The closure property for rational numbers means that adding, subtracting, or multiplying any two rational numbers always results in another rational number. The set of rational numbers is closed under these three operations, but not for division by zero.
Closure property in detail
Addition: The sum of any two rational numbers is a rational number.
Example: 14+23=3+812=1112one-fourth plus two-thirds equals the fraction with numerator 3 plus 8 and denominator 12 end-fraction equals 11 over 12 end-fraction14+23=3+812=1112, which is a rational number.
Subtraction: The difference between any two rational numbers is a rational number.
Example: 47−59=36−3563=163four-sevenths minus five-nineths equals the fraction with numerator 36 minus 35 and denominator 63 end-fraction equals 1 over 63 end-fraction47−59=36−3563=163, which is a rational number.
Multiplication: The product of any two rational numbers is a rational number.
Example: 47×59=2063four-sevenths cross five-nineths equals 20 over 63 end-fraction47×59=2063, which is a rational number.
Division: The quotient of two rational numbers is a rational number, with the exception of division by zero.
Example: 47÷59=47×95=3635four-sevenths divided by five-nineths equals four-sevenths cross nine-fifths equals 36 over 35 end-fraction47÷59=47×95=3635, which is a rational number.
Exception: Division by zero is undefined, so rational numbers are not closed under division by zero.
Whoever gives you Rs.100, you give him a piece of paper titled “blah blah blah..these are the terms and conditions, repayment dates, interest rates etc.”
This piece of paper is called Debenture.
In this case, you need 100 crores, meaning print 1 crore papers (debentures) each worth Rs.100.
Whoever holds such paper units is called Debenture holder.
The cash thus collected is a loan for the company. (=debt)
Difference between Bonds and Debentures?
Overall, the principle behind Bonds and Debentures is same: They offer fixed interest rate + principal repaid at the specified date.
1.Bond
2.Debenture
Issued byUnion GovernmentState GovernmentPSUs
Issued by companies.
Second difference: the different rates of Stamp Duty applied on each of them.
Third difference: The interest rate offered by Debenture is (usually) higher than Government Bonds. Because Government more likely to repay = no need to seduce customers with higher interest rate.
Types of Debentures
Based on ‘convertibility’ the Debentures are of two types
1.Convertible debentures
They can be converted into shares of the company on the expiry of xyz date.
2.Non-Convertible Debentures
They cannot be converted into shares.
When debenture is converted into shares, it means debt holder becomes an equity holder.
Both debt vs equity have their own advantages and disadvantages. We’ve discussed it in the earlier article (click ME)
But by and large, from the investor’s point of view, Debt is safer than Equity.
What is Optionally fully-convertible debentures (OFCD)?
These debentures can be converted into shares, when debt holder (investor) wishes (after expiry of xyz pre-decided date).
But the “rate”, will be decided by the company e.g. 20 debentures =>1 share.
From investor’s view, this “option” to convert Debenture into Shares is good ONLY IF
Company is likely to make huge profit (so you, the shareholder can earn more dividend.) OR
Company’s share-price is likely to rise in the share market (then you can sell shares to third-party and make profit).
BUT if the Company is going bankrupt, then it is better to avoid converting the Debenture into shares. Because when a company is liquidated (i.e. its assets sold off), the Debenture holders get the money before the shareholders.
It means Sahara OFCD is a bit tricky game. Investors should have some knowledge and understanding of share prices, company performance etc. else they could lose money. (or end up not getting maximum profit out of their investment). Now let’s move to the SEBI-SAHARA case.
2008-09: The game begins
Two firms of Sahara Conglomerate:
Sahara Housing Investment Corporation
Sahara India Real Estate Corporation. (aka Sahara Commodities)
These ^two companies Issued OFCD to collect money from investors.
~23 million people, mostly from villages and small towns subscribed to this scheme. They invested ~24,000 crores rupees in these OFCDs of SAHARA.
