Molality

The reciprocal function: y = 1/x. For every x except 0, y represents its multiplicative inverse. The graph forms a rectangular hyperbola.

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x⁻¹, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The reciprocal function, the function f(x) that maps x to 1/x, is one of the simplest examples of a function which is its own inverse (an involution).

The term reciprocal was in common use at least as far back as the third edition of Encyclopædia Britannica (1797) to describe two numbers whose product is 1; geometrical quantities in inverse proportion are described as reciprocall in a 1570 translation of Euclid's Elements.

In the phrase multiplicative inverse, the qualifier multiplicative is often omitted and then tacitly understood (in contrast to the additive inverse). Multiplicative inverses can be defined over many mathematical domains as well as numbers. In these cases it can happen that ab ≠ ba; then "inverse" typically implies that an element is both a left and right inverse.

The notation x⁻¹ is sometimes also used for the inverse function, which usually is not equal to the multiplicative inverse. For example, 1/sin x = (sin x)⁻¹ is very different from the inverse of sin x, denoted sin⁻¹ x or arcsin x. Only for linear maps are they strongly related (see below). The terminology difference reciprocal versus inverse is not sufficient to make this distinction, since many authors prefer the opposite naming convention, probably for historical reasons (for example in French, the inverse function is preferably called application réciproque).

Examples and counterexamples

In the real numbers, zero does not have a reciprocal because no real number multiplied by 0 produces 1 (the product of any number with zero is zero). With the exception of zero, reciprocals of every real number are real, reciprocals of every rational number are rational, and reciprocals of every complex number are complex. The property that every element other than zero has a multiplicative inverse is part of the definition of a field, of which these are all examples. On the other hand, no integer other than 1 and -1 has an integer reciprocal, and so the integers are not a field.

In modular arithmetic, the modular multiplicative inverse of a is also defined: it is the number x such that ax ≡ 1 (mod n). This multiplicative inverse exists if and only if a and n are coprime. For example, the inverse of 3 modulo 11 is 4 because 4 · 3 ≡ 1 (mod 11). The extended Euclidean algorithm may be used to compute it.

The sedenions are an algebra in which every nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, i.e. nonzero elements x, y such that xy = 0.

A square matrix has an inverse if and only if its determinant has an inverse in the coefficient ring. The linear map that has the matrix A⁻¹ with respect to some base is then the reciprocal function of the map having A as matrix in the same base. Thus, the two distinct notions of the inverse of a function are strongly related in this case, while they must be carefully distinguished in the general case (as noted above).

The trigonometric functions are related by the reciprocal identity: the cotangent is the reciprocal of the tangent; the secant is the reciprocal of the cosine; the cosecant is the reciprocal of the sine.

A ring in which every nonzero element has a multiplicative inverse is a division ring; likewise an algebra in which this holds is a division algebra.

Complex numbers

As mentioned above, the reciprocal of every nonzero complex number Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is complex. It can be found by multiplying both top and bottom of 1/z by its complex conjugate ${\bar {z}}=a-bi$ and using the property that $z{\bar {z}}=\|z\|^{2}$, the absolute value of z squared, which is the real number Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.:

${\frac {1}{z}}={\frac {\bar {z}}{z{\bar {z}}}}={\frac {\bar {z}}{\|z\|^{2}}}={\frac {a-bi}{a^{2}+b^{2}}}={\frac {a}{a^{2}+b^{2}}}-{\frac {b}{a^{2}+b^{2}}}i.$

In particular, if ||z||=1 (z has unit magnitude), then $1/z={\bar {z}}$. Consequently, the imaginary units, ±Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer., have additive inverse equal to multiplicative inverse, and are the only complex numbers with this property. For example, additive and multiplicative inverses of Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. are −(Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.) = −Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. and 1/Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. = −Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer., respectively.

For a complex number in polar form Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer., the reciprocal simply takes the reciprocal of the magnitude and the negative of the angle:

${\frac {1}{z}}={\frac {1}{r}}\left(\cos(-\varphi )+i\sin(-\varphi )\right).$

Geometric intuition for the integral of 1/x. The three integrals from 1 to 2, from 2 to 4, and from 4 to 8 are all equal. Each region is the previous region scaled vertically down by 50%, then horizontally by 200%. Extending this, the integral from 1 to 2^k is k times the integral from 1 to 2, just as ln 2^k = k ln 2.

