# 1.1: Introduction to concept of a limit

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Learning Objectives

• Using correct notation, describe the limit of a function.
• Use a table of values to estimate the limit of a function or to identify when the limit does not exist.
• Use a graph to estimate the limit of a function or to identify when the limit does not exist.
• Define one-sided limits and provide examples.
• Explain the relationship between one-sided and two-sided limits.
• Using correct notation, describe an infinite limit.
• Define a vertical asymptote.

## Limits

Two key problems led to the initial formulation of calculus:

(1) the tangent problem, or how to determine the slope of a line tangent to a curve at a point;

and (2) the area problem, or how to determine the area under a curve.

The concept of a limit or limiting process, essential to the understanding of calculus, has been around for thousands of years. In fact, early mathematicians used a limiting process to obtain better and better approximations of areas of circles. Yet, the formal definition of a limit—as we know and understand it today—did not appear until the late 19th century. We, therefore, begin our quest to understand limits, as our mathematical ancestors did, by using an intuitive approach.

We begin our exploration of limits by taking a look at the graphs of the functions

• (f(x)=dfrac{x^2−4}{x−2}),
• (g(x)=dfrac{|x−2|}{x−2}), and
• (h(x)=dfrac{1}{(x−2)^2}),

which are shown in Figure (PageIndex{1}). In particular, let’s focus our attention on the behavior of each graph at and around (x=2).

Each of the three functions is undefined at (x=2), but if we make this statement and no other, we give a very incomplete picture of how each function behaves in the vicinity of (x=2). To express the behavior of each graph in the vicinity of (2) more completely, we need to introduce the concept of a limit.

## Intuitive Definition of a Limit

Let’s first take a closer look at how the function (f(x)=(x^2−4)/(x−2)) behaves around (x=2) in Figure (PageIndex{1}). As the values of (x) approach (2) from either side of (2), the values of (y=f(x)) approach (4). Mathematically, we say that the limit of (f(x)) as (x) approaches (2) is (4). Symbolically, we express this limit as

(displaystyle lim_{x o 2} f(x)=4).

From this very brief informal look at one limit, let’s start to develop an intuitive definition of the limit. We can think of the limit of a function at a number a as being the one real number (L) that the functional values approach as the (x)-values approach a, provided such a real number (L) exists. Stated more carefully, we have the following definition:

Definition(Intuitive): Limit

Let (f(x)) be a function defined at all values in an open interval containing (a), with the possible exception of a itself, and let (L) be a real number. If all values of the function (f(x)) approach the real number (L) as the values of (x(≠a)) approach the number a, then we say that the limit of (f(x)) as (x) approaches (a) is (L). (More succinct, as (x) gets closer to (a), (f(x)) gets closer and stays close to (L).) Symbolically, we express this idea as

[lim_{x o a} f(x)=L.]

We can estimate limits by constructing tables of functional values and by looking at their graphs. This process is described in the following Problem-Solving Strategy.

Problem-Solving Strategy: Evaluating a Limit Using a Table of Functional Values

1. To evaluate (displaystyle lim_{x o a} f(x)), we begin by completing a table of functional values. We should choose two sets of (x)-values—one set of values approaching (a) and less than (a), and another set of values approaching (a) and greater than (a). Table (PageIndex{1}) demonstrates what your tables might look like.

Table (PageIndex{1})
(x)(f(x))(x)(f(x))
(a-0.1)(f(a-0.1))(a+0.1)(f(a+0.1))
(a-0.01)(f(a-0.01))(a+0.001)(f(a+0.001))
(a-0.001)(f(a-0.001))(a+0.0001)(f(a+0.001))
(a-0.0001)(f(a-0.0001))(a+0.00001)(f(a+0.0001))

2. Next, let’s look at the values in each of the (f(x)) columns and determine whether the values seem to be approaching a single value as we move down each column. In our columns, we look at the sequence (f(a−0.1)), (f(a−0.01)), (f(a−0.001)), (f(a−0.0001)), and so on, and (f(a+0.1), ;f(a+0.01), ;f(a+0.001), ;f(a+0.0001)), and so on. (Note: Although we have chosen the (x)-values (a±0.1, ;a±0.01, ;a±0.001, ;a±0.0001), and so forth, and these values will probably work nearly every time, on very rare occasions we may need to modify our choices.)

3. If both columns approach a common (y)-value (L), we state (displaystyle lim_{x o a}f(x)=L). We can use the following strategy to confirm the result obtained from the table or as an alternative method for estimating a limit.

4. Using a graphing calculator or computer software that allows us graph functions, we can plot the function (f(x)), making sure the functional values of (f(x)) for (x)-values near a are in our window. We can use the trace feature to move along the graph of the function and watch the (y)-value readout as the (x)-values approach a. If the (y)-values approach (L) as our (x)-values approach (a) from both directions, then (displaystyle lim_{x o a}f(x)=L). We may need to zoom in on our graph and repeat this process several times.

We apply this Problem-Solving Strategy to compute a limit in Examples (PageIndex{1A}) and (PageIndex{1B}).

Example (PageIndex{1A}): Evaluating a Limit Using a Table of Functional Values

Evaluate (displaystyle lim_{x o 0}frac{sin x}{x}) using a table of functional values.

Solution

We have calculated the values of (f(x)=dfrac{sin x}{x}) for the values of (x) listed in Table (PageIndex{2}).

Table (PageIndex{2})
(x)(frac{sin x}{x})(x)(frac{sin x}{x})
-0.10.9983341664680.10.998334166468
-0.010.9999833334170.010.999983333417
-0.0010.9999998333330.0010.999999833333
-0.00010.9999999983330.00010.999999998333

Note: The values in this table were obtained using a calculator and using all the places given in the calculator output.

As we read down each (dfrac{sin x}{x}) column, we see that the values in each column appear to be approaching one. Thus, it is fairly reasonable to conclude that (displaystyle lim_{x o0}frac{sin x}{x}=1). A calculator-or computer-generated graph of (f(x)=dfrac{sin x}{x}) would be similar to that shown in Figure (PageIndex{2}), and it confirms our estimate.

Example (PageIndex{1B}): Evaluating a Limit Using a Table of Functional Values

Evaluate (displaystyle lim_{x o4}frac{sqrt{x}−2}{x−4}) using a table of functional values.

Solution

As before, we use a table—in this case, Table (PageIndex{3})—to list the values of the function for the given values of (x).

Table (PageIndex{3})
(x)(frac{sqrt{x}−2}{x−4})(x)(frac{sqrt{x}−2}{x−4})
3.90.2515823418694.10.248456731317
3.990.250156445624.010.24984394501
3.9990.2500156274.0010.249984377
3.99990.2500015634.00010.249998438
3.999990.250000164.000010.24999984

After inspecting this table, we see that the functional values less than 4 appear to be decreasing toward 0.25 whereas the functional values greater than 4 appear to be increasing toward 0.25. We conclude that (displaystyle lim_{x o4}frac{sqrt{x}−2}{x−4}=0.25). We confirm this estimate using the graph of (f(x)=dfrac{sqrt{x}−2}{x−4}) shown in Figure (PageIndex{3}).

