In late 2006, Finland’s Nokia announced Wibree, a new short range wireless technology it had developed to enable short-range communication between electronic devices. The crucial difference between Wibree and existing Bluetooth technology is that Wibree ran on as little as one tenth the power of Bluetooth.

Wibree is a digital radio technology (intended to become an open standard of wireless communications) designed for ultra low power consumption (button cell batteries) within a short range (10 meters / 30ft) based around low-cost transceiver microchips in each device.

Wibree, also known as *Bluetooth **U**ltra **L**ow **P*** ower**, consumes only a fraction of the power compared to other such radio technologies, enabling smaller and less costly implementations and it is easy to integrate with Bluetooth solutions.

In 2001, Nokia researchers determined that there were various scenarios that contemporary wireless technologies did not address. To address the problem, Nokia Research Center started the development of a wireless technology adapted from the Bluetooth standard which would provide lower power usage and price while minimizing difference between Bluetooth and the new technology. The results were published in 2004 using the name Bluetooth Low End Extension. After further development with partners, e.g., within EU FP6 project MIMOSA, the technology was released to public in October 2006 with brand name Wibree. After negotiations with Bluetooth SIG members, in June 2007, an agreement was reached to include Wibree in future. Bluetooth specification as a Bluetooth ultra-low-power technology, now known as Bluetooth low energy technology

Wibree is similar in many respects to the now prevalent Bluetooth standard. Both use the 2.45 GHz band to transfer data and have a 1 Mbps transfer rate (although the newer Bluetooth 2.0 standard already incorporates a 3.0 Mbps transfer rate) and a rage of about 10 meters (m). The two complementary technologies differ in size, price, and most of all power consumption. Wibree would use only a fraction of the power consumed by today’s Bluetooth chips, resulting in a much longer battery life and more compact devices. While Bluetooth can be used to transmit audio and media files, Wibree is designed to extend this network by serving applications that transmit only small amounts of data and where size and cost are priorities. Many applications that were not cost-effective using existing Bluetooth technology, such as wirelessly controlled toys, watches, medical and sports sensors, and a range of other applications that have not been conceived yet, might be developed using Wibree technology

There are two types of Wibree implementations –one based on the Wibree standalone chip, and another based on the Wibree-Bluetooth dual-mode chip – which serve different purposes and are installed on different devices. Stand-alone Wibree chips would be implemented in small, low cost devices such as wireless mouse and keyboards, sensors, and toys. The Wibree-Bluetooth dual-mode chips are in mobile phones, allowing users to benefit from both worlds – Bluetooth 2.0 high speed and Wibree’s low power and extended ability to communicate with a new generation of smaller wireless devices.

There is an irony in the fact that the origins of Wibree were the alternative proposal for the radio and Media Access Controller (MAC) for the 802.15.4 standard, which is now the basis of ZigBee and other short range radio networks.

Wibree comprises a physical layer, light weight protocol stack and application-specific profiles. It is designed to connect mobile phones and PCs to a range of coin-cell battery-powered devices that require battery lifetimes of years. Nordic Semiconductor became one of the first members joining the Wibree open initiative and is member of the Wibree specification group. Other members include CSR, Broadcom, Epson, Suunto, and Taiyo Yuden.

The specification details a short-range RF communication technology featuring ultra low power consumption, a lightweight protocol stack and integration with Bluetooth. Wibree operates in the globally accepted 2.4 GHz ISM (Industrial, Scientific & Medical) band. It features a physical layer bit rate of 1Mbit/s over a range of 5 to 10 meters. The specification features two implementations: dual-mode and stand-alone. In the dual-mode implementation, Wibree functionality is integrated into Bluetooth circuitry.

Wibree is designed to work side-by-side with and complement Bluetooth. It operates in 2.4 GHz ISM band with physical layer bit rate of 1 Mbit/s. Main applications include devices such as wrist watches, wireless keyboards, toys and sports sensors where low power consumption is a key design requirement.

Wibree is not designed to replace Bluetooth, but rather to complement the technology in supported devices. Wibree-enabled devices will be smaller and more energy-efficient than their Bluetooth counterparts. This is especially important in devices such as wristwatches, where Bluetooth models may be too large and heavy to be comfortable. Replacing Bluetooth with Wibree will make the devices closer in dimensions and weight to current standard wristwatches.

