* The perimeter of a two-dimensional object is the total length of all its sides. For example, if these sides are straight, it"s a matter of simple addition: A rectangle measuring 4 inches by 5 inches has two sides 4 inches long and two sides 5 inches long, so its perimeter is 4 + 5 + 4 + 5 = 18 inches.
* The area of a four-sided figure is calculated by multiplying the length by the width: In the above example 4 x 5 = 20 square inches (in.2).
* Volume is calculated in the same way, by multiplying the length by the width by the height (or, if you prefer, the area by the height). For instance, a box that is 6 inches high, whose base measures 4 inches by 5 inches, has a volume of 4 x 5 x 6 = 120 cubed inches (in3).
The volume of a pyramid is the area of the base multiplied by the height, divided by 3: It"s when you get to circles that it all becomes more complicated, because then you have to start dealing with...
* PI Pi () is the Greek equivalent of the Roman p and is used in math to represent the ratio of the circ.u.mference of a circle to its diameter. Depending on how sophisticated you are as a mathematician, you can say that = 3.142, 3.14159, or 3.1415926535897932384626433832795, but even then it is not 100 percent exact. Expressed as a fraction, pi is roughly 3, or 22/7.
Before we go on, three more quick definitions:* The circ.u.mference of a circle is its perimeter, the distance around the outside.
* The diameter is the length of a straight line through the middle, from one point on the circ.u.mference to another.
* The radius is half the diameter; that is, the distance from the center of the circle to the circ.u.mference.
So to calculate the circ.u.mference of a circle, you multiply the diameter by : a circle that is, say, 7 inches in diameter has a circ.u.mference of 7 x 22/7 = approximately 22 inches. The formula for this can be expressed as d, but is usually given as 2(r).
Area is r2-that is, times the radius squared. So a circle of 6 inches radius has an area of 22/7 x (6 x 6) = approximately 113 square inches.
The three-dimensional equivalent to a circle is a sphere, and its volume is calculated by the formula 4/3r3-that is, four thirds (or one and one third) of the product of and the radius cubed (multiplied by itself and then by itself again). So a sphere with a radius of 6 inches has a volume of 4/3 x x(6 x 6 x 6) = approximately 905 cubic inches.
A cone is effectively a pyramid with a circular base, so the pyramid formula applies: A cone with a base 6 inches in diameter and a height of 10 inches has a base area of x (6 x 6) = approximately 113 square inches, and a volume of:
which equals approximately 377 cubic inches.
* TRIANGLES The area of a triangle is calculated by: There are three types of triangles, depending on the length of their sides:* An equilateral triangle has three sides of equal length.
* An isosceles triangle has two sides of equal length.
* A scalene triangle has three sides that are all of different lengths.
The sum total of the angles of a triangle, whatever its shape, is 180. A right angle is 90; any angle smaller than 90 is called an acute angle, while anything above 90 but lower than 180 is obtuse. In a right-angled triangle the side opposite the right angle (also always the longest side) is called the hypotenuse, which brings us neatly to...
* THE PYTHAGOREAN THEOREM This theorem states that the square on the hypotenuse is equal to the sum of the squares on the other two sides. The simplest example of this is what is called a 3:4:5 triangle, in which the hypotenuse is 5 inches (or centimeters or miles, it doesn"t matter) and the other two sides are 3 inches and 4 inches.
The square on the side that is 3 inches long is 9 in.2 (3 x 3), the square on the 4-inch side is 16 in.2 (4 x 4), and when you add them together, you get 25 in.2, which is the square of the hypotenuse (5 x 5).
This can also be remembered using the formula a2 + b2 = c2, where c is the hypotenuse.
The burning question, of course, is, Why does it matter? Well, it could have had some practical value in the ancient world. It has been suggested, for example, that the Egyptians could have used ropes in the proportion 3:4:5 to produce right angles when building the pyramids. Unfortunately, there isn"t the remotest sc.r.a.p of evidence that they did any such thing. In fact, the Pythagorean theorem matters most to mathematicians because it is fundamental to our next topic.
Trigonometry
Trigonometry is "the branch of mathematics that deals with the relations between the sides and angles of a triangle," and a trigonometric function is "any function of an angle that is defined by the relationship between the sides and angles of a right-angled triangle."