SEBI
You (SAHARA) have violated rules. If OFCDs are issued then whole process should be completed within 10 working days, but here you continue collecting money from people for more than two years!
SAHARA
This fund-raising was in the form of a private placement. I.e. we offered the schemes only to our select clients, this wasn’t meant a “Public Offer”! So what’s your problem?
SEBI
Dude if this is private placement, then maximum only 50 people can invest money in it.Here ~23 million people have parked their hard earned cash! Hell the number of investors in this case, is even more than the total number of people investing in the conventional stock-exchanges of India!India’s biggest IPO till date was of Coal India worth Rs.15000+ crores, and you’ve made 24,000 crores out of these OFCDs! It is my responsibility to protect the investors’ in Capital market.Hence, By the powers given to me under SEBI Act, I hereby order you to stop collecting money and refund all the money to those investors with 15% interest rate.
SAHARA
This is not right!
SEBI
Well, if you’re unhappy with my order you can go to the Securities Appellate Tribunal (SAT)
SAHARA
Pleads before SAT.
SAT
SEBI is right. You refund money to those people.
SAHARA
Now, I’ll appeal in Supreme Court.
2012: Supreme Court hearing
SC
What are your arguments?
SAHARA
Those two companies are unlisted. Meaning, their shares are not listed on any Stock Exchange of India.Therefore, their conduct is outside the jurisdiction of SEBI. Because SEBI is regulator for listed firms only.Our matter falls under Union Corporate Affairs Ministry and not under SEBI.
SEBI
Nope, this matter comes under my jurisdiction, becauseOFCD is a “security” under the Securities Act= it comes under the Sebi Act= I’ve the jurisdiction= Hence I can pass a special order to regulate unlisted companies!
Order of Supreme Court
Saare sabuto aur gawaaho ko madde nazar rakhte hue (in the light of all evidence and witnesses)
To SAHARA
To SEBI
We are unconvinced with your logic that OFCD schemes don’t come under the scope of SEBI.Mostly rural people have Invested money in your schemes and they’re not aware of OFCD.At the end of day, they would come and say that they were cheated. You know Harshad Mehta’s case, same modus operandi was there. Investors were not aware of the scheme.It seems you have no intention of returning the investors’ money. Your intentions are shady.We order you to refund the money.
If those two companies of SAHARA donot refund money, you’re free to attach their properties and freeze their bank accounts.Also conduct a probe against those two Sahara companies to find out their actual subscriber base. (to make sure some funny game or money laundering isnot going on.)Check the genuineness of the investors and if the investors are not traceable, the amount will go to the government.
Government’s response
As you’ve seen in ^this case, SAHARA’s main argument is “SEBI doesn’t have jurisdiction over our OFCD investment scheme, because this money was meant for our “unlisted” companies.”
Government has decided to fix this ambiguity in the new Companies Act.
According to Companies Bill 2012 (passed in Lok Sabha): SEBI will have undisputed jurisdiction over any investment scheme involving more than 50 investors-It doesn’t matter whether you’re a listed company or an unlisted company.
SEBI ordinance
In 2013, Government had introduced Securities Laws (amendment) Bill, 2013. This was to empower SEBI against Sahara like litigations.
although it couldn’t be passed, but since the matter was of urgent importance, government decided to implement the provisions as an ordinance.
In March 2014, For the third time, Government extended this ordinance.
The silent features of the bill/ordinance:
Empowers SEBI to investigate, raid and attach bank account and properties.
If a person doesn’t comply with SEBI order, SEBI can arrest him without approaching the court.
If any entity gathers funds of 100 crore OR above from juntaa, it’ll be called Collective investment scheme. (=indirectly it means company will fall under jurisdiction of SEBI, because SEBI regulates after Collective investment schemes.) This is necessary because Sahara says, “our scheme was above 100 crore but we are not a CIS, hence SEBI doesn’t have jurisdiction over us”
SEBI can further expand the definition of Collective Investment funds through regulations.
SEBI can sign agreements with foreign financial regulators for exchange of information.