There will be two types of Wibree implementations: – one based on the ** Wibree standalone chip**, and another based on the

** Stand-alone Wibree chips** would be implemented in small, low cost devices such as wireless mouse and keyboards, sensors, and toys. The Wibree stand-alone chip is designed for use with applications which require extremely low power consumption, small size, low cost and where only small quantities of data are transferred. It’s an ideal solution for small devices (like heart rate monitors) that use only short data message and must have long battery life. Examples of devices that would benefit from the Wibree stand-alone chip are: watches, sports and wellness devices and human interface devices (HID) such as wireless keyboards.

The ** Wibree-Bluetooth dual-mode chips** would probably be implemented in future mobile phones, allowing users to benefit from both worlds – Bluetooth 2.0 high speed and Wibree’s low power and extended ability to communicate with a new generation of smaller wireless devices. The Bluetooth-Wibree dual-mode chip is designed for use in Bluetooth devices. In this type of implementation, Wibree functionality can be integrated with Bluetooth for a minor incremental cost by utilizing key Bluetooth components and the existing Bluetooth RF

Wibree differs from Bluetooth in several fundamental ways.

: Recent Bluetooth specifications, notably 2.0, are designed with an emphasis on throughput, or data transfer speed. Bluetooth 2.0 devices can exceed speeds of 350 kb/s under ideal conditions. This is about three times the maximum speed of planned Wibree devices, which transfer data no faster than.128 kb/s. The tradeoff comes to light in terms of power, space and weight savings. Current Bluetooth-enabled wristwatches must replace their large, specialty batteries on a monthly basis.*Data Transfer Speed*Bluetooth uses a frequency hopping technology to avoid interference from other devices operating in the same frequency. Wibree does not use frequency hopping.*Use of Frequency Hopping:*Bluetooth uses fixed packet length. This increases power usage as unnecessary transmission occurs. Wibree has a variable packet length and transmits only when necessary.*Packet Length:*Bluetooth drains your cell phone battery as it needs quite a lot of power to remain active. Wibree aims to survive for a full year on a button sized battery. In contrast to Bluetooth, Wibree goes into sleep mode when not transmitting. In sleep mode the radio will be off and will save a lot of power. Wibree devices wake up only when they want to transmit.*Battery Usage:*The major usage difference between Wibree and Bluetooth is the traffic characteristics. Bluetooth is useful when transferring files, using the hands free etc where the volume of data that needs to transferred is considerable.*Traffic Characteristics:*Wibree is used in areas where only short bursts of data need to be transmitted. Remotes, sensor data etc.*Type of Data Transferred:*

Imagine a wireless keyboard and mouse with battery lifetimes exceeding one year communicating with a PC without using a fragile dongle. Imagine a watch equipped with a wireless link communicating with both a tiny sports sensor embedded within the user’s shoe and mobile phone. Imagine a range of personal devices communicating with mobile phones or PCs, but without the inconvenience of changing or charging batteries every week. Imagine no longer, because Wibree will make all of these applications – and many more – a reality.

*Mobile phones* equipped with Wibree will enable a range of new accessories such as call control/input devices, sports and health sensors, and security and payment devices with battery lifetimes of up to three years (depending on usage pattern). Wibree will also bring wireless connectivity to high-performance PC accessories such as mice, keyboards and multimedia remote controls with battery lifetimes of up to a year. And Wibree will add wireless connectivity to watches and sports sensors (for example, heart rate monitors) without significantly compromising battery lifetime.

*Mobile Phone Accessories* – Mobile phones equipped with Wibree technology will enable a range of new accessories such as call control/input devices, sports and health sensors, security and payment devices. These devices will benefit from the ultra-low power consumption of Wibree making possible compact, coin cell battery operated devices with battery lifetimes up to 3 years (depending on the actual application).

*PC Accessories* – Wibree is designed to offer wireless connectivity to high performance PC accessories such as mice, keyboards and multimedia remote controls. The ultra-low power consumption of Wibree extends battery lifetimes to over a year. Nordic will build on its position as a leading provider of ultra-low power 2.4 GHz technology to encourage Wibree’s adoption into PC accessories enabling the next generation of wireless mice and keyboards.

*Watches* – Imagine your watch equipped with a wireless link communicating with both a tiny sports sensor embedded in your shoe and your mobile phone.

