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The majority of Bolivians belong to indigenous groups. Many are Aymara and Quechua.
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Peru has a large indigenous population, around 80% of Peru's population identify as indigenous or mestizo.
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In the later half of the 20th century, many Native Americans started to protest the unfair treatment they experienced from the societies they lived in.
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Some Native Americans have become famous in politics. For example, an Aymara man named Evo Morales was elected as president of Bolivia in 2005. He was the first indigenous presidential candidate in Bolivia and South America.
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Apple
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An apple is the edible fruit of a number of trees, known for its juicy, green, or red fruits. The tree (Malus spp.) is grown worldwide. Its fruit is low-cost, popular, and common all over the earth.
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Applewood is a type of wood that comes from this tree.
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The apple tree comes from southern Kazakhstan, Kyrgyzstan, Uzbekistan, and northwestern part of China. Apples have been grown for thousands of years in Asia and Europe. They were brought to North America by European settlers. Apples have religious and mythological significance in many cultures.
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Apples are generally grown by grafting, although wild apples grow readily from seed. Apple trees are large if grown from seed, but small if grafted onto roots (rootstock). There are more than 10000 known variants of apples, with a range of desired characteristics. Different variants are bred for various tastes and uses: cooking, eating raw and cider production are the most common uses.
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Trees and fruit are attacked by fungi, bacteria and pests. In 2010, the fruit's genome was sequenced as part of research on disease control and selective breeding in apple production.
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Worldwide production of apples in 2013 was 90.8 million tonnes. China grew 49% of the total.
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The apple has a small, leaf-shedding tree that grows up to tall. The apple tree has a broad crown with thick twigs.
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The wild ancestor of apple trees is "Malus sieversii". They grow wild in the mountains of Central Asia in the north of Kazakhstan, Kyrgyzstan, Tajikistan, and Xinjiang, China, and possibly also "Malus sylvestris". Unlike domesticated apples, their leaves become red in autumn. They are being used recently to develop "Malus domestica" to grow in colder climates.
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The apple tree was possibly the earliest tree to be cultivated. Its fruits have become better over thousands of years. It is said that Alexander the Great discovered dwarf apples in Asia Minor in 300 BC. Asia and Europe have used winter apples as an important food for thousands of years. From when Europeans arrived, Argentina and the United States have used apples as food as well. Apples were brought to North America in the 1600s. The first apple orchard on the North American continent was said to be near Boston in 1625. In the 1900s, costly fruit industries, where the apple was a very important species, began developing.
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Sometimes apples are eaten after they are cooked. Often, apples are eaten uncooked. Apples can also be made into drinks. Apple juice and apple cider are drinks made with apples.
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The flesh of the fruit is firm with a taste anywhere from sour to sweet. Apples used for cooking are sour, and need to be cooked with sugar, while other apples are sweet, and do not need cooking. There are some seeds at the core, that can be removed with a tool that removes the core, or by carefully using a knife.
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The scientific name of the apple tree genus in the Latin language is "Malus". Most apples that people grow are of the "Malus domestica" species.
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Most apples are good to eat raw (not cooked), and are also used in many kinds of baked foods, such as apple pie. Apples are cooked until they are soft to make apple sauce.
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Apples are also made into the drinks apple juice and cider. Usually, cider contains a little alcohol, about as much as beer. The regions of Brittany in France and Cornwall in England are known for their apple ciders.
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If one wants to grow a certain type of apple, it is not possible to do this by planting a seed from the wanted type. The seed will have DNA from the apple that the seeds came from, but it will also have DNA from the apple flower that pollinated the seeds, which might be a different variant of apple. This means that the tree which would grow from planting would be a mixture of two, or a hybrid. In order to grow a certain type of apple, a small twig, or 'scion', is cut from the tree that grows the type of apple desired, and then added on to a specially grown stump called a rootstock. The tree that grows will create apples of the type needed.
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There are more than 7,500 known variants of apples. Different variants are available for temperate and subtropical climates. One large collection of over 2,100 apple variants is at the National Fruit Collection in England. Most of these variants are grown for eating fresh (dessert apples). However, some are grown simply for cooking or making cider. Cider apples are usually too tart to eat immediately. However, they give cider a rich flavor that dessert apples cannot.