Exercise (PageIndex{1})

Estimate (displaystyle lim_{x o 1} frac{frac{1}{x}−1}{x−1}) using a table of functional values. Use a graph to confirm your estimate.

Hint

Use 0.9, 0.99, 0.999, 0.9999, 0.99999 and 1.1, 1.01, 1.001, 1.0001, 1.00001 as your table values.

[lim_{x o1}frac{frac{1}{x}−1}{x−1}=−1 onumber]

At this point, we see from Examples (PageIndex{1A}) and (PageIndex{1b}) that it may be just as easy, if not easier, to estimate a limit of a function by inspecting its graph as it is to estimate the limit by using a table of functional values. In Example (PageIndex{2}), we evaluate a limit exclusively by looking at a graph rather than by using a table of functional values.

Example (PageIndex{2}): Evaluating a Limit Using a Graph

For (g(x)) shown in Figure (PageIndex{4}), evaluate (displaystyle lim_{x o−1}g(x)).

Solution:

Despite the fact that (g(−1)=4), as the (x)-values approach (−1) from either side, the (g(x)) values approach (3). Therefore, (displaystyle lim_{x o−1}g(x)=3). Note that we can determine this limit without even knowing the algebraic expression of the function.

Based on Example (PageIndex{2}), we make the following observation: It is possible for the limit of a function to exist at a point, and for the function to be defined at this point, but the limit of the function and the value of the function at the point may be different.

Exercise (PageIndex{2})

Use the graph of (h(x)) in Figure (PageIndex{5}) to evaluate (displaystyle lim_{x o 2}h(x)), if possible.

Hint

What (y)-value does the function approach as the (x)-values approach (2)?

Solution

(displaystyle lim_{x o 2}h(x)=−1.)

Looking at a table of functional values or looking at the graph of a function provides us with useful insight into the value of the limit of a function at a given point. However, these techniques rely too much on guesswork. We eventually need to develop alternative methods of evaluating limits. These new methods are more algebraic in nature and we explore them in the next section; however, at this point we introduce two special limits that are foundational to the techniques to come.

Two Important Limits

Let (a) be a real number and (c) be a constant.

1. (displaystyle lim_{x o a}x=a)
2. (displaystyle lim_{x o a}c=c)

We can make the following observations about these two limits.

1. For the first limit, observe that as (x) approaches (a), so does (f(x)), because (f(x)=x). Consequently, (displaystyle lim_{x o a}x=a).
2. For the second limit, consider Table (PageIndex{4}).
Table (PageIndex{4})
(x)(f(x)=c)(x)(f(x)=c)
(a-0.1)(c)(a+0.1)(c)
(a-0.01)(c)(a+0.01)(c)
(a-0.001)(c)(a+0.001)(c)
(a-0.0001)(c)(a+0.0001)(c)

Observe that for all values of (x) (regardless of whether they are approaching (a)), the values (f(x)) remain constant at (c). We have no choice but to conclude (displaystyle lim_{x o a}c=c).

## The Existence of a Limit

As we consider the limit in the next example, keep in mind that for the limit of a function to exist at a point, the functional values must approach a single real-number value at that point. If the functional values do not approach a single value, then the limit does not exist.

Example (PageIndex{3}): Evaluating a Limit That Fails to Exist

Evaluate (displaystylelim_{x o 0}sin(1/x)) using a table of values.

Solution

Table (PageIndex{5}) lists values for the function (sin(1/x)) for the given values of (x).

Table (PageIndex{5})
(x)(sin(1/x))(x)(sin(1/x))
-0.10.5440211108890.1−0.544021110889
-0.010.506365641110.01−0.50636564111
-0.001−0.82687954053120.0010.8268795405312
-0.00010.3056143888880.0001−0.305614388888
-0.00001−0.0357487979870.000010.035748797987
-0.0000010.3499935041870.000001−0.349993504187

After examining the table of functional values, we can see that the (y)-values do not seem to approach any one single value. It appears the limit does not exist. Before drawing this conclusion, let’s take a more systematic approach. Take the following sequence of (x)-values approaching (0):

[frac{2}{π},;frac{2}{3π},;frac{2}{5π},;frac{2}{7π},;frac{2}{9π},;frac{2}{11π},;…. onumber]

The corresponding (y)-values are

[1,;-1,;1,;-1,;1,;-1,;.... onumber]

At this point we can indeed conclude that (displaystyle lim_{x o 0} sin(1/x)) does not exist. (Mathematicians frequently abbreviate “does not exist” as DNE. Thus, we would write (displaystyle lim_{x o 0} sin(1/x)) DNE.) The graph of (f(x)=sin(1/x)) is shown in Figure (PageIndex{6}) and it gives a clearer picture of the behavior of (sin(1/x)) as (x) approaches (0). You can see that (sin(1/x)) oscillates ever more wildly between (−1) and (1) as (x) approaches (0).

Exercise (PageIndex{3})

Use a table of functional values to evaluate (displaystyle lim_{x o 2}frac{∣x^2−4∣}{x−2}), if possible.

Hint

Use (x)-values 1.9, 1.99, 1.999, 1.9999, 1.9999 and 2.1, 2.01, 2.001, 2.0001, 2.00001 in your table.

(displaystyle lim_{x o 2}frac{∣x^2−4∣}{x−2}) does not exist.

## Limit (mathematics)

In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. [1] Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.

The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory.

In formulas, a limit of a function is usually written as

and is read as "the limit of f of x as x approaches c equals L ". The fact that a function f approaches the limit L as x approaches c is sometimes denoted by a right arrow (→ or → ), as in

Now, what is a mathematical way of saying "close" . could we subtract one value from the other?

Example 1: 4.01 − 4 = 0.01 (that looks good)
Example 2: 3.8 − 4 = −0.2 (negatively close?)

So how do we deal with the negatives? We don't care about positive or negative, we just want to know how far . which is the absolute value.

Example 1: |4.01−4| = 0.01
Example 2: |3.8−4| = 0.2

And when |a−b| is small we know we are close, so we write:

"|f(x)−L| is small when |x−a| is small"

And this animation shows what happens with the function

f(x) = (x 2 −1) (x−1)

f(x) approaches L=2 as x approaches a=1,
so |f(x)−2| is small when |x−1| is small.

## 1.1: Introduction to concept of a limit

The topic that we will be examining in this chapter is that of Limits. This is the first of three major topics that we will be covering in this course. While we will be spending the least amount of time on limits in comparison to the other two topics limits are very important in the study of Calculus. We will be seeing limits in a variety of places once we move out of this chapter. In particular we will see that limits are part of the formal definition of the other two major topics.

