FUNDAMENTAL NUMBERS
FUNDAMENTAL NUMBERS
GLOSSARY
π Greek letter ‘pi’ representing a mathematical constant defined as the ratio of a circle’s circumference to its diameter. It is approximately equal to 3.14159 and is an example of an irrational number.
√ Mathematical symbol for the square root. The square root of a number, x, is another number, y, with the property that y × y = x.
Avogadro’s number The number of molecules in a ‘mole’ of that chemical substance (usually atoms, molecules or ions), equal to 6.022140857 × 1023.
binary digits In the binary numerical system there are only two digits: 1 and 0. This system is used by all computers and digital devices.
composite number A positive integer that is made by multiplying two smaller positive integers. A composite number cannot be a prime number.
conjugate A mathematical conjugate is when the sign is changed in a binomial, for example x - y is the conjugate of x + y.
cryptography The mathematical study of methods used to encrypt and decrypt information ensuring its secure transmission.
denominator In a simple fraction (one number over another) the denominator is the bottom number and cannot be zero.
e A mathematical constant, also called Euler’s number, which is approximately equal to 2.71828. It is the number whose natural logarithm is 1.
Fundamental Theorem of Arithmetic Theory presented by ancient Greek mathematician Euclid in Elements c. 300 BCE. It states that every number greater than 1 is either a prime or is the product of primes.
imaginary number A non-real number that cannot be computed, called i; the square root of – 1.
integer A whole number that can be positive or negative.
logarithm The opposite of an exponent. The resulting number raised by an exponent has the equivalent logarithm using the number as base and the exponent as the result: 8 = 23 which is log2 8 = 3.
natural number Any positive integer.
numerator In a simple fraction the numerator is the top number.
polynomial An expression of at least two algebraic terms with different powers in a sum.
Sieve of Eratosthenes A simple method for finding all prime numbers up to a specified number, originally attributed to ancient Greek scholar Eratosthenes of Cyrene. Write all the integers to the number and then sequentially remove all numbers wholly divisible by 2, 3, 5, etc., which are greater than these numbers.
REAL NUMBERS
the 30-second calculation
René Descartes introduced the term ‘real number’ in the seventeenth century, to distinguish between the real and imaginary solutions of polynomials (expressions formed of sums of several terms that contain different powers of the same variable; for example, x2 + 12x + 1 = 0). In contrast to the natural numbers (numbers used in everyday life for counting, 0, 1, 2, 3, etc.) and integer numbers (the natural numbers together with their opposites, –2, –1, 0, 1, 2, etc.), the real numbers are difficult to define. They are basically any values on the continuous number line that can be used to measure quantities. Real numbers include natural numbers and integers, as well as numbers that contain decimal points. Thus the real numbers include and are classified in terms of both rational numbers (which can be written as a ratio of two integers) and irrational numbers (which cannot be written as a ratio of two integers). The real numbers form the largest set of numbers that we encounter in real-world applications. The fact that the sum, difference, multiplication and division (not by 0) of two real numbers give rise to another real number, allows us to use them to count, order and measure different quantities in everyday life: from buying groceries or calculating monthly mortgage payments to measuring the speed at which we drive our cars.
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A real number is any value we can choose on the continuous number line, with the purpose of using it in real-world applications for measuring various quantities.
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Most constants in physical sciences (such as the gravity constant, or the speed of light constant), as well as physical variables (such as position, speed) are described using real numbers, as are most constants (such as Avogadro’s number) and physiological variables (such as blood volume) in biological sciences and medicine.
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3-SECOND BIOGRAPHY
RENÉ DESCARTES
1596–1650
French mathematician and philosopher who first used the term ‘real’ number, to distinguish it from an imaginary number in the context of solutions of polynomial equations
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Raluca Eftimie
Positive real numbers are used, for example, to describe distances, vehicle speeds and grocery costs.
RATIONAL NUMBERS
the 30-second calculation
The term ‘rational number’ comes from the fact that these numbers are expressed as the ‘ratio’ of two integer numbers (whole numbers that can be positive, negative or zero; although one cannot divide by zero). Since any integer a can be written as fractions of the form a/1, it implies that all integers are also rational numbers. The following types of numbers are rational: whole numbers (1.0; -22.0), numbers with exact decimals (1/2 = 0.5), numbers with pure repeating decimals (1/3 = 0.333333…). In contrast, those numbers with non-repeating decimals are known as irrational numbers. One of the earliest rigorous studies of rational numbers is Euclid’s Elements, which deals (among other topics) with the ratios of numbers. Rational numbers have various properties that allow us to use them in everyday life. For example, for any two rational numbers, their sum, difference and product (as well as division by a non-zero number) is also a rational number (and therefore we can add/subtract/multiply such numbers to measure things). These simple properties can be used to show another important property of rational numbers: between any two such numbers there is always another rational number. For example, if we choose two rational numbers, p and q, it is clear that (p + q)/2 is another rational number which lies between p and q.
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Rational numbers can be expressed as the ratio of two integers, a/b, where a is called the numerator while b is called the denominator and must be non-zero.