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Most popular apple cultivars are soft but crisp. Colorful skin, easy shipping, disease resistance, 'Red Delicious' apple shape, and popular flavor are also needed. Modern apples are usually sweeter than older cultivars. This is because popular tastes in apples have become different. Most North Americans and Europeans enjoy sweet apples. Extremely sweet apples with hardly any acid taste are popular in Asia and India.
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Apples are grown around the world. China produces more than half of all commercially grown apples. In 2020/2021, China produced 44,066,000 metric tons. Other important producers were the European Union (EU) (11,719,000 metric tons, the United States (4,490,000 metric tons), and Turkey (4,300,000 metric tons). Total world production was 80,522,000 metric tons.
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In the United Kingdom there are about 3000 different types of apples. The most common apple type grown in England is the 'Bramley seedling', which is a popular cooking apple.
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Apple orchards are not as common as they were in the early 1900s, when apples were rarely brought in from other countries. Organizations such as Common Ground teach people about the importance of rare and local varieties of fruit.
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Many apples are grown in temperate parts of the United States and Canada. "Washington State currently produces over half the Nation's domestically grown apples and has been the leading apple-growing State since the early 1920s." New York and Michigan are the next two leading states in apple production. "The total reported area dedicated to the crop in the United States is 336,940 acres or 526.47 square miles."
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In many areas where apple growing is important, people have huge celebrations:
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There are many different varieties of apples, including
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Apples are in the group Maloideae. This is a subfamily of the family "Rosaceae". They are in the same subfamily as pears.
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Abrahamic religions
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True Abrahamic religions are monotheistic (the belief that there is only one God). They also all believe that people should pray to God and worship God often. Among monotheistic religions, the Abrahamic religions have the world's largest number of followers.
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Algebra
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Besides equations, there are inequalities ("less than" and "greater than"). A special type of equation is called the function. This is often used in making graphs because it always turns one input into one output.
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Algebra can be used to solve real problems because the rules of algebra work in real life and numbers can be used to represent the values of real things. Physics, engineering and computer programming are areas that use algebra all the time. It is also useful to know in surveying, construction and business, especially accounting.
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People who do algebra use the rules of numbers and mathematical operations used on numbers. The simplest are adding, subtracting, multiplying, and dividing. More advanced operations involve exponents, starting with squares and square roots.
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Algebra was first used to solve equations and inequalities. Two examples are linear equations (the equation of a straight line, formula_5 or formula_6) and quadratic equations, which has variables that are squared (multiplied by itself, for example: formula_7, formula_8, or formula_9).
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Here is a simple example of an algebra problem:
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These are the steps you can use to solve the problem:
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With practice, algebra can be used when faced with a problem that is too hard to solve any other way. Problems such as building a freeway, designing a cell phone, or finding the cure for a disease all require algebra.
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As in most parts of mathematics, adding formula_22 to formula_23 (or formula_22 plus formula_23) is written as formula_26;
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subtracting formula_23 from formula_22 (or formula_22 minus formula_23) is written as formula_31;
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and dividing formula_22 by formula_23 (or formula_22 over formula_23) is written as formula_36 or formula_37.
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In algebra, multiplying formula_22 by formula_23 (or formula_22 times formula_23) can be written in 3 different ways: formula_42, formula_43 or just formula_44. All of these notations mean the same thing: formula_22 times formula_23. The symbol "formula_47" used in arithmetic is not used in algebra, because it looks too much like the letter formula_3, which is often used as a variable.
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When we multiply a number and a variable in algebra, we can simply write the number in front of the letter: formula_49. When the number is 1, then it is not written because 1 times any number is that number (formula_50) and so it is not needed. And when it is 0, we can completely remove the terms, because 0 times any number is zero (formula_51).
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As a side note, you do not have to use the letters formula_3 or formula_22 in algebra. Variables are just symbols that mean some unknown number or value, so you can use any letter for a variable (except formula_54 (Euler's number) and formula_55 (Imaginary unit), because these are mathematical constants). formula_3 and formula_22 are the most common, though.