There are six basic trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant, and they are calculated as follows. In a right-angled triangle where the other two angles are valued at x and y degrees, the side opposite x is a, the side opposite y is b, and the hypotenuse is c:sin x = a/c
cos x = b/c
tan x = a/b
cot x = b/a
sec x = c/b
cosec x = c/a
Why do we care? Well, the point is that the functions or ratios remain the same whatever the size of the triangle. So if you know the sine of a 90 angle in a triangle whose sides measure 3, 4, and 5 inches, you can extrapolate all sorts of measurements for a much larger triangle with the same proportions.
The trigonometric version of the Pythagorean theorem tells us that for any angle x,sin2x + cos2x = 1,
where sin2x is a conventional way of writing (sin x)2 without the need for brackets. If you know the sine of an angle, you can use this formula to calculate all the rest of the trigonometric functions given above.
Trigonometry is vital to the study of higher mathematics and the sciences. At a more comprehensible and practical level, it is used in land surveying, mapmaking, engineering, astronomy, geography, satellite navigation systems, and so on.
SCIENCE.
The world of science is so vast and expanding that to condense it into 30 pages seems like a futile experiment. Every school system teaches the topic differently, so what may seem familiar and commonplace to one person can remain a mystery to others. Consider this chapter the foundation on which you can build.
Biology
The term biology comes from the Greek, meaning study of life; therefore, this field of learning concerns plants and animals and how the human body works.
* PHOTOSYNTHESIS This is the process by which plants convert carbon dioxide and water into the carbohydrates they need for growth, using energy that they absorb from light (hence, the photo element). Light is absorbed into the plant by the green pigment called chlorophyll, stored mainly in the leaves, which provides the green color of so many plants. In fact, plants need only the hydrogen element from water (H2O), so photosynthesis releases oxygen back into the atmosphere, enabling the rest of us to breathe.
* THE STRUCTURE OF A PLANT The flower contains the plant"s reproductive organs. The stigma, style, and ovary make up the carpel, which contains the female cells; if a flower has more than one carpel, these combine to form the pistil. The male organ is called the stamen and consists of an anther that contains the pollen sacs and is supported on a filament. Most plants self-pollinate, but some, such as certain hollies and the kiwifruit, require a male and female plant of the same species in order to reproduce.
The leaves enable the plant to feed and breathe. They contain the chlorophyll that is essential to photosynthesis, which absorbs light. Leaves also contain pores (stoma), through which gases and water are absorbed and released back into the atmosphere. The shape of the leaf reflects the plant"s needs: big, broad leaves are designed to absorb maximum light; the fleshy, succulent leaves of a cactus store water in case of drought.
The stem is the plant"s support and the conduit between roots, leaves, and plants. It contains phloem, a tissue that transports food within the plant; and xylem, which princ.i.p.ally transports water. It is the xylem that hardens to form the trunks of trees and shrubs.
The roots anchor the plant in the ground and absorb nutrients and water from the soil. A tap root system has a single main root; a fibrous system has-well, lots of fibers. In root vegetables, such as turnips and carrots, the vegetable part is, in fact, a swollen root. Advent.i.tious roots are less common; the name means coming from the outside, and these roots grow in unusual places, such as from the stem.
* THE CARBON CYCLE The process by which carbon (in the form of carbon dioxide) is absorbed from the atmosphere during photosynthesis and is then transferred from one organism to another and eventually released back into the atmosphere is known as the carbon cycle. For example, a plant takes in carbon dioxide; the plant is eaten by a herbivorous animal, which is in turn eaten by a carnivore; when the animal dies, its rotting body releases carbon dioxide. Alternatively, the herbivorous animal excretes its waste, which also degrades to give off carbon dioxide.
This provides a smooth transition from plants to the human body.
* CHROMOSOMES A normal human body has 46 chromosomes composed of 22 matched pairs and two s.e.x chromosomes. Half of each pair, along with a single s.e.x chromosome, is found in the sperm. The other half is in the egg. Fusion of the two creates the human embryo. s.e.x chromosomes are of two types, called X (female) and Y (male). The egg always contains an X chromosome, so the s.e.x of the embryo is determined by whether a sperm is carrying an X or Y chromosome. Other chromosomes dictate other genetic factors, such as hair and eye color.
Chromosomes are made up of DNA, RNA, and protein.
DNA stands for deoxyribonucleic acid and is fundamental to the organization and functioning of living cells. It consists of the famous "double helix" (identified by the scientists Crick and Watson in 1953), with two strands coiled around each other. When the strands of a helix separate, each provides a template for the synthesis of an identical strand, containing the same genetic information. This enables normal growth, cell repair, and the production of cells that will turn into the next generation-which is why humans produce babies rather than tiger cubs, and why tigers produce tiger cubs rather than roses.