Wibree is the first wireless technology to solve the following needs in a single solution.

- Ultra low peak and average power consumption in both active and idle modes.
- Ultra low cost and small size for accessories and human interface devices (HID).
- Minimal cost and size addition to mobile phones and PCs.
- Global, intuitive and secure multi-vendor interoperability.

Data transmission is very slow, i.e., only 1 megabit per second. Also, Wibree cannot be used in high bandwidth required applications.

Wibree will also dramatically extend the battery lifetime of existing wireless device such as keyboards, mice and remote controls. It’s up to 10 times more energy efficient then Bluetooth. Nokia said it expected the first commercial version of the standard to be available during the second quarter of 2007. The firm said it expected dual Bluetooth-Wibree devices such as mobile phones to hit the market within two years.

Wibree and Bluetooth both operate at 2.4GHz and have a similar range of around 10m, but the difference between the two lies in the continuity of the data being transferred. While Bluetooth is suited to constant uses such as streaming data or voice connectivity, Wibree is being touted as ideal for infrequent bursts of data — where the connected device will need to consume much less power.

Bluetooth and Wi-Fi are both wireless networking standards that provide connectivity via radio waves. The main difference: Bluetooth’s primary use is to replace cables, while Wi-Fi is largely used to provide wireless, high-speed access to the Internet or a local area network.

Wibree radio technology complements other local connectivity technologies, consuming only a fraction of the power compared to other such radio technologies, enabling smaller and less costly implementations and being easy to integrate with Bluetooth solutions.

Wibree is the first open technology offering connectivity between mobile devices or Personal Computers, and small, button cell battery power devices such as watches, wireless keyboards, toys and sports sensors. By extending the role mobile devices can play in consumers’ lives, this technology increases the growth potential in these market segments.

Bluetooth wireless technology is a short-range communications technology intended to replace the cables connecting portable and/or fixed devices while maintaining high levels of security. The key features of Bluetooth technology are robustness, low power, and low cost. The Bluetooth specification defines a uniform structure for a wide range of devices to connect and communicate with each other.

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*Standard **D*** eviation (SD)** and

Interestingly, an SE has nothing to do with standards, with errors, or with the communication of scientific data.

A detailed look at the origin and the explanation of SD and SE will reveal, why professional statisticians and those who use it cursorily, both tend to err.

*Standard Deviation (SD)*

An SD is a **descriptive** statistic describing the spread of a distribution. As a metric, it is useful when the data are normally distributed. However, it is less useful when data are highly skewed or bimodal because it doesn’t describe very well the shape of the distribution. Typically, we use SD when reporting the characteristics of the sample, because we intend to **describe** how much the data varies around the mean. Other useful statistics for describing the spread of the data are inter-quartile range, the 25th and 75th percentiles, and the range of the data.

*Figure 1. SD is a measure of the spread of the data. When data are a sample from a normally distributed distribution, then one expects two-thirds of the data to lie within 1 standard deviation of the mean.*

Variance is a **descriptive** statistic also, and it is defined as the square of the standard deviation. It is not usually reported when describing results, but it is a more mathematically tractable formula (a.k.a. the sum of squared deviations) and plays a role in the computation of statistics.

For example, if we have two statistics ** P** &

But standard deviations carry an important meaning for spread, particularly when the data are normally distributed: The interval mean ** +/- 1 SD** can be expected to capture 2/3 of the sample, and the interval mean

SD provides an indication of how far the individual responses to a question vary or “deviate” from the mean. SD tells the researcher how spread out the responses are — are they concentrated around the mean, or scattered far & wide? Did all of your respondents rate your product in the middle of your scale, or did some approve it and some disapprove it?

Consider an experiment where respondents are asked to rate a product on a series of attributes on a 5-point scale. The mean for a group of ten respondents (labeled ‘A’ through ‘J’ below) for “good value for the money” was 3.2 with a SD of 0.4 and the mean for “product reliability” was 3.4 with a SD of 2.1.

At first glance (looking at the means only) it would seem that reliability was rated higher than value. But the higher SD for reliability could indicate (as shown in the distribution below) that responses were very polarized, where most respondents had no reliability issues (rated the attribute a “5”), but a smaller, but important segment of respondents, had a reliability problem and rated the attribute “1”. Looking at the mean alone tells only part of the story, however, more often than not, this is what researchers focus on. The distribution of responses is important to consider and the SD provides a valuable descriptive measure of this.