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An important part of algebra is the study of functions, since they often appear in equations that we are trying to solve. A function is like a machine you can put a number (or numbers) into and get a certain number (or numbers) out. When using functions, graphs can be powerful tools in helping us to study the solutions to equations.
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A graph is a picture that shows all the values of the variables that make the equation or inequality true. Usually this is easy to make when there are only one or two variables. The graph is often a line, and if the line does not bend or go straight up-and-down it can be described by the basic formula formula_5. The variable formula_59 is the y-intercept of the graph (where the line crosses the vertical axis) and formula_60 is the slope or steepness of the line. This formula applies to the coordinates of a graph, where each point on the line is written formula_61.
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In some math problems like the equation for a line, there can be more than one variable (formula_3 and formula_22 in this case). To find points on the line, one variable is changed. The variable that is changed is called the "independent" variable. Then the math is done to make a number. The number that is made is called the "dependent" variable. Most of the time the independent variable is written as formula_3 and the dependent variable is written as formula_22, for example, in formula_66. This is often put on a graph, using an formula_3 axis (going left and right) and a formula_22 axis (going up and down). It can also be written in function form: formula_69. So in this example, we could put in 5 for formula_3 and get formula_71. Put in 2 for formula_3 would get formula_73. And 0 for formula_3 would get formula_75. So there would be a line going through the points formula_76, formula_77, and formula_78 as seen in the graph to the right.
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If formula_3 has a power of 1, it is a straight line. If it is squared or some other power, it will be curved. If it uses an inequality (formula_80 or formula_81), then usually part of the graph is shaded, either above or below the line.
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In algebra, there are a few rules that can be used for further understanding of equations. These are called the rules of algebra. While these rules may seem senseless or obvious, it is wise to understand that these properties do not hold throughout all branches of mathematics. Therefore, it will be useful to know how these axiomatic rules are declared, before taking them for granted. Before going on to the rules, reflect on two definitions that will be given.
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"Commutative" means that a function has the same result if the numbers are swapped around. In other words, the order of the terms in an equation does not matter. When two terms (addends) are being added, the "commutative property of addition" is applicable. In algebraic terms, this gives formula_86.
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Note that this does not apply for subtraction (i.e. formula_87 except if formula_88).
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When two terms (factors) are being multiplied, the "commutative property of multiplication" is applicable. In algebraic terms, this gives formula_89.
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Note that this does not apply for division (i.e. formula_90, when formula_91 and formula_92, except if formula_88).
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"Associative" refers to the grouping of numbers. The associative property of addition implies that, when adding three or more terms, it doesn't matter how these terms are grouped. Algebraically, this gives formula_94. Note that this does not hold for subtraction, e.g. formula_95 (see distributive property).
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The associative property of multiplication implies that, when multiplying three or more terms, it doesn't matter how these terms are grouped. Algebraically, this gives formula_96. Note that this does not hold for division, e.g. formula_97.
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The distributive property states that the multiplication of a term by another term can be distributed. For instance: formula_98. (Do not confuse this with the associative properties! For instance: formula_99.)
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"Identity" refers to the property of a number that it is equal to itself. In other words, there exists an operation of two numbers so that it equals the variable of the sum. The additive identity property states that any number plus 0 is that number: formula_100. This also holds for subtraction: formula_101.
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The multiplicative identity property states that any number times 1 is that number: formula_102. This also holds for division: formula_103.
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The additive inverse property is somewhat like the inverse of the additive identity. When we add a number and its opposite, the result is 0. Algebraically, it states the following: formula_104, which is the same as formula_105. For example, the additive inverse (or opposite) of 1 is -1.
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The multiplicative inverse property means that when we multiply a number and its reciprocal, the result is 1. Algebraically, it states the following: formula_106, which is the same as formula_107. For example, the multiplicative inverse (or reciprocal) of 2 is 1/2. To get the reciprocal of a fraction, switch the numerator and the denominator: the reciprocal of formula_108 is formula_109.