RNA stands for ribonucleic acid, which occurs as a single strand and contains different sugars and bases but is otherwise structurally similar to DNA. It"s vital to the synthesis of...
Proteins, which fulfill many important roles in a living organism-they are involved in the makeup of tissue; the properties of muscles; and the functioning of hormones, the immune system, and the digestive system, to name a few. They are manufactured within cells using information conveyed by the DNA and RNA.
* THE SKELETAL SYSTEM The human skeleton is made up of more than 200 bones, held together by fibrous tissue called ligaments, and linked at the joints. Joints allow varying degrees of movement from none (between the bones that make up the skull) through some (the hinge joints at the elbow and knee) to lots (the ball-and-socket joints at the hip and shoulder).
The princ.i.p.al bones of the body, starting at the top and working down, are:* cranium: skull * spine: made up of 26 smaller bones called vertebrae * clavicle: collar bone * scapula: shoulder blade * humerus: upper arm * radius and ulna: lower arm-the radius is the broader one on the thumb side, the ulna the narrower one on the little finger side * carpus: a collective name for the bones of the wrist, individually known as carpals * metacarpus: ditto for the five long bones of the hand * phalanges: fingers * sacrum: actually a fusion of five vertebrae attached to the * hip bone * coccyx: tail bone, a fusion of the lowest four vertebrae * femur: thigh bone * patella: knee cap * tibia and fibula: lower leg-the tibia is the broader one that runs down toward the big toe; the fibula the narrower one that runs toward the little toe * tarsus: a collective name for the bones of the ankle and heel, individually known as tarsals * metatarsus: ditto for the five long bones of the foot * phalanges: toes * THE CIRCULATORY SYSTEM Blood is the body"s transportation system-everything from oxygen to hormones is transported around the body in the bloodstream, and its waste products, from carbon dioxide to urea, are carried away for disposal.
In order for blood to do its job, it needs to be pumped around, and that is the primary purpose of the heart. The heart is two pumps, each consisting of two chambers-an auricle and a ventricle-with a valve in between. The left side of the heart receives oxygen-rich blood from the lungs and forces it throughout the body; the right side receives the oxygen-depleted blood and returns it to the lungs to be re-oxygenated. (Oxygen, of course, comes into the lungs in the air that we breathe, and without it the cells in the body would die.) All this requires a well-organized system of blood vessels. These are divided into arteries, which are strong and muscular and carry fast-flowing blood away from the heart, and veins, which are weaker and more sluggish and bring it back. The princ.i.p.al artery, the aorta, divides into smaller arteries and arterioles. Smaller veins are called venules, and really tiny blood vessels-whether veins or arteries-are called capillaries.
The exception to the useful mnemonia-arteries go away-is the pulmonary artery-the one that goes from the lung to the heart. The pulmonary vein runs from the heart to the lungs. Therefore, the truth is that the arteries simply carry the oxygen-rich blood.
Blood has four major components:* red blood cells, which carry hemoglobin, made up of heme (an iron-containing pigment) and globin (a protein) (This combines with oxygen to form oxyhemoglobin, the means by which oxygen is transported throughout the body. Oxyhemoglobin also gives the blood its red color, which is why arterial blood is bright red; venous blood, having deposited oxygen in cells all over the body, has a bluish tinge.) * white blood cells, or leukocytes, which fight infection * platelets, which are necessary for the clotting process * plasma, the liquid that makes the blood... well, liquid * THE DIGESTIVE SYSTEM The digestive process is divided into four parts:* ingestion: eating food * digestion: breaking the food down into const.i.tuent parts * absorption: extracting nutrients from the food * elimination: disposing of waste Once you swallow food or drink, it enters the esophagus, or gullet, and pa.s.ses (through a process of muscular contraction called peristalsis) into the stomach. From there it continues into the small intestine (comprising the duodenum, jejunum, and ileum), where digested food is absorbed into the bloodstream. The whole process is helped by the secretion of enzymes. One of the effects of the digestion of protein (which enters the body via meat, fish, eggs, etc.) is the release of amino acids, which are the building blocks of the protein the body needs for all sorts of different purposes.
Anything undigested after this stage pa.s.ses into the colon (the beginning of the large intestine), where water is extracted from it. What remains are the feces, which pa.s.s through the r.e.c.t.u.m and out of the body via the a.n.u.s.