Here is a list of topics that are in this chapter.

Tangent Lines and Rates of Change –In this section we will introduce two problems that we will see time and again in this course : Rate of Change of a function and Tangent Lines to functions. Both of these problems will be used to introduce the concept of limits, although we won't formally give the definition or notation until the next section.

The Limit – In this section we will introduce the notation of the limit. We will also take a conceptual look at limits and try to get a grasp on just what they are and what they can tell us. We will be estimating the value of limits in this section to help us understand what they tell us. We will actually start computing limits in a couple of sections.

One-Sided Limits – In this section we will introduce the concept of one-sided limits. We will discuss the differences between one-sided limits and limits as well as how they are related to each other.

Limit Properties – In this section we will discuss the properties of limits that we’ll need to use in computing limits (as opposed to estimating them as we've done to this point). We will also compute a couple of basic limits in this section.

Computing Limits – In this section we will looks at several types of limits that require some work before we can use the limit properties to compute them. We will also look at computing limits of piecewise functions and use of the Squeeze Theorem to compute some limits.

Infinite Limits – In this section we will look at limits that have a value of infinity or negative infinity. We’ll also take a brief look at vertical asymptotes.

Limits At Infinity, Part I – In this section we will start looking at limits at infinity, i.e. limits in which the variable gets very large in either the positive or negative sense. We will concentrate on polynomials and rational expressions in this section. We’ll also take a brief look at horizontal asymptotes.

Limits At Infinity, Part II – In this section we will continue covering limits at infinity. We’ll be looking at exponentials, logarithms and inverse tangents in this section.

Continuity – In this section we will introduce the concept of continuity and how it relates to limits. We will also see the Intermediate Value Theorem in this section and how it can be used to determine if functions have solutions in a given interval.

The Definition of the Limit – In this section we will give a precise definition of several of the limits covered in this section. We will work several basic examples illustrating how to use this precise definition to compute a limit. We’ll also give a precise definition of continuity.

## How Limit Calculator determine Limits?

For any chosen degree of nearness &epsilon, multivariable limit calculator determine an interval nearby x0(or previously assumed b). Because, the given values of f(x) may varies from L by a quantity less than &epsilon (i.e., if &epsilon= |x &minus x 0 | < &delta, then |f (x) &minus L| < &epsilon).

Limit calculator step by step determine whether a given number is a limit or not. The estimation of limit quotients, involves adjustments of the function in order to write it in an obvious form. After determining & evaluating, limit solver uses limit formula to calculate limit of a function online.

You can also try our other math related calculators like cross product calculator or area of a sector calculator in order to learn and practice online.

## Introduction to Mechanisms

This chapter introduces the basic physical principles behind mechanisms as well as basic concepts and principles required for this course.

Every day we deal with forces of one kind or another. A pressure is a force. The earth exerts a force of attraction for all bodies or objects on its surface. To study the forces acting on objects, we must know how the forces are applied, the direction of the forces and their value. Graphically, forces are often represented by a vector whose end represents the point of action.

A mechanism is what is responsible for any action or reaction. Machines are based on the idea of transmitting forces through a series of predetermined motions. These related concepts are the basis of dynamic movement.

Torque: Something that produces or tends to produce rotation and whose effectiveness is measured by the product of the force and the perpendicular distance from the line of action of the force to the axis of rotation.

Consider the lever shown in Figure 1-1. The lever is a bar that is free to turn about the fixed point, A, called the fulcrum a weight acts on the one side of the lever, and a balancing force acts on the other side of the lever.

#### Figure 1-1 A lever with balanced forces

To analyze levers, we need to find the torques of the forces acting on the lever. To get the torque of force W about point A, multiply W by l 1 , its distance from A. Similarly F x l 2 is the torque of F about fulcrum A.

Motion : a change of position or orientation.

We begin our study of motion with the simplest case, motion in a straight line.

where t = t2 - t1 is the time interval during which the displacement occurred. When velocity varies, we can let the time interval become infinitesimally small, thus

More generally, acceleration is

The picture becomes more complicated when the motion is not merely along a straight line, but rather extends into a plane. Here we can describe the motion with a vector which includes the magnitude and the direction of movement.

Position vector and displacement vector
The directed segment which describes the position of an object relative to an origin is the position vector , as d 1 and d 2 in Figure 1-2

#### Figure 1-2 Position vector and displacement vector

If we wish to describe a motion from position d 1 to position d 2 , for example, we can use vector d 1 , the vector starts at the point described by d 1 and goes to the point described by d 2 , which is called the displacement vector .

Clearly V ave has the direction of d .

In the limit as delta t approaches zero, the instantaneous velocity is

The direction of V is the direction of d for a very small displacement it is therefore along, or tangent to, the path.

The previous sections discuss the motion of particles. For a rigid body in a plane, its motion is often more complex than a particle because it is comprised of a linear motion and a rotary motion. Generally, this kind of motion can be decomposed into two motions (Figure 1-3), they are:

1. The linear motion of the center of the mass of the rigid body. In this part of the motion, the motion is the same as the motion of a particle on a plane.
2. The rotary motion of the rigid body relative to its center of mass .

#### Figure 1-3 Motion of a rigid body in a plane

When no force is exerted on a body, it stays at rest or moves in a straight line with constant speed. This principle of inertia is also known as Newton's first law. It is from this law that Newton was able to build up our present understanding of dynamics.

1. When a force F is applied on an object, V , the change of the velocity of the object, increases with the length of time delta t increases
2. The greater the force F , the greater V and
3. The larger the body (object) is, the less easily accelerated by forces.

It is convenient to write the proportionality between F t and V in the form:

The proportionality constant m varies with the object. This constant m is refered to as the inertial mass of the body. The relationship above embodies Newton's law of motion ( Newton's second law ). As

in which a is the acceleration of the object. We have

If m = 1 kg and a = 1m/sec 2 , than F = 1 newton .

Forces and accelerations are vectors, and Newton's law can be written in vector form.

Try to make a baseball and a cannon ball roll at the same speed. As you can guess, it is harder to get the cannon ball going. If you apply a constant force F for a time t, the change in velocity is given by Equation 1-9. So, to get the same v , the product F t must be greater the greater the mass m you are trying to accelerate.

To throw a cannon ball from rest and give it the same final velocity as a baseball (also starting from rest), we must push either harder or longer. What counts is the product F t. This product F t is the natural measure of how hard and how long we push to change a motion. It is called the impulse of the force.

Suppose we apply the same impulse to a baseball and a cannon ball, both initially at rest. Since the initial value of the quantity m v is zero in each case, and since equal impulses are applied, the final values m v will be equal for the baseball and the cannon ball. Yet, because the mass of the cannon ball is much greater than the mass of the baseball, the velocity of the cannon ball will be much less than the velocity of the baseball. The product m v , then, is quite a different measure of the motion than simply v alone. We call it the momentum p of the body, and measure it in kilogram-meters per second.