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In addition to "elementary algebra", or basic algebra, there are advanced forms of algebra, taught in colleges and universities, such as abstract algebra, linear algebra, and universal algebra. This includes how to use a matrix to solve many linear equations at once. Abstract algebra is the study of things that are found in equations, going beyond numbers to the more abstract with groups of numbers.
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Many math problems are about physics and engineering. In many of these physics problems time is a variable. The letter used for time is formula_110. Using the basic ideas in algebra can help reduce a math problem to its simplest form making it easier to solve difficult problems. Energy is formula_54, force is formula_112, mass is formula_60, acceleration is formula_82 and speed of light is sometimes formula_115. This is used in some famous equations, like formula_116 and formula_117 (although more complex math beyond algebra was needed to come up with that last equation).
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Atom
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Atoms are very small, but their exact size depends on the type. Atoms are from 0.1 to 0.5 nanometers across. One nanometer is about 100,000 times smaller than the width of a human hair. This makes one atom impossible to see without special tools. Scientists learn how they work by doing experiments.
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Atoms are made of three kinds of subatomic particles. These are protons, neutrons, and electrons. Protons and neutrons have much more mass. These are in the middle of the atom, called the nucleus. Lightweight electrons move quickly around them. The electromagnetic force holds the nucleus and electrons together.
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Atoms with the same number of protons belong to the same chemical element. Examples of elements are carbon and gold. Atoms with the same number of protons, but different numbers of neutrons, are called isotopes. Usually an atom has the same number of electrons as protons. If an atom has more or less electrons than protons, it is called an ion, and has an electric charge.
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Atoms can join by chemical bonds. Many things are made of more than one kind of atom. These are chemical compounds or mixtures. A group of atoms connected by chemical bonds is called a molecule. For example, a water molecule is made of two hydrogen atoms and one oxygen atom. The forming or breaking of bonds is a chemical reaction.
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Atoms split if the forces inside are too weak to hold them together. This is what causes radioactivity. Atoms can also join to make larger atoms at very high temperatures, such as inside a star. These changes are studied in nuclear physics. Most atoms on Earth are not radioactive. They are rarely made, destroyed, or changed into another kind of atom.
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In 1777 French chemist Antoine Lavoisier defined the term "element" as we now use it. He said that an element was any substance that could not be broken down into other substances by the methods of chemistry. Any substance which could be broken down was a "compound".
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In 1803, English philosopher John Dalton suggested that elements were made of tiny, solid balls called atoms. Dalton believed that all atoms of the same element have the same mass. He said that compounds are formed when atoms of more than one element combine. In any one compound, the atoms would always combine in the same numbers.
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In 1827, British scientist Robert Brown looked at pollen grains in water under his microscope. The pollen grains appeared to be shaking. Brown used Dalton's atomic theory to describe patterns in how they moved. This was called "Brownian motion". In 1905 Albert Einstein used mathematics to prove that the pollen particles were being moved by the motion, or heat, of individual water molecules. By doing this, he proved that atoms are real without question.
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In 1869, Russian scientist Dmitri Mendeleev published the first periodic table. The periodic table groups elements by their atomic number (how many protons they have; this is usually the same as the number of electrons). Elements in the same column, or group, usually have similar qualities. For example, helium, neon, argon, krypton, and xenon are all in the same column and are very similar. All these elements are gases that have no color or smell. Also, they cannot combine with other atoms to form compounds. Together they are known as noble gases.
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The physicist J.J. Thomson was the first person to discover electrons. This happened while he was working with cathode rays in 1897. He learned they had a negative charge, and the rest of the atom had a positive charge. Thomson made the plum pudding model, which said that an atom was like plum pudding: the dried fruit (electrons) were stuck in a mass of pudding (having a positive charge).
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In 1913, Niels Bohr created the Bohr model. This model showed that electrons travel around the nucleus in fixed circular orbits. This was better than the Rutherford model, but it was still not completely true.