Calculus

In real calculus, the derivative of Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is given by the power rule with the power −1:

${\frac {d}{dx}}x^{-1}=(-1)x^{(-1)-1}=-x^{-2}=-{\frac {1}{x^{2}}}.$

The power rule for integrals (Cavalieri's quadrature formula) cannot be used to compute the integral of 1/x, as it is an exceptional case. Instead the integral is given by:

$\int _{1}^{a}{\frac {1}{x}}\,dx=\ln a,$

$\int {\frac {1}{x}}\,dx=\ln x+C.$

where ln is the natural logarithm. To show this, note that ${\frac {d}{dx}}e^{x}=e^{x}$, so if $y=e^{x}$ and $x=\ln y$, we have:

${\frac {dy}{dx}}=y\quad \Rightarrow \quad {\frac {dy}{y}}=dx\quad \Rightarrow \quad \int {\frac {1}{y}}\,dy=\int 1\,dx\quad \Rightarrow \quad \int {\frac {1}{y}}\,dy=x+C=\ln y+C.$

Algorithms

Computing the reciprocal is important in many division algorithms, since the quotient a/b can be computed by first computing 1/b and then multiplying it by a. Noting that $f(x)=1/x-b$ has a zero at x = 1/b, Newton's method can find that zero, starting with a guess $x_{0}$ and iterating using the rule:

$x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}=x_{n}-{\frac {1/x_{n}-b}{-1/x_{n}^{2}}}=2x_{n}-bx_{n}^{2}=x_{n}(2-bx_{n}).$

This continues until the desired precision is reached. For example, suppose we wish to compute 1/17 ≈ 0.0588 with 3 digits of precision. Taking x₀ = 0.1, the following sequence is produced:

x₁ = 0.1(2 - 17 × 0.1) = 0.03

x₂ = 0.03(2 - 17 × 0.03) = 0.0447

x₃ = 0.0447(2 - 17 × 0.0447) ≈ 0.0554

x₄ = 0.0554(2 - 17 × 0.0554) ≈ 0.0586

x₅ = 0.0586(2 - 17 × 0.0586) ≈ 0.0588

A typical initial guess can be found by rounding b to a nearby power of 2, then using bit shifts to compute its reciprocal.

In constructive mathematics, for a real number x to have a reciprocal, it is not sufficient that x ≠ 0. There must instead be given a rational number r such that 0 < r < |x|. In terms of the approximation algorithm described above, this is needed to prove that the change in y will eventually become arbitrarily small.

Graph of f(x) = x^x showing the minimum at (1/e, e^-1/e).

This iteration can also be generalised to a wider sort of inverses, e.g. matrix inverses.

Reciprocals of irrational numbers

Every number excluding zero has a reciprocal, and reciprocals of certain irrational numbers often can prove useful for reasons linked to the irrational number in question. Examples of this are the reciprocal of e (≈ 0.367879), which is special because no other positive number can produce a lower number when put to the power of itself; and the golden ratio's reciprocal (≈ 0.618034), which is exactly one less than the golden ratio; the golden ratio is the only positive number with this property.

There are an infinite number of irrational reciprocal pairs that differ by an integer (giving the curious effect that the pairs share their infinite mantissa). These pairs can be found by simplifying n+√(n²+1) for any integer n, and taking the reciprocal. For example, n = 2 produces 2+√5, whose reciprocal 1/(2+√5) is -2+√5, exactly 4 less.

Further remarks

If the multiplication is associative, an element x with a multiplicative inverse cannot be a zero divisor (meaning for some y, xy = 0 with neither x nor y equal to zero). To see this, it is sufficient to multiply the equation xy = 0 by the inverse of x (on the left), and then simplify using associativity. In the absence of associativity, the sedenions provide a counterexample.

The converse does not hold: an element which is not a zero divisor is not guaranteed to have a multiplicative inverse. Within Z, all integers except −1, 0, 1 provide examples; they are not zero divisors nor do they have inverses in Z. If the ring or algebra is finite, however, then all elements a which are not zero divisors do have a (left and right) inverse. For, first observe that the map ƒ(x) = ax must be injective: ƒ(x) = ƒ(y) implies x = y:

${\begin{aligned}ax&=ay&\quad \Rightarrow &\quad ax-ay=0\\&&\quad \Rightarrow &\quad a(x-y)=0\\&&\quad \Rightarrow &\quad x-y=0\\&&\quad \Rightarrow &\quad x=y.\end{aligned}}$

Distinct elements map to distinct elements, so the image consists of the same finite number of elements, and the map is necessarily surjective. Specifically, ƒ (namely multiplication by a) must map some element x to 1, ax = 1, so that x is an inverse for a.

Applications

The expansion of the reciprocal 1/q in any base can also act as a source of pseudo-random numbers, if q is a "suitable" safe prime, a prime of the form 2p + 1 where p is also a prime. A sequence of pseudo-random numbers of length q − 1 will be produced by the expansion.

See also

• Division (mathematics)
• Fraction (mathematics)
• Group (mathematics)
• Ring (mathematics)
• Division algebra
• Exponential decay
• Unit fractions – reciprocals of integers
• Hyperbola
• Repeating decimal

References

• Maximally Periodic Reciprocals, Matthews R.A.J. Bulletin of the Institute of Mathematics and its Applications vol 28 pp 147–148 1992