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Rational numbers have numerous applications in everyday life, from grocery shopping (the cost of 1kg of apples versus the cost of a 0.5kg bag of apples), to driving (measuring car speeds) and cartography (scales on maps to represent distances between places).
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EUCLID
fl c. 300 BCE
Ancient Greek mathematician (often referred to as the founder of geometry) who discussed rational numbers in his work Elements
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Raluca Eftimie
Positive rational numbers can be used to compare different quantities when shopping and cooking, or to measure distances.
IRRATIONAL NUMBERS
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Irrational numbers have been used and approximated since ancient times by mathematicians interested in astronomy and engineering. One of their first mentions comes from Hippasus of Metapontum, who discovered that the diagonal of a unit square (i.e. a diagonal of length √2) could not be expressed as the ratio of two whole numbers. In other words, √2 is not a rational number. More than two centuries later, Euclid gave a geometric proof of this result in his Elements. Nowadays, an irrational number is usually defined as a limit of a sequence of rational numbers, or defined by partitioning the rational numbers into subsets of numbers greater/less than the irrational number of interest. Examples of irrational numbers are π = 3.141592654…, √2 = 1.414213562…, e = 2.718281828… In fact, all square roots of rational numbers (which are not perfect squares) are irrational numbers: e.g. -√5 = -2.2360679 … The main characteristic of irrational numbers is that they have non-repeating decimal expansions (see above the decimal expansions for π, e, √2, – √5). This aspect is mentioned by Stifel in his 1544 work Aritmetica integra, where he discussed the irrational numbers that could not be considered true numbers because they were ‘concealed under a fog of infinity’ (this ‘fog’ referring to the number of non-repeating decimals).
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Irrational numbers are those numbers that cannot be expressed as the ratio of two integers, namely a/b, with a, b integers and b non-zero.
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Irrational numbers have various applications in engineering, architecture, physics, etc., being used, for example, to calculate the lengths and areas of various geometric objects, or to calculate angles between different trajectories in space.
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3-SECOND BIOGRAPHIES
KARL WEIERSTRASS
1815–97
German mathematician who presented the construction of irrational numbers in a series of lectures delivered in the 1860s in Berlin
RICHARD DEDEKIND
1831–1916
German mathematician who defined the rational and irrational numbers in his 1858 paper, ‘Continuity and Irrational Numbers’, where he also showed how these numbers combine to form real numbers
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Raluca Eftimie
Irrational numbers are used, for example, to calculate angles between different trajectories (using pi).
PRIME NUMBERS
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A prime number p is a natural (discrete) number that has exactly two positive divisors: 1 and itself. Examples of prime numbers are 2, 3, 5, 7. In Elements, Euclid proved that there are infinitely many prime numbers. If a number p is not prime (i.e. p can be written as the product of two other numbers each strictly between 1 and p), then p is called a ‘composite number’. The number ‘1’ is neither prime nor composite. The prime numbers are the multiplicative building blocks of the natural numbers, meaning that each natural number can be written uniquely as the product of finitely many prime numbers, a result known as the Fundamental Theorem of Arithmetic. The fact that every number (no matter how large it is) can be factorized into primes, or the fact that two very large prime numbers can be multiplied to give rise to another number, has implications for computer security since it is very difficult to figure out quickly how an extremely large number can be decomposed into two prime numbers. This aspect is used in the encryption of messages, where in order to undo the encryption and read the message, one needs to know the prime numbers used for encryption.
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A prime number is a natural number (greater than 1) that cannot be obtained by multiplying two strictly smaller natural numbers.
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As well as their importance in cryptography, prime numbers also appear in biology where, for example, some species of cicadas have life cycles of 13 years or 17 years, which gives them an evolutionary advantage.
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3-SECOND BIOGRAPHY
ERATOSTHENES OF CYRENE
c. 276 BCE–c.194 BCE
Ancient Greek mathematician known as the founder of geography who proposed a simple method to calculate prime numbers
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Raluca Eftimie
Prime numbers are used in cryptography. They can also describe the life cycle of cicadas.
COMPLEX NUMBERS
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Complex numbers are expressions of the form a + ib, where a and b are real numbers and i is a symbol defined as i = √(-1). This symbol i is called the imaginary number (or the imaginary unit) and has the property that when it is squared it becomes a (negative) real number. The number a is called the real part of a + ib, while the number b is called the imaginary part of a + ib. For any complex number a + ib, the expression a – ib is called its complex conjugate. The product of a complex number with its complex conjugate leads to real numbers (i.e. (a + ib)(a - ib)=a2 + b2). Even though the square root of a negative number was recorded around 50 ce when Heron of Alexandria studied the volume of a section of a pyramid, problems that involved square roots of negative numbers were deemed impossible to solve until the sixteenth century. The first major step was made in 1545 by Girolamo Cardano who, in his book Ars Magna, tried to solve the equation x(10 – x) = 40 and found two solutions: x = 5 + √(-15) and x = 5 – √(-15). René Descartes first introduced the term ‘imaginary number’ in 1637 in the context of solutions of polynomial equations. More than one century later, Leonhard Euler associated the symbol i with √(-1). In the 1830s, Carl Friedrich Gauss introduced the term ‘complex number’ for an expression of the form a + ib.