Velocity and momentum are quite different concepts: velocity is a kinematical quantity, whereas momentum is a dynamic one, connected with the causes of changes in the motion of masses.

Because of its connection with the impulse which occurs naturally in Newton's law (Equation 1-9), we expect momentum to fit naturally into Newtonian dynamics. Newton did express his law of motion in terms of the momentum , which he called the quantity of motion . We can express Newton's law in terms of the change in momentum instead of change in velocity :

where v and v ' are the velocities before and after the impulse. The right-hand side of the last equation can be written as

the change in the momentum . Therefore

or, in other words, the impulse equals the change in the momentum .

In Figure 1-4 a moving billiard ball collides with a billiard ball at rest. The incident ball stops and the ball it hits goes off with the same velocity with which the incident ball came in. The two billiard balls have the same mass. Therefore, the momentum of the second ball after the collision is the same as that of the incident ball before collision. The incident ball has lost all its momentum , and the ball it struck has gained exactly the momentum which the incident ball lost.

#### Figure 1-4 Collision of billiard balls

This phenomenon is consistent with the law of conservation of momentum which says that the total momentum is constant when two bodies interact.

Work is a force applied over a distance. If you drag an object along the floor you do work in overcoming the friction between the object and the floor. In lifting an object you do wor k against gravity which tends to pull the object toward the earth. Steam in a locomotive cylinder does work when it expands and moves the piston against the resisting forces. Work is the product of the resistance overcome and the distance through which it is overcome.

Power is the rate at which work is done.

In the British system, power is expressed in foot-pounds per second. For larger measurements, the horsepower is used.

1horsepower = 550ft *lb/s = 33,000ft*lb/min

In SI units, power is measured in joules per second, also called the watt (W).

All object possess energy. This can come from having work done on it at some point in time. Generally, there are two kinds of energy in mechanical systems, potential and kinetic . Potential energy is due to the position of the object and kinetic energy is due to its movement.

For example, an object set in motion can overcome a certain amount of resistance before being brought to rest, and the energy which the object has on account of its motion is used up in overcoming the resistance, bring the object to rest. Fly wheels on engines both receive and give up energy and thus cause the energy to return more smoothly throughout the stroke .

Elevated weights have power to do work on account of their elevated position, as in various types of hammers, etc.

## Graphical Approach to Limits

Example 3:
The graph below shows that as x approaches 1 from the left, y = f(x) approaches 2 and this can be written as
limx𔾵 - f(x) = 2
As x approaches 1 from the right, y = f(x) approaches 4 and this can be written as
limx𔾵 + f(x) = 4
Note that the left and right hand limits and f(1) = 3 are all different.

Example 4:
This graph shows that
lim x𔾵 - f(x) = 2
As x approaches 1 from the right, y = f(x) approaches 4 and this can be written as
limx𔾵 + f(x) = 4
Note that the left hand limit and f(1) = 2 are equal.

Example 5:
This graph shows that
limx𔾴 - f(x) = 1
and
limx𔾴 + f(x) = 1
Note that the left and right hand limits are equal and we cvan write
limx𔾴 f(x) = 1
In this example, the limit when x approaches 0 is equal to f(0) = 1.

Example 6:
This graph shows that as x approaches - 2 from the left, f(x) gets smaller and smaller without bound and there is no limit. We write
limx→-2 - f(x) = - ∞
As x approaches - 2 from the right, f(x) gets larger and larger without bound and there is no limit. We write
limx→-2 + f(x) = + ∞
Note that - ∞ and + ∞ are symbols and not numbers. These are symbols used to indicate that the limit does not exist.

Example 7:
The graph below shows a periodic function whose range is given by the interval [-1 , 1]. If x is allowed to increase without bound, f(x) take values within [-1 , 1] and has no limit. This can be written
limx→ + ∞ f(x) = does not exist
If x is allowed to decrease without bound, f(x) take values within [-1 , 1] and has no limit again. This can be written
limx→ - ∞ f(x) = does not exist

Example 8:
If x is allowed to increase without bound, f(x) in the graph below approaches 2. This can be written
lim x→ + ∞ f(x) = 2
If x is allowed to decrease without bound, f(x) approaches 2. This can be written
limx→ - ∞ f(x) = 2

## Contents

The technology of code-division multiple access channels has long been known. In the Soviet Union (USSR), the first work devoted to this subject was published in 1935 by Dmitry Ageev. [3] It was shown that through the use of linear methods, there are three types of signal separation: frequency, time and compensatory. [ clarification needed ] The technology of CDMA was used in 1957, when the young military radio engineer Leonid Kupriyanovich in Moscow made an experimental model of a wearable automatic mobile phone, called LK-1 by him, with a base station. [4] LK-1 has a weight of 3 kg, 20–30 km operating distance, and 20–30 hours of battery life. [5] [6] The base station, as described by the author, could serve several customers. In 1958, Kupriyanovich made the new experimental "pocket" model of mobile phone. This phone weighed 0.5 kg. To serve more customers, Kupriyanovich proposed the device, which he called "correlator." [7] [8] In 1958, the USSR also started the development of the "Altai" national civil mobile phone service for cars, based on the Soviet MRT-1327 standard. The phone system weighed 11 kg (24 lb). It was placed in the trunk of the vehicles of high-ranking officials and used a standard handset in the passenger compartment. The main developers of the Altai system were VNIIS (Voronezh Science Research Institute of Communications) and GSPI (State Specialized Project Institute). In 1963 this service started in Moscow, and in 1970 Altai service was used in 30 USSR cities. [9]

• Synchronous CDM (code-division 'multiplexing', an early generation of CDMA) was implemented in the Global Positioning System (GPS). This predates and is distinct from its use in mobile phones.
• The Qualcomm standard IS-95, marketed as cdmaOne.
• The Qualcomm standard IS-2000, known as CDMA2000, is used by several mobile phone companies, including the Globalstar network. [nb 1]
• The UMTS 3G mobile phone standard, which uses W-CDMA. [nb 2]
• CDMA has been used in the OmniTRACS satellite system for transportation logistics.

Each user in a CDMA system uses a different code to modulate their signal. Choosing the codes used to modulate the signal is very important in the performance of CDMA systems. The best performance occurs when there is good separation between the signal of a desired user and the signals of other users. The separation of the signals is made by correlating the received signal with the locally generated code of the desired user. If the signal matches the desired user's code, then the correlation function will be high and the system can extract that signal. If the desired user's code has nothing in common with the signal, the correlation should be as close to zero as possible (thus eliminating the signal) this is referred to as cross-correlation. If the code is correlated with the signal at any time offset other than zero, the correlation should be as close to zero as possible. This is referred to as auto-correlation and is used to reject multi-path interference. [14] [15]

An analogy to the problem of multiple access is a room (channel) in which people wish to talk to each other simultaneously. To avoid confusion, people could take turns speaking (time division), speak at different pitches (frequency division), or speak in different languages (code division). CDMA is analogous to the last example where people speaking the same language can understand each other, but other languages are perceived as noise and rejected. Similarly, in radio CDMA, each group of users is given a shared code. Many codes occupy the same channel, but only users associated with a particular code can communicate.