Respondent |
Good Value for the Money |
Product Reliability |

A | 3 | 1 |

B | 3 | 1 |

C | 3 | 1 |

D | 3 | 1 |

E | 4 | 5 |

F | 4 | 5 |

G | 3 | 5 |

H | 3 | 5 |

I | 3 | 5 |

J | 3 | 5 |

Mean |
3.2 |
3.4 |

Std. Dev. |
0.4 |
2.1 |

*First Survey: Respondents rating a product on a 5-point scale*

Two very different distributions of responses to a 5-point rating scale can yield the same mean. Consider the following example showing response values for two different ratings.

In the first example (Rating “A”), SD is zero because ALL responses were exactly the mean value. The individual responses did not deviate at all from the mean.

In Rating “B”, even though the group mean is the same (3.0) as the first distribution, the Standard Deviation is higher. The Standard Deviation of 1.15 shows that the individual responses, on average*, were a little over 1 point away from the mean.

Respondent |
Rating “A” |
Rating “B” |

A | 3 | 1 |

B | 3 | 2 |

C | 3 | 2 |

D | 3 | 3 |

E | 3 | 3 |

F | 3 | 3 |

G | 3 | 3 |

H | 3 | 4 |

I | 3 | 4 |

J | 3 | 5 |

Mean |
3.0 |
3.0 |

Std. Dev. |
0.00 |
1.15 |

*Second Survey: Respondents rating a product on a 5-point scale*

Another way of looking at SD is by plotting the distribution as a histogram of responses. A distribution with a low SD would display as a tall narrow shape, while a large SD would be indicated by a wider shape.

SD generally does not indicate “right or wrong” or “better or worse” — a lower SD is not necessarily more desirable. It is used purely as a descriptive statistic. It describes the distribution in relation to the mean.

*T**echnical disclaimer relating to SD*

Thinking of SD as an “average deviation” is an excellent way of conceptually understanding its meaning. However, it is not actually calculated as an average (if it were, we would call it the “average deviation”). Instead, it is “standardized,” a somewhat complex method of computing the value using the sum of the squares.

For practical purposes, the computation is not important. Most tabulation programs, spreadsheets or other data management tools will calculate the SD for you. More important is to understand what the statistics convey.

*Standard Error*

A standard error is an **inferential** statistic that is used when comparing sample means (averages) across populations. It is a measure of **precision** of the sample mean. The sample mean is a statistic derived from data that has an underlying distribution. We can’t visualize it in the same way as the data, since we have performed a single experiment and have only a single value. Statistical theory tells us that the sample mean (for a large “enough” sample and under a few regularity conditions) is approximately normally distributed. The standard deviation of this normal distribution is what we call the standard error.

*Figure 2.** The distribution at the bottom repre**sents the distribution of the data, whereas the distribution at the top is the theoretical distribution of the sample mean. The SD of 20 is a measure of the spread of the data, whereas the SE of 5 is a measure of uncertainty around the sample mean.*

When we want to compare the means of outcomes from a two-sample experiment of Treatment A vs. Treatment B, then we need to estimate how precisely we’ve measured the means.

Actually, we are interested in how precisely we’ve measured the difference between the two means. We call this measure the standard error of the difference. You may not be surprised to learn that the standard error of the difference in the sample means is a function of the standard errors of the means:

Now that you’ve understood that the standard error of the mean (SE) and the standard deviation of the distribution (SD) are two different beasts, you may be wondering how they got confused in the first place. Whilst they differ conceptually, they have a simple relationship mathematically:

,where n is the number of data points.

Notice that the standard error depends upon two components: the standard deviation of the sample, and the size of the sample ** n**. This makes intuitive sense: the larger the standard deviation of the sample, the less precise we can be about our estimate of the true mean.

Also, the large the sample size, the more information we have about the population and the more precisely we can estimate the true mean.

SE is an indication of the reliability of the mean. A small SE is an indication that the sample mean is a more accurate reflection of the actual population mean. A larger sample size will normally result in a smaller SE (while SD is not directly affected by sample size).