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In 1925, chemist Frederick Soddy discovered that some elements had more than one kind of atom, called isotopes. Soddy believed that each different isotope of an element has a different mass. To prove this, chemist Francis William Aston built the mass spectrometer, which measures the mass of single atoms. Aston proved that Soddy was right. He also found that the mass of each atom is a whole number times the mass of the proton. This meant that there must be some particles in the nucleus other than protons. In 1932, physicist James Chadwick shot alpha particles at beryllium atoms. He saw that a particle shot out of the beryllium atoms. This particle had no charge, but about the same mass as a proton. He named this particle the neutron.
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In 1937, German chemist Otto Hahn became the first person to make nuclear fission in a laboratory. He discovered this by chance when shooting neutrons at a uranium atom, hoping to make a new isotope. However, instead of a new isotope, the uranium changed into a barium atom, a smaller atom than uranium. Hahn had "broken" the uranium atom. This was the world's first recorded nuclear fission reaction. This discovery led to the creation of the atomic bomb and nuclear power, where fission happens over and over again, creating a chain reaction.
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Later in the 20th century, physicists went deeper into the mysteries of the atom. Using particle accelerators, they discovered that protons and neutrons were made of other particles, called quarks.
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An atom is made of three main particles: the proton, the neutron, and the electron. Protons and neutrons have nearly the same size and mass (about grams). The mass of an electron is about 1800 times smaller (about grams). Protons have a positive charge, electrons have a negative charge, and neutrons have no charge. Most atoms have no charge. The number of protons (positive) and electrons (negative) are the same, so the charges balance out to zero. However, ions have a different number of electrons than protons, so they have a positive or negative charge.
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Scientists believe that electrons are elementary particles: they are not made of any smaller pieces. Protons and neutrons are made of quarks of two kinds: up quarks and down quarks. A proton is made of two up quarks and one down quark, and a neutron is made of two down quarks and one up quark.
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The nucleus is in the middle of the atom. It is made of protons and neutrons. The nucleus makes up more than 99.9% of the mass of the atom. However, it is very small: about 1 femtometer (10 m) across, which is around 100,000 times smaller than the width of an atom, so it has a very high density.
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Usually in nature, two things with the same charge repel or shoot away from each other. So for a long time, scientists did not know how the positively charged protons in the nucleus stayed together. We now believe that the attraction between protons and neutrons comes from the "strong nuclear force". This force also holds together the quarks that make up the protons and neutrons. Particles called mesons travel back and forth between protons and neutrons, and carry the force.
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The number of neutrons in relation to protons defines whether the nucleus stays together or goes through radioactive decay. When there are too many neutrons or protons, the atom tries to make the numbers smaller or more equal by removing the extra particles. It sends out radiation in the form of alpha, beta, or gamma decay. Nuclei can also change in other ways. Nuclear fission is when the nucleus breaks into two smaller nuclei, releasing a lot of energy. This release of energy makes nuclear fission useful for making bombs, and electricity in the form of nuclear power.
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The other way nuclei can change is through nuclear fusion, when two nuclei join or fuse to make a larger nucleus. This process requires very high amounts of energy to overcome the electric repulsion between the protons, as they have the same charge. Such high energies are most common in stars like our Sun, which fuses hydrogen for fuel. However, once fusion happens, far more energy is released, because some of the mass becomes energy.
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The energy needed to break a nucleus into protons and neutrons is called its nuclear binding energy. This energy can be converted to mass, as stated by Einstein's famous formula "E" = "mc". Medium-sized nuclei, such as iron-56 and nickel-62, have the highest binding energy per proton or neutron. They will probably not go through fission or fusion, because they cannot release energy in this way. Very small and very large atoms have low binding energy, so they are most willing to go through fission or fusion.
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Electrons orbit, or travel around, the nucleus. They are called the atom's "electron cloud". They are attracted to the nucleus because of the electromagnetic force. Electrons have a negative charge, and the nucleus always has a positive charge, so they attract each other.
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The Bohr model shows that some electrons are farther from the nucleus than others in different levels. These are called "electron shells". Only the electrons in the outer shell can make chemical bonds. The number of electrons in the outer shell determines whether the atom is stable or which atoms it will bond with in a chemical reaction. If an atom has only one shell, it needs two electrons to be complete. Otherwise, the outer shell needs eight electrons to be complete.