In general, CDMA belongs to two basic categories: synchronous (orthogonal codes) and asynchronous (pseudorandom codes).

The digital modulation method is analogous to those used in simple radio transceivers. In the analog case, a low-frequency data signal is time-multiplied with a high-frequency pure sine-wave carrier and transmitted. This is effectively a frequency convolution (Wiener–Khinchin theorem) of the two signals, resulting in a carrier with narrow sidebands. In the digital case, the sinusoidal carrier is replaced by Walsh functions. These are binary square waves that form a complete orthonormal set. The data signal is also binary and the time multiplication is achieved with a simple XOR function. This is usually a Gilbert cell mixer in the circuitry.

Synchronous CDMA exploits mathematical properties of orthogonality between vectors representing the data strings. For example, binary string 1011 is represented by the vector (1, 0, 1, 1). Vectors can be multiplied by taking their dot product, by summing the products of their respective components (for example, if u = (a, b) and v = (c, d), then their dot product u·v = ac + bd). If the dot product is zero, the two vectors are said to be orthogonal to each other. Some properties of the dot product aid understanding of how W-CDMA works. If vectors a and b are orthogonal, then a ⋅ b = 0 =0> and:

Each user in synchronous CDMA uses a code orthogonal to the others' codes to modulate their signal. An example of 4 mutually orthogonal digital signals is shown in the figure below. Orthogonal codes have a cross-correlation equal to zero in other words, they do not interfere with each other. In the case of IS-95, 64-bit Walsh codes are used to encode the signal to separate different users. Since each of the 64 Walsh codes is orthogonal to all other, the signals are channelized into 64 orthogonal signals. The following example demonstrates how each user's signal can be encoded and decoded.

### Example Edit

Start with a set of vectors that are mutually orthogonal. (Although mutual orthogonality is the only condition, these vectors are usually constructed for ease of decoding, for example columns or rows from Walsh matrices.) An example of orthogonal functions is shown in the adjacent picture. These vectors will be assigned to individual users and are called the code, chip code, or chipping code. In the interest of brevity, the rest of this example uses codes v with only two bits.

Each user is associated with a different code, say v. A 1 bit is represented by transmitting a positive code v, and a 0 bit is represented by a negative code −v. For example, if v = (v0, v1) = (1, −1) and the data that the user wishes to transmit is (1, 0, 1, 1), then the transmitted symbols would be

(v, −v, v, v) = (v0, v1, −v0, −v1, v0, v1, v0, v1) = (1, −1, −1, 1, 1, −1, 1, −1).

For the purposes of this article, we call this constructed vector the transmitted vector.

Each sender has a different, unique vector v chosen from that set, but the construction method of the transmitted vector is identical.

Now, due to physical properties of interference, if two signals at a point are in phase, they add to give twice the amplitude of each signal, but if they are out of phase, they subtract and give a signal that is the difference of the amplitudes. Digitally, this behaviour can be modelled by the addition of the transmission vectors, component by component.

If sender0 has code (1, −1) and data (1, 0, 1, 1), and sender1 has code (1, 1) and data (0, 0, 1, 1), and both senders transmit simultaneously, then this table describes the coding steps:

 Step Encode sender0 Encode sender1 0 code0 = (1, −1), data0 = (1, 0, 1, 1) code1 = (1, 1), data1 = (0, 0, 1, 1) 1 encode0 = 2(1, 0, 1, 1) − (1, 1, 1, 1) = (1, −1, 1, 1) encode1 = 2(0, 0, 1, 1) − (1, 1, 1, 1) = (−1, −1, 1, 1) 2 signal0 = encode0 ⊗ code0= (1, −1, 1, 1) ⊗ (1, −1)= (1, −1, −1, 1, 1, −1, 1, −1) signal1 = encode1 ⊗ code1= (−1, −1, 1, 1) ⊗ (1, 1)= (−1, −1, −1, −1, 1, 1, 1, 1)

Because signal0 and signal1 are transmitted at the same time into the air, they add to produce the raw signal

(1, −1, −1, 1, 1, −1, 1, −1) + (−1, −1, −1, −1, 1, 1, 1, 1) = (0, −2, −2, 0, 2, 0, 2, 0).

This raw signal is called an interference pattern. The receiver then extracts an intelligible signal for any known sender by combining the sender's code with the interference pattern. The following table explains how this works and shows that the signals do not interfere with one another:

 Step Decode sender0 Decode sender1 0 code0 = (1, −1), signal = (0, −2, −2, 0, 2, 0, 2, 0) code1 = (1, 1), signal = (0, −2, −2, 0, 2, 0, 2, 0) 1 decode0 = pattern.vector0 decode1 = pattern.vector1 2 decode0 = ((0, −2), (−2, 0), (2, 0), (2, 0)) · (1, −1) decode1 = ((0, −2), (−2, 0), (2, 0), (2, 0)) · (1, 1) 3 decode0 = ((0 + 2), (−2 + 0), (2 + 0), (2 + 0)) decode1 = ((0 − 2), (−2 + 0), (2 + 0), (2 + 0)) 4 data0=(2, −2, 2, 2), meaning (1, 0, 1, 1) data1=(−2, −2, 2, 2), meaning (0, 0, 1, 1)

Further, after decoding, all values greater than 0 are interpreted as 1, while all values less than zero are interpreted as 0. For example, after decoding, data0 is (2, −2, 2, 2), but the receiver interprets this as (1, 0, 1, 1). Values of exactly 0 means that the sender did not transmit any data, as in the following example:

Assume signal0 = (1, −1, −1, 1, 1, −1, 1, −1) is transmitted alone. The following table shows the decode at the receiver:

 Step Decode sender0 Decode sender1 0 code0 = (1, −1), signal = (1, −1, −1, 1, 1, −1, 1, −1) code1 = (1, 1), signal = (1, −1, −1, 1, 1, −1, 1, −1) 1 decode0 = pattern.vector0 decode1 = pattern.vector1 2 decode0 = ((1, −1), (−1, 1), (1, −1), (1, −1)) · (1, −1) decode1 = ((1, −1), (−1, 1), (1, −1), (1, −1)) · (1, 1) 3 decode0 = ((1 + 1), (−1 − 1), (1 + 1), (1 + 1)) decode1 = ((1 − 1), (−1 + 1), (1 − 1), (1 − 1)) 4 data0 = (2, −2, 2, 2), meaning (1, 0, 1, 1) data1 = (0, 0, 0, 0), meaning no data

When the receiver attempts to decode the signal using sender1's code, the data is all zeros, therefore the cross-correlation is equal to zero and it is clear that sender1 did not transmit any data.

