap )q = apq
(iii) ap / aq = ap-q
(iv) ap . bp = (ab)p
(v) a0= 1.
(vi) a−p=1/ap
(vii) a1/q =q√a
| Irrational Numbers | |