Most survey research involves drawing a sample from a population. We then make inferences about the population from the results obtained from that sample. If a second sample was drawn, the results probably won’t exactly match the first sample. If the mean value for a rating attribute was 3.2 for one sample, it might be 3.4 for a second sample of the same size. If we were to draw an infinite number of samples (of equal size) from our population, we could display the observed means as a distribution. We could then calculate an average of all of our sample means. This mean would equal the true population mean. We can also calculate the SD of the distribution of sample means. The SD of this distribution of sample means is the SE of each individual sample mean.

We, thus, have our most significant observation: *SE is the SD of the population mean.*

Sample |
Mean |

1st | 3.2 |

2nd | 3.4 |

3rd | 3.3 |

4th | 3.2 |

5th | 3.1 |

…. | …. |

…. | …. |

…. | …. |

…. | …. |

…. | …. |

Mean |
3.3 |

Std. Dev. |
0.13 |

*Table illustrating the relation between SD and SE*

It is now clear that if the SD of this distribution helps us to understand how far a sample mean is from the true population mean, then we can use this to understand how accurate any individual sample mean is in relation to the true mean. That is the essence of SE.

In actuality, we have only drawn a single sample from our population, but we can use this result to provide an estimate of the reliability of our observed sample mean.

In fact, SE tells us that we can be 95% confident that our observed sample mean is plus or minus roughly 2 (actually 1.96) Standard Errors from the population mean.

The below table shows the distribution of responses from our first (and only) sample used for our research. The SE of 0.13, being relatively small, gives us an indication that our mean is relatively close to the true mean of our overall population. The margin of error (at 95% confidence) for our mean is (roughly) twice that value (+/- 0.26), telling us that the true mean is most likely between 2.94 and 3.46.

Respondent |
Rating |

A | 3 |

B | 3 |

C | 3 |

D | 3 |

E | 4 |

F | 4 |

G | 3 |

H | 3 |

I | 3 |

J | 3 |

Mean |
3.2 |

Std. Err |
0.13 |

*Summary*

Many researchers fail to understand the distinction between Standard Deviation and Standard Error, even though they are commonly included in data analysis. While the actual calculations for Standard Deviation and Standard Error look very similar, they represent two very different, but complementary, measures. SD tells us about the shape of our distribution, how close the individual data values are from the mean value. SE tells us how close our sample mean is to the true mean of the overall population. Together, they help to provide a more complete picture than the mean alone can tell us.

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*Fig. 1: Tethering refers to literally tethering your phone to the computer through USB to act as a USB modem**.*

*Fig. 2: Hotspot is the act of creating a Wi-Fi network where the phone acts as a modem/router**.*

__Approaches to Tethering__

** Mobile hotspot** is the most pervasive approach to tethering. It is easy to setup, and the presence of Wi-Fi modules on most devices makes it require no extra components.

** Tethering via Bluetooth** is comparatively difficult to setup and also the speed is less than Wi-Fi. Currently, Bluetooth tethering is not often though it was common before Wi-Fi became widely available.

** Tethering over USB** is very fast and power consumption is minimal as the device can be charged over USB. However, not many devices support this USB tethering capability. Also, it will need special drivers or software on both sides and probably some configuration stuff.

__Connectivity Protocols and Requirements__

Tethering usually uses ** NAT (Network Address Translation)** to share internet. In this case, only the device that is connected to the internet (one which its internet connection is shared) has a public IP. Other devices connected by tethering have private IPs and the technique called NAT is used to identify different devices from the point of view of the single public IP.

Mobile hotspots offered by various telecom providers consist of an ** adapter** or device that will allow computer users to hook up to the internet from wherever they happen to be. Mobile hotspots are promoted as an alternative to the conventional method of logging onto a local area network or other wireless network from a PC. Although mobile hotspots could be used for other kinds of devices, they are most commonly associated with laptop computers, because laptop computers are a type of “hybrid” device that may roam, but doesn’t usually come with built-in mobile Wi-Fi.

Apart from hardware, nowadays, software as well can create hotspots. Software such as Connectify Virtual Router® and also built-in tools in operating systems let you share internet by turning the Wi-Fi module on your laptop or mobile phone into a virtual hotspot.