When mobile-to-base links cannot be precisely coordinated, particularly due to the mobility of the handsets, a different approach is required. Since it is not mathematically possible to create signature sequences that are both orthogonal for arbitrarily random starting points and which make full use of the code space, unique "pseudo-random" or "pseudo-noise" sequences called spreading sequences are used in asynchronous CDMA systems. A spreading sequence is a binary sequence that appears random but can be reproduced in a deterministic manner by intended receivers. These spreading sequences are used to encode and decode a user's signal in asynchronous CDMA in the same manner as the orthogonal codes in synchronous CDMA (shown in the example above). These spreading sequences are statistically uncorrelated, and the sum of a large number of spreading sequences results in multiple access interference (MAI) that is approximated by a Gaussian noise process (following the central limit theorem in statistics). Gold codes are an example of a spreading sequence suitable for this purpose, as there is low correlation between the codes. If all of the users are received with the same power level, then the variance (e.g., the noise power) of the MAI increases in direct proportion to the number of users. In other words, unlike synchronous CDMA, the signals of other users will appear as noise to the signal of interest and interfere slightly with the desired signal in proportion to number of users.

All forms of CDMA use the spread-spectrum spreading factor to allow receivers to partially discriminate against unwanted signals. Signals encoded with the specified spreading sequences are received, while signals with different sequences (or the same sequences but different timing offsets) appear as wideband noise reduced by the spreading factor.

Since each user generates MAI, controlling the signal strength is an important issue with CDMA transmitters. A CDM (synchronous CDMA), TDMA, or FDMA receiver can in theory completely reject arbitrarily strong signals using different codes, time slots or frequency channels due to the orthogonality of these systems. This is not true for asynchronous CDMA rejection of unwanted signals is only partial. If any or all of the unwanted signals are much stronger than the desired signal, they will overwhelm it. This leads to a general requirement in any asynchronous CDMA system to approximately match the various signal power levels as seen at the receiver. In CDMA cellular, the base station uses a fast closed-loop power-control scheme to tightly control each mobile's transmit power.

In 2019, schemes to precisely estimate the required length of the codes in dependence of Doppler and delay characteristics have been developed. Soon after, machine learning based techniques that generate sequences of a desired length and spreading properties have been published as well. These are highly competitive with the classic Gold and Welch sequences. These are not generated by linear-feedback-shift-registers, but have to be stored in lookup tables.

### Advantages of asynchronous CDMA over other techniques Edit

#### Efficient practical utilization of the fixed frequency spectrum Edit

In theory CDMA, TDMA and FDMA have exactly the same spectral efficiency, but, in practice, each has its own challenges – power control in the case of CDMA, timing in the case of TDMA, and frequency generation/filtering in the case of FDMA.

TDMA systems must carefully synchronize the transmission times of all the users to ensure that they are received in the correct time slot and do not cause interference. Since this cannot be perfectly controlled in a mobile environment, each time slot must have a guard time, which reduces the probability that users will interfere, but decreases the spectral efficiency.

Similarly, FDMA systems must use a guard band between adjacent channels, due to the unpredictable Doppler shift of the signal spectrum because of user mobility. The guard bands will reduce the probability that adjacent channels will interfere, but decrease the utilization of the spectrum.

#### Flexible allocation of resources Edit

Asynchronous CDMA offers a key advantage in the flexible allocation of resources i.e. allocation of spreading sequences to active users. In the case of CDM (synchronous CDMA), TDMA, and FDMA the number of simultaneous orthogonal codes, time slots, and frequency slots respectively are fixed, hence the capacity in terms of the number of simultaneous users is limited. There are a fixed number of orthogonal codes, time slots or frequency bands that can be allocated for CDM, TDMA, and FDMA systems, which remain underutilized due to the bursty nature of telephony and packetized data transmissions. There is no strict limit to the number of users that can be supported in an asynchronous CDMA system, only a practical limit governed by the desired bit error probability since the SIR (signal-to-interference ratio) varies inversely with the number of users. In a bursty traffic environment like mobile telephony, the advantage afforded by asynchronous CDMA is that the performance (bit error rate) is allowed to fluctuate randomly, with an average value determined by the number of users times the percentage of utilization. Suppose there are 2N users that only talk half of the time, then 2N users can be accommodated with the same average bit error probability as N users that talk all of the time. The key difference here is that the bit error probability for N users talking all of the time is constant, whereas it is a random quantity (with the same mean) for 2N users talking half of the time.

In other words, asynchronous CDMA is ideally suited to a mobile network where large numbers of transmitters each generate a relatively small amount of traffic at irregular intervals. CDM (synchronous CDMA), TDMA, and FDMA systems cannot recover the underutilized resources inherent to bursty traffic due to the fixed number of orthogonal codes, time slots or frequency channels that can be assigned to individual transmitters. For instance, if there are N time slots in a TDMA system and 2N users that talk half of the time, then half of the time there will be more than N users needing to use more than N time slots. Furthermore, it would require significant overhead to continually allocate and deallocate the orthogonal-code, time-slot or frequency-channel resources. By comparison, asynchronous CDMA transmitters simply send when they have something to say and go off the air when they do not, keeping the same signature sequence as long as they are connected to the system.

### Spread-spectrum characteristics of CDMA Edit

Most modulation schemes try to minimize the bandwidth of this signal since bandwidth is a limited resource. However, spread-spectrum techniques use a transmission bandwidth that is several orders of magnitude greater than the minimum required signal bandwidth. One of the initial reasons for doing this was military applications including guidance and communication systems. These systems were designed using spread spectrum because of its security and resistance to jamming. Asynchronous CDMA has some level of privacy built in because the signal is spread using a pseudo-random code this code makes the spread-spectrum signals appear random or have noise-like properties. A receiver cannot demodulate this transmission without knowledge of the pseudo-random sequence used to encode the data. CDMA is also resistant to jamming. A jamming signal only has a finite amount of power available to jam the signal. The jammer can either spread its energy over the entire bandwidth of the signal or jam only part of the entire signal. [14] [15]

CDMA can also effectively reject narrow-band interference. Since narrow-band interference affects only a small portion of the spread-spectrum signal, it can easily be removed through notch filtering without much loss of information. Convolution encoding and interleaving can be used to assist in recovering this lost data. CDMA signals are also resistant to multipath fading. Since the spread-spectrum signal occupies a large bandwidth, only a small portion of this will undergo fading due to multipath at any given time. Like the narrow-band interference, this will result in only a small loss of data and can be overcome.

Another reason CDMA is resistant to multipath interference is because the delayed versions of the transmitted pseudo-random codes will have poor correlation with the original pseudo-random code, and will thus appear as another user, which is ignored at the receiver. In other words, as long as the multipath channel induces at least one chip of delay, the multipath signals will arrive at the receiver such that they are shifted in time by at least one chip from the intended signal. The correlation properties of the pseudo-random codes are such that this slight delay causes the multipath to appear uncorrelated with the intended signal, and it is thus ignored.