Fast track courts for speedy trials of pending SEBI cases.
When you are drawing a graph of a line segment, then it is necessary to find the midpoint. In math the midpoint is the halfway of a line segment, it can be fun to find the midpoint by the Midpoint calculator. You can say the mid points are the average of the points in cartesian coordinates. You may be considering how to find the midpoint of points (x1, x2), (y1, y2), find the midpoint by adding the points x1 and x2 and then dividing them by 2. The average of the points actually enable us to find the midpoint in 2 dimension and 3 dimensional plan.
What is the midpoint formula?
The midpoint formula is adding the two points x1 and x2 and then dividing them by 2. The same process is done with the y1 and y2 and then divide them by 2. It is actually the average of the points, and we would be able to find the middle of the line segment. It can be tricky to find the middle point when we have the decimal or the fractional points, but when you are using our Midpoint calculator. You are simply able to find the middle of the line segment by the online calculator. So it would become simple, how to find the midpoint of a line segment of two dimensions and three dimensional plan.
In the two dimensional plan, the midpoint formula:
Point (x1,x2) and (y1,y2)
The simple formula for the mid point is:
M = (x₁ + x₂/2, y₁ + y₂/2)
We can write the midpoint formula as
Xm = x₁ + x₂/2
Ym = y₁ + y₂/2
Where;
M indicated as midpoint
Xm = x coordinate of midpoint
Ym = y coordinate of midpoint
In three dimension plan we can find the answer as:
Point (x1,x2) (y1,y2) (z1,z2)
M = (x₁ + x₂/2, y₁ + y₂/2, z₁ + z₂/2)
We can write the midpoint formula as
Xm = x₁ + x₂/2
Ym = y₁ + y₂/2
Zm = z₁ + z₂/2
It can be problematic to find the middle point, when we are dealing with the decimal or the fractional points, but when you are using our Midpoint calculator, the task becomes simple and fast.
Example 1:
Consider a cartesian plane, where (x₁ , x₂) are (6 , 8) and (y₁ , y₂) are ( 10, 12). The midpoint formula calculator is used to find the answer of the midpoint in the mid point.
The points are:
(x₁ , x₂) = (6 , 8)
(y₁ , y₂) = (10 , 12)
Now find the midpoint by the following formula:
M = (x₁ + x₂/2, y₁ + y₂/2)
M = (6 + 8)/2, (10+12)/2
M = (14)/2, (22)/2
M =7, 11
(x , y) = (7 , 11)
How to find midpoint distance?
It can be difficult to find the midpoint distance when we are determining the midpoint of decimal points. The Midpoint calculator is simple and fast and makes the task for the user.
Examples 2:
Consider (x,y,z) plan, here (x₁ , x₂) are (5 , 10) , (y₁ , y₂) are (8, 20) and the (z₁ , z₂) (20 , 25).
Then by answer of question is:
(x₁ , x₂) = (5 , 10)
(y₁ , y₂) = (8, 20)
(z₁ , z₂) = (20 , 25)
Solution:
M = (x₁ + x₂/2, y₁ + y₂/2, z₁ + z₂/2)
M = (5+10)/2, (8+20)/2, (20+25)/2
M = (15)/2, (28)/2, (45)/2
Mid points = ( 7.5 ,14, 22.5 )
Where x= 7.5 , y = 14, z = 22.5
The distance of midpoint:
The distance calculated by the Midpoint calculator by the following formula and the we are adding the three axis in the formula, the
The distance describes the average distance between the points in a three dimensional plan.
Conclusion:
The midpoint is used to draw the graph and the drawing a straight line. It can be tricky to find the midpoint of fractional or decimal points.But you can find the answer in a matter of moments when using the Midpoint calculator. You are even able to find the distance between the points and their exact location in cartesian coordinates and three dimensional plan.You can determine the midpoint of decimal points. The Midpoint calculator is simple and fast and makes the task for the user.It is also a way to learn the midpoints of various geometrical points.