Organs encountered along the way include:* the liver, which in adult life often copes with our alcohol intake, but which has many more functions to do with digestion and keeping the blood healthy * the gall bladder, which stores bile, needed in the digestion of fats * the pancreas, which secretes various enzymes and the hormones insulin and glucagon, which regulate levels of blood sugar * the kidneys, which control the amount of salt and water in the blood. (Excess fluid containing waste products is filtered through the kidneys down to the bladder and leaves the body in the form of urine.) * THE RESPIRATORY SYSTEM Air pa.s.ses into the body through the trachea or windpipe via the mouth and nose. With the help of contractions from the diaphragm, which is a large muscle extending across the bottom of the rib cage, it is carried down into the lungs via two smaller tubes, called bronchi, which then split into even smaller bronchioles. Inside the lungs are lots of little air sacs, or alveoli. Within the alveoli oxygen is extracted from the air, absorbed into the bloodstream, and carried off to the heart via the pulmonary artery. The pulmonary vein brings "used" blood back to the alveoli, and the process is reversed as we breathe out air that now has a high carbon dioxide content.
* THE NERVOUS SYSTEM The brain, spinal cord, and nerves make up perhaps the most important and intricately complex system in the human body. The nervous system essentially controls all the other systems in your body. It is what allows you to remember things, or at least remember that you used to know something. It tells your muscles and organs what to do and how to do it. The three interconnected parts of the nervous system are:* the central nervous system, composed of the brain and spinal cord, which sends nerve impulses and a.n.a.lyzes information from the sense organs (eyes, ears, nose, mouth, skin, etc.). These organs enable you to see, touch, taste, hear, and feel.
* the peripheral nervous system, which includes the craniospinal nerves, a vast network of nerves that extends from your brain and spinal cord to all parts of your body and carries signals back and forth. It carries nerve impulses from the central nervous system to the muscles and glands.
* the autonomic nervous system (ANS), which regulates involuntary actions, such as pulse rate and digestion. The ANS is broken into the sympathetic nervous system (fight or flight), the parasympathetic nervous system (rest so you can digest), and the enteric nervous system (the digestive system"s personal messenger).
However, no discussion of the nervous system is complete without those trusty neurons, the nerve cells that send and carry the signals throughout your body. A neuron consists of a main cell body with a long nerve fiber, called an axon, branching from it. Electrical signals pa.s.s from axon to axon through small gaps called synapses. In order to do this, these electrical signals turn into chemical ones, called neurotransmitters. In fact, right now the neurons in the temporal lobe of your brain (which interprets language), your frontal lobe (which involves reasoning), and your occipital lobe (which controls sight) are firing away!
Chemistry
This is the study of elements and compounds and the reactions they undergo-which is a definition that surely cries out for a few more definitions.
atom: the smallest particle in an element that can take part in a chemical reaction, made up of a nucleus, which is containing positively charged protons and neutral neutrons; and a number of electrons, which are negatively charged particles that orbit the nucleus. Each atom normally has the same number of protons and electrons, leaving it with a neutral charge. The movement of electrons is responsible for most commonly observed chemical, electrical, or magnetic reactions. If an atom loses or gains an electron, it becomes either positively or negatively charged and is known as an ion.
element: a substance that cannot be decomposed into a simpler substance by a chemical process. Groups of elements come together to form a compound. So, for example, a combination of the element hydrogen (H) and the element oxygen (O) can form the compound water (H2O).
mole: also known as Avogadro"s number or Avogadro"s constant, a mole contains the same number of particles as there are in 12 g of carbon-12 atoms-that is, 6.022 x 1023 particles. Carbon has three naturally occurring isotopes (forms of the same substance with different numbers of neutrons), and one of these is carbon-12.
molecule: the smallest particle of a compound that can exist independently and retain its properties. So in the previous example, the smallest imaginable quant.i.ty of hydrogen and oxygen joined together in the right conditions and right proportions will still produce a molecule of water. Only when the hydrogen and oxygen are chemically separated again do they lose the properties that make them water and return to being atoms of hydrogen and oxygen.
* THE PERIODIC TABLE OF THE ELEMENTS The periodic table was first devised in 1889 by the Russian chemist Dmitri Mendeleev. When putting the table together, Mendeleev realized there were gaps between some of the elements. Based on this, he predicted that some elements had yet to be discovered.
The table arranges the elements in ascending order of atomic number (the number of protons that each possesses) in such a way that the vertical columns contain groups or families with similar chemical properties. The horizontal rows represent periods, with the most electropositive (an alkali metal) on the left and the so-called inert gases on the right, and the whole thing proves that "the chemical properties of the elements are periodic functions of their atomic weights"-or, in other words, that similar properties in an element recur at regular intervals.