Some CDMA devices use a rake receiver, which exploits multipath delay components to improve the performance of the system. A rake receiver combines the information from several correlators, each one tuned to a different path delay, producing a stronger version of the signal than a simple receiver with a single correlation tuned to the path delay of the strongest signal. [1] [2]

Frequency reuse is the ability to reuse the same radio channel frequency at other cell sites within a cellular system. In the FDMA and TDMA systems, frequency planning is an important consideration. The frequencies used in different cells must be planned carefully to ensure signals from different cells do not interfere with each other. In a CDMA system, the same frequency can be used in every cell, because channelization is done using the pseudo-random codes. Reusing the same frequency in every cell eliminates the need for frequency planning in a CDMA system however, planning of the different pseudo-random sequences must be done to ensure that the received signal from one cell does not correlate with the signal from a nearby cell. [1]

Since adjacent cells use the same frequencies, CDMA systems have the ability to perform soft hand-offs. Soft hand-offs allow the mobile telephone to communicate simultaneously with two or more cells. The best signal quality is selected until the hand-off is complete. This is different from hard hand-offs utilized in other cellular systems. In a hard-hand-off situation, as the mobile telephone approaches a hand-off, signal strength may vary abruptly. In contrast, CDMA systems use the soft hand-off, which is undetectable and provides a more reliable and higher-quality signal. [2]

## What are Nutrients?

Nutrients are substances required by the body to perform its basic functions. Most nutrients must be obtained from our diet, since the human body does not synthesize or produce them. Nutrients have one or more of three basic functions: they provide energy, contribute to body structure, and/or regulate chemical processes in the body. These basic functions allow us to detect and respond to environmental surroundings, move, excrete wastes, respire (breathe), grow, and reproduce.

There are six classes of nutrients required for the body to function and maintain overall health. These are: carbohydrates, lipids, proteins, water, vitamins, and minerals. Nutritious foods provide nutrients for the body. Foods may also contain a variety of non-nutrients. Some non-nutrients such as as antioxidants (found in many plant foods) are beneficial to the body, whereas others such as natural toxins (common in some plant foods) or additives (like certain dyes and preservatives found in processed foods) are potentially harmful.

### Macronutrients

Nutrients that are needed in large amounts are called macronutrients. There are three classes of macronutrients: carbohydrates, lipids, and proteins. Macronutrients are carbon-based compounds that can be metabolically processed into cellular energy through changes in their chemical bonds. The chemical energy is converted into cellular energy known as ATP, that is utilized by the body to perform work and conduct basic functions.

The amount of energy a person consumes daily comes primarily from the 3 macronutrients. Food energy is measured in kilocalories. For ease of use, food labels state the amount of energy in food in &ldquocalories,&rdquo meaning that each calorie is actually multiplied by one thousand to equal a kilocalorie. (Note: Using scientific terminology, &ldquoCalorie&rdquo (with a capital &ldquoC&rdquo) is equivalent to a kilocalorie. Therefore: 1 kilocalorie = 1 Calorie - 1000 calories

Water is also a macronutrient in the sense that the body needs it in large amounts, but unlike the other macronutrients, it does not contain carbon or yield energy.

Note: Consuming alcohol also contributes energy (calories) to the diet at 7 kilocalories/gram, so it must be counted in daily energy consumption. However, alcohol is not considered a "nutrient" because it does not contribute to essential body functions and actually contains substances that must broken down and excreted from the body to prevent toxic effects.

Figure (PageIndex<2>): The Macronutrients: Carbohydrates, Lipids, Protein, and Water.

#### Carbohydrates

Carbohydrates are molecules composed of carbon, hydrogen, and oxygen that provide energy to the body. The major food sources of carbohydrates are milk, grains, fruits, and starchy vegetables, like potatoes. Non-starchy vegetables also contain carbohydrates, but in lesser quantities. Carbohydrates are broadly classified into two forms based on their chemical structure: simple carbohydrates (often called simple sugars) and complex carbohydrates.

Simple carbohydrates consist of one or two basic sugar units linked together. Their scientific names are "monosaccharides" (1 sugar unit) and disaccharides (2 sugar units). They are broken down and absorbed very quickly in the digestive tract and provide a fast burst of energy to the body. Examples of simple sugars include the disaccharide sucrose, the type of sugar you would have in a bowl on the breakfast table, and the monosaccharide glucose, the most common type of fuel for most organisms including humans. Glucose is the primary sugar that circulates in blood to provide energy to cells. The terms "blood sugar" and "blood glucose" can be substituted for each other.

Complex carbohydrates are long chains of sugars units that can link in a straight chair or a branched chain. During digestion, the body breaks down digestible complex carbohydrates into simple sugars, mostly glucose. Glucose is then absorbed into the bloodstream and transported to all our cells where it is stored, used to make energy, or used to build macromolecules. Fiber is also a complex carbohydrate, but it cannot be broken down by digestive enzymes in the human intestine. As a result, it passes through the digestive tract undigested unless the bacteria that inhabit the colon or large intestine break it down.

One gram of digestible carbohydrates yields 4 kilocalories of energy for the cells in the body to perform work. In addition to providing energy and serving as building blocks for bigger macromolecules, carbohydrates are essential for proper functioning of the nervous system, heart, and kidneys. As mentioned, glucose can be stored in the body for future use. In humans, the storage molecule of carbohydrates is called glycogen, and in plants, it is known as starch. Glycogen and starch are complex carbohydrates.

#### Lipids

Lipids are also a family of molecules composed of carbon, hydrogen, and oxygen, but unlike carbohydrates, they are insoluble in water. Lipids are found predominantly in butter, oils, meats, dairy products, nuts, and seeds, and in many processed foods. The three main types of lipids are triglycerides (triacylglycerols), phospholipids, and sterols. The main job of triacylglycerols is to provide or store energy. Lipids provide more energy per gram than carbohydrates (9 kilocalories per gram of lipids versus 4 kilocalories per gram of carbohydrates). In addition to energy storage, lipids serve as a major component of cell membranes, surround and protect organs (in fat-storing tissues), provide insulation to aid in temperature regulation. Phospholipds and sterols have a somewhat different chemical structure and are used to regulate many other functions in the body.

#### Proteins

Proteins are macromolecules composed of chains of basic subunits called amino acids. Amino acids are composed of carbon, oxygen, hydrogen, and nitrogen. Food sources of proteins include meats, dairy products, seafood, and a variety of different plant-based foods, most notably soy. The word protein comes from a Greek word meaning &ldquoof primary importance,&rdquo which is an apt description of these macronutrients they are also known colloquially as the &ldquoworkhorses&rdquo of life. Proteins provide the basic structure to bones, muscles and skin, enzymes and hormones and play a role in conducting most of the chemical reactions that take place in the body. Scientists estimate that greater than one-hundred thousand different proteins exist within the human body. The genetic codes in DNA are basically protein recipes that determine the order in which 20 different amino acids are bound together to make thousands of specific proteins. Because amino acids contain carbon, they can be used by the body for energy and supply 4 kilocalories of energy per gram however providing energy is not protein&rsquos most important function.