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The Bohr model is important because it has the idea of energy levels. The electrons in each shell have a certain amount of energy. Shells that are farther from the nucleus have more energy. When a small burst of energy called a photon hits an electron, the electron can jump into a "higher-energy" shell. This photon must carry exactly the right amount of energy to bring the electron to the new energy level. A photon is a burst of light, and the amount of energy determines the color of light. So each kind of atom will absorb certain colors of light, called the absorption spectrum. An electron can also send out, or emit, a photon, and fall into a "lower energy" shell. For the same reason, the atom will only send out certain colors of light, called the emission spectrum.
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The complete picture is more complicated. Unlike the Earth moving around the Sun, electrons do not move in a circle. We cannot know the exact place of an electron. We only know the probability, or chance, that it will be in any place. Each electron is part of an "orbital", which describes where it is likely to be. No more than two electrons can be in one orbital; these two electrons have different "spin".
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For each shell, numbered 1, 2, 3, and so on, there may be a number of different orbitals. These have different shapes, or point in different directions. Each orbital can be described by its three "quantum numbers". The "principal quantum number" is the electron shell number. The "azimuthal quantum number" is represented by a letter: s, p, d, or f. Depending on the principal and azimuthal quantum numbers, the electron can have more or less energy. There is also a "magnetic quantum number", but it does not usually affect the energy level. As more electrons are added, they join orbitals in order from lowest to highest energy. This order starts as follows: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d. For example, a chlorine atom has 17 electrons. So, it will have:
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In other words, it has 2 electrons in the first shell, 8 in the second shell, and 7 in the third shell.
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The number of protons in an atom is called its "atomic number". Atoms of the same element have the same atomic number. For example, all carbon atoms have six protons, so the atomic number of carbon is six. Today, 118 elements are known. Depending on how the number is counted, 90 to 94 elements exist naturally on earth. All elements above number 94 have only been made by humans. These elements are organized on the periodic table.
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Because protons and neutrons have nearly the same mass, and the mass of electrons is very small, we can call the number of protons and neutrons in an atom its "mass number". Most elements have several isotopes with different mass numbers. To name an isotope, we use the name of the element, followed by its mass number. So an atom with six protons and seven neutrons is called carbon-13.
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Sometimes, we need a more exact measurement. The exact mass of an atom is called its "atomic mass". This is usually measured with the atomic mass unit (amu), also called the dalton. One amu is exactly 1/12 of the mass of a carbon-12 atom, which is grams. Hydrogen-1 has a mass of about 1 amu. The heaviest atom known, oganesson, has a mass of about 294 amu, or grams. The average mass of all atoms of a particular element is called its "atomic weight".
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The size of an atom depends on the size of its electron cloud. Moving down the periodic table, more electron shells are added. As a result, atoms get bigger. Moving to the right on the periodic table, more protons are added to the nucleus. This more positive nucleus pulls electrons more strongly, so atoms get smaller. The biggest atom is caesium, which is about 0.596 nanometers wide according to one model. The smallest atom is helium, which is about 0.062 nanometers wide.
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When atoms are far apart, they attract each other. This attraction is stronger for some kinds of atoms than others. At the same time, the heat, or kinetic energy, of atoms makes them always move. If the attraction is strong enough, relative to the amount of heat, atoms will form a solid. If the attraction is weaker, they will form a liquid, and if it is even weaker, they will form a gas.
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Chemical bonds are the strongest kinds of attraction between atoms. The movement of electrons explains all chemical bonds.
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Atoms usually bond with each other in a way that fills or empties their outer electron shell. The most reactive elements have an almost full or almost empty outer shell. Atoms with a full outer shell, called noble gases, do not usually form bonds.
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There are three main kinds of bonds: ionic bonds, covalent bonds, and metallic bonds.
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All atoms attract each other by Van der Waals forces. These forces are weaker than chemical bonds. They are caused when electrons move to one side of an atom. This movement gives a negative charge to that side. It also gives a positive charge to the other side. When two atoms line up their sides with negative and positive charges, they will attract.
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