__Provider Models for Tethering and Hotspot__

Another fundamental difference between tethering and hotspot is in ** provider models**. Most telecom operators offering mobile hotspots sell a box or adapter for a fixed price, and offer the mobile hotspot service on a monthly basis. With tethering, the offer could involve simple cable connectors to hook up an existing mobile wireless device to a laptop, without any monthly charge. However, mobile hotspots seem to be a popular option because of convenience.

__Cost Considerations__

When you have the option of using either one of these services, you may want to consider the ** potential costs** involved. If you use tethering to access the internet, you may have to pay for each kilobyte of data that is transferred over the cellular network. If you use the internet frequently, this could amount to a large monthly bill on your cell phone. By comparison, with a traditional hotspot, the internet can be used as much as you want without having to worry about the amount of data that you are accessing. The owner of the hotspot will pay a monthly service fee to the internet service provider.

Typically, a mobile hotspot doesn’t come with contracts and fees. The best ones are pay-as-you-use, so you only pay for the data you use, and refill when necessary. The carriers with the best coverage and speed (Verizon Wireless, for example) are usually the most expensive, and may demand a contract. The ones with the best prices and value (Karma, FreedomPop etc.) sometimes suffer less-than-stellar coverage and speed.

__Connection Availability__

Wi-Fi hotspots are found in public places as well as private places. Today many public places in the world such as airports, stores, restaurants, hotels, hospitals, libraries, public payphones, train stations, schools and universities have hotspots. Many provide free access to the internet while there are commercial ones as well. Hotspots can be setup at home as well by simply connecting a wireless router to the internet via ADSL or 3G. This is the most pervasive method used these days to share the internet connection at home across various devices.

__Advantages of Mobile Hotspots over Tethering__

As a technology offering, mobile hotspots have multiple advantages over tethering.

When involved in work that consumes a lot of data bandwidth, you may land in a situation wherein you exceed the data transfer limits. Using a hotspot is one of the first options in this scenario.*Data bandwidth:*Using a hotspot means never draining your battery just because you needed to get some work done. This comes with the benefit of better long-term usage, since you’re not taxing your phone just to stay connected.*Phone’s battery life:*You can reliably tether multiple devices. While you can tether multiple devices to your phone, the more you add, the worse the experience usually is. While most hotspots will put a limit on the number of devices you can connect, you can always connect more than one or two without performance problems.*Using multiple devices:*Tethering is prone to frequent call drops especially when used over long. This is usually attributed to the firmware used. Likewise, even “unlimited” data plans—depending on carrier—get throttled after a certain point. Hotspots have greater reliability with the added advantage that you pay-as-you-use.*Work continuity:*This varies by carrier, but with Verizon Wireless and Sprint, tethering (via 3G, not via LTE) conflicts with talking on the phone. Although the phone may ring, data would be disconnected the moment you answer it and vice versa.*Choosing between data and voice:*

** Diversifying carriers**: Frequent travellers have the option of whichever carrier offers the better service where you roam. Even if you’re at home, you now have the option to pick the carrier with the best performance, or switch off when you need to.

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Extending the classifications of numbers further, we encounter **rational** and **irrational** numbers. A rational number is a number that can be written as a fraction. In other words, the rational number can be written as a ratio of two numbers.

Consider, for example, the number ** 6**. It can be written as the ratio of two numbers viz.

We can, thus, define a rational number, as a number written in the form of a fraction, wherein both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers. By definition, therefore, every whole number is also a rational number.

A ratio of two large numbers such as (*129,367,871**)**/**(**547,724,863*** )** would also constitute an example of a rational number for the simple reason that both the numerator and the denominator are whole numbers.

Conversely, any number that cannot be expressed in the form of a fraction or a ratio is termed as irrational. The most commonly cited example of an irrational number is *√**2** (**1.414213*** …)**. Another popular example of an irrational number is the numerical constant

An irrational number can be written as a decimal, but not as a fraction. Irrational numbers are not often used in daily life although they do exist on the number line. There are an infinite number of irrational numbers between ** 0** and

Note that the oft-cited value of ** 22/7** for the constant

Only the square roots of the square numbers; i.e., the square roots of the *perfect squares* are rational.

*√1*** = 1** (Rational)

** √2** (Irrational)

** √3** (Irrational)

*√4*** = 2** (Rational)

** √5, √6, √7, √8** (Irrational)

*√9*** = 3** (Rational) and so on.