#### Water

There is one other nutrient that we must have in large quantities: water. Water does not contain carbon, but is composed of two hydrogen atoms and one oxygen atom per molecule of water. More than 60 percent of your total body weight is water. Without water, nothing could be transported in or out of the body, chemical reactions would not occur, organs would not be cushioned, and body temperature would widely fluctuate. On average, an adult consumes just over two liters of water per day from both eating foods and drinking liquids. Since water is so critical for life&rsquos basic processes, total water intake and output is supremely important. This topic will be explored in detail in Chapter 4.

Table (PageIndex<1>): Functions of Nutrients.
Nutrients Primary Function
Carbohydrates Provide a ready source of energy for the body (4 kilocalories/gram) and structural constituents for the formation of cells.
Fat Provides stored energy for the body (9 kilocalories/gram), functions as structural components of cells and also as signaling molecules for proper cellular communication. It provides insulation to vital organs and works to maintain body temperature.
Protein Necessary for tissue and organ formation, cellular repair and hormone and enzyme production. Provide energy, but not a primary function (4 kilocalories/gram)
Water Transports essential nutrients to all body parts, transports waste products for disposal and aids with body temperature regulation
Minerals Regulate body processes, are necessary for proper cellular function, and comprise body tissue.
Vitamins Regulate body processes and promote normal body-system functions.

### Micronutrients

Micronutrients are also essential for carrying out bodily functions, but they are required by the body in lesser amounts. Micronutrients include all the essential minerals and vitamins. There are sixteen essential minerals and thirteen essential vitamins (See Table (PageIndex<1>) and Table (PageIndex<2>) for a complete list and their major functions).

In contrast to carbohydrates, lipids, and proteins, micronutrients are not sources of energy (calories) for the body. Instead they play a role as cofactors or components of enzymes (i.e., coenzymes) that facilitate chemical reactions in the body. They are involved in all aspects of body functions from producing energy, to digesting nutrients, to building macromolecules. Micronutrients play many essential roles in the body.

#### Minerals

Minerals are solid inorganic substances that form crystals and are classified depending on how much of them we need. Trace minerals, such as molybdenum, selenium, zinc, iron, and iodine, are only required in a few milligrams or less. Macrominerals, such as calcium, magnesium, potassium, sodium, and phosphorus, are required in hundreds of milligrams. Many minerals are critical for enzyme function, while others are used to maintain fluid balance, build bone tissue, synthesize hormones, transmit nerve impulses, contract and relax muscles, and protect against harmful free radicals in the body that can cause health problems such as cancer.

Table (PageIndex<1>): Minerals and Their Major Functions.
Minerals Major Functions
Macro
Sodium Fluid balance, nerve transmission, muscle contraction
Chloride Fluid balance, stomach acid production
Potassium Fluid balance, nerve transmission, muscle contraction
Calcium Bone and teeth health maintenance, nerve transmission, muscle contraction, blood clotting
Phosphorus Bone and teeth health maintenance, acid-base balance
Magnesium Protein production, nerve transmission, muscle contraction
Sulfur Protein production
Trace
Iron Carries oxygen, assists in energy production
Zinc Protein and DNA production, wound healing, growth, immune system function
Iodine Thyroid hormone production, growth, metabolism
Selenium Antioxidant
Copper Coenzyme, iron metabolism
Manganese Coenzyme
Fluoride Bone and teeth health maintenance, tooth decay prevention
Chromium Assists insulin in glucose metabolism
Molybdenum Coenzyme

#### Vitamins

The thirteen vitamins are categorized as either water-soluble or fat-soluble. The water-soluble vitamins are vitamin C and all the B vitamins, which include thiamine, riboflavin, niacin, pantothenic acid, pyridoxine, biotin, folate and cobalamin. The fat-soluble vitamins are A, D, E, and K. Vitamins are required to perform many functions in the body such as assisting in energy production, making red blood cells, synthesizing bone tissue, and supporting normal vision, nervous system function, and immune system function.

Vitamin deficiencies can cause severe health problems and even death. For example, a deficiency in niacin causes a disease called pellagra, which was common in the early twentieth century in some parts of America. The common signs and symptoms of pellagra are known as the &ldquo4D&rsquos&mdashdiarrhea, dermatitis, dementia, and death.&rdquo Until scientists discovered that better diets relieved the signs and symptoms of pellagra, many people with the disease ended up hospitalized in insane asylums awaiting death. Other vitamins were also found to prevent certain disorders and diseases such as scurvy (vitamin C), night blindness (vitamin A), and rickets (vitamin D).

Table (PageIndex<2>): Vitamins and Their Major Functions.
Vitamins Major Functions
Water-soluble
Thiamin (B1) Coenzyme, energy metabolism assistance
Riboflavin (B2 ) Coenzyme, energy metabolism assistance
Niacin (B3) Coenzyme, energy metabolism assistance
Pantothenic acid (B5) Coenzyme, energy metabolism assistance
Pyridoxine (B6) Coenzyme, amino acid synthesis assistance
Biotin (B7) Coenzyme, amino acid and fatty acid metabolism
Folate (B9) Coenzyme, essential for growth
Cobalamin (B12) Coenzyme, red blood cell synthesis
C (ascorbic acid) Collagen synthesis, antioxidant
Fat-soluble
A Vision, reproduction, immune system function
D Bone and teeth health maintenance, immune system function
E Antioxidant, cell membrane protection
K Bone and teeth health maintenance, blood clotting

### Contributor

University of Hawai&rsquoi at Mānoa Food Science and Human Nutrition Program: Allison Calabrese, Cheryl Gibby, Billy Meinke, Marie Kainoa Fialkowski Revilla, and Alan Titchenal

## Nanostructured materials: basic concepts and microstructure ☆

Nanostructured Materials (NsM) are materials with a microstructure the characteristic length scale of which is on the order of a few (typically 1–10) nanometers. NsM may be in or far away from thermodynamic equilibrium. NsM synthesized by supramolecular chemistry are examples of NsM in thermodynamic equilibrium. NsM consisting of nanometer-sized crystallites (e.g. of Au or NaCl) with different crystallographic orientations and/or chemical compositions are far away from thermodynamic equilibrium. The properties of NsM deviate from those of single crystals (or coarse-grained polycrystals) and/or glasses with the same average chemical composition. This deviation results from the reduced size and/or dimensionality of the nanometer-sized crystallites as well as from the numerous interfaces between adjacent crystallites. An attempt is made to summarize the basic physical concepts and the microstructural features of equilibrium and non-equilibrium NsM.