Further, we note that, only the *n*th roots of *n*th powers are rational. Thus, the ** 6th** root of

Inevitably, the decimal representation of irrationals comes into picture and poses some interesting results.

When we express a *rational* number as a decimal, then either the decimal will be ** exact** (as in

Moreover, there will not be a predictable pattern of digits. For example,

*√2 ≈**1.4142135623730950488016887242097*

Now, with rational numbers, we occasionally encounter ** 1/11 = 0.0909090**.

The use of both the equal sign (** =**) and three dots (

Thus, the decimal form of ** 1/11** is deemed inexact. By the same token, the decimal form of

Coming to the decimal form for irrational numbers, they are going to be always inexact. Continuing with the example of *√*** 2**, when we write

Any note on rational and irrational numbers cannot end without the obligatory proof as to why √2 is irrational. In doing so, we also elucidate, the classic example of a *proof by cont**radiction.*

Suppose √2 is rational. This leads us to represent it as a ratio of two integers, say ** p** and

*√2 = p/q*

Needless to say, ** p** and

Squaring both sides of the equation, we end up with,

*2 = p**2**/ q**2*

This can be conveniently written as,

*p**2 **= 2q**2*

The last equation suggests that *p*** 2** is even. This is possible only if

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__Definite Integral__

The definite integral of ** f(x)** is a NUMBER and represents the area under the curve

A definite integral has upper and lower limits on the integrals, and it’s called definite because, at the end of the problem, we have a number – it is a definite answer.

__Indefinite Integral__

The indefinite integral of f(x) is a FUNCTION and answers the question, “What function when differentiated gives ** f(x)**?”

With an indefinite integral there are no upper and lower limits on the integral here, and what we’ll get is an answer that still has ** x**‘s in it and will also have a constant (usually denoted by

Indefinite integral usually gives a general solution to the differential equation.

Indefinite integral is more of a general form of integration, and it can be interpreted as the anti-derivative of the considered function.

Suppose differentiation of function ** F** leads to another function

*F(x)=∫ƒ(x)dx*

or

*F=∫ƒ dx*

where both ** F** and

An indefinite integral often produces a family of functions; therefore, the integral is indefinite.

Integrals and integration process are at the heart of solving differential equations. However, unlike the steps in differentiation, steps in integration do not always follow a clear and standard routine. Occasionally, we see that the solution cannot be expressed explicitly in terms of elementary function. In that case, the analytic solution is often given in the form of an indefinite integral.

__Fundamental Theorem of Calculus__

The definite and the indefinite integral are linked by the Fundamental Theorem of Calculus as follows: In order to compute a ** definite integral**, find the

The difference between definite and indefinite integrals will be evident once we evaluate the integrals for the same function.

Consider the following integral:

OK. Let’s do both of them and see the difference.

For integration, we need to add one to the index which leads us to the following expression:

At this point of time ** C** is merely a constant to us. Additional information is needed in the problem to determine the precise value of

Let us evaluate the same integral in its definite form i.e., with the upper and lower limits included.

Graphically speaking, we are now computing the area under the curve ** f(x) = y^{3}** between

The first step in this evaluation is the same as the indefinite integral evaluation. The only difference is that this time around we do not add the constant ** C**.

The expression in this case looks as follows:

This is turn leads to:

Essentially, we substituted 3 and then 2 in the expression and obtained the difference between them.

This is the definite value as opposed to the usage of constant ** C** earlier.

Let’s explore the constant factor (with regard to indefinite integral) in some more detail.

If the differential of ** y^{3}** is

*∫**3y ^{2}dy = y^{3}*

However, ** 3y^{2}** could be the differential of many expressions some of which include

So in general, ** 3y^{2}** is the differential of

We write this as:

*∫** 3y ^{2}.dx = y^{3} + C*

Integration techniques for an indefinite integral, such as table lookup or Risch integration, can add new discontinuities during the integration process. These new discontinuities appear because the anti-derivatives can require the introduction of complex logarithms.

Complex logarithms have a jump discontinuity when the argument crosses the negative real axis, and the integration algorithms sometimes cannot find a representation where these jumps cancel.

If the definite integral is evaluated by first computing an indefinite integral and then substituting the integration boundaries into the result, we must be aware that indefinite integration might produce discontinuities. If it does, additionally, we must investigate the discontinuities in the integration interval.

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