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Complex numbers are numbers which have been introduced to solve equations that involve square roots of negative real numbers.
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Due to Euler’s identity eix = cos(x)+isin(x) (which holds for any real number x; where constant e is an irrational number defined as e = 2.71828…), nowadays complex numbers are essential to describe phenomena that behave like a wave such as the alternating current, where both the current and the voltage vary in time like a sinusoidal wave.
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LEONHARD EULER
1707–83
Swiss mathematician who made important discoveries in many areas of mathematics and introduced various notations including the letter i to denote the imaginary number, and e to denote the base of the exponential function and the natural logarithm
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Raluca Eftimie
Complex numbers can describe physical phenomena that behave like waves.
DISCRETE NUMBERS
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Discrete numbers can take only certain particular values, and these values cannot be meaningfully subdivided. One example would be the number of marbles in a jar. More formally, discrete numbers are either numbers taken from a finite set or numbers taken from an infinite set that is ‘countable’. For this reason, discrete numbers are the first ones we meet when we start learning about mathematical concepts as children, since they can be used for counting. If we take the former (conceptually simpler) case first, a finite set is simply a set with a given finite number of entries, for example the whole numbers between 1 and 10. The numbers don’t have to be whole, though; the set {1.1, 2.34, 5.987, 6} is also a finite set. Discrete numbers can also come in infinite sets, the most familiar example being the whole numbers 1, 2, 3, 4, etc. The important thing is that for these numbers to be discrete the set should be ‘countable’. This means that we can put the numbers in the set in order and match them up one-to-one with the whole numbers (even if this matching process goes on to infinity). Discrete numbers should be contrasted with continuous numbers, which cannot be counted because between any two numbers in a continuous set is another number.
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Discrete numbers are numbers taken from a set that contains only certain values, a typical example being the whole numbers 1, 2, 3, 4, etc.
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Discrete numbers form the basis for ‘discrete mathematics’. They are used for counting, and as such form the basis for the most elementary ideas of quantity. Common examples are the integers, whole numbers (positive integers) or the bits (binary digits) in a binary number used in computing and digital communication, which may take the value either 0 or 1.
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David Pontin
Learning to count is one of the first things we do with numbers – in particular discrete numbers.
CONTINUOUS NUMBERS
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Continuous numbers can be used to quantify concepts that can be subdivided in an infinite number of ways. Typical examples might be the weight of an apple, the temperature of the air at a point or the distance between two arbitrarily chosen points in space. Think of the latter example, and for concreteness take the distance between the points A and B. There is no restriction to the values that this distance can take, because we can move A any small or large amount closer to B, without restriction on how small or large the change in position of A. The most common example of a set of continuous numbers are the ‘real numbers’: these are the set of all positive and negative numbers. They can be represented by decimal numbers, where the number of decimal places can go on forever. Unlike discrete numbers, continuous numbers are not countable. We cannot even put them in order for the purpose of counting because between any two numbers in a continuous set is another number. Thinking of our representation using decimals, given any two decimal numbers we can always find a number between them by making use of extra decimal places after those in which the two numbers match in the decimal representation.
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Continuous numbers are numbers that can take any value and are used to represent things that cannot be counted, but are instead measured, for example distance.
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To see that the real numbers are continuous, look for a number between 1 and 2. We can take, say, 1.1. We can also find a real number between 1 and 1.1, e.g. 1.01. Now, between 1 and 1.01 is 1.001, between 1 and 1.001 is 1.0001… And we could keep doing this forever.
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3-SECOND BIOGRAPHY
GEORG CANTOR
1845–1914
German mathematician who, in 1891, put forward his ‘diagonal argument’ which shows that the real numbers are not countable
30-SECOND TEXT
David Pontin
Continuous numbers are used to represent things that can be measured, such as distance, weight or temperature.
ZERO & NEGATIVE NUMBERS
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Numbers are often first understood by counting objects, where zero and negative numbers are not used, but they facilitate more than counting. Zero represents the amount ‘nothing’ and is essential for writing numbers in the positional number system; for example, to distinguish between 301 and 31. Zero may also commonly represent a base point – such as the freezing point of water, or sea level in maps. In these settings, positive and negative numbers represent opposing directions away from the zero point. Basic arithmetic can be done using zero or negative numbers. Starting with any number and adding or subtracting 0 leaves the original number unchanged. Multiplying by 0 gives 0. However, division by 0 is undefined. Addition or subtraction with negative numbers may be thought of in terms of direction. To add –17 and (+)12 is equivalent to moving 17 steps in one direction, and then 12 in the opposite direction, ending up 5 steps in the first (negative) direction. Multiplication (or division) can be done by first ignoring any minus signs to obtain an answer, then adjusting by the rule that multiplying two numbers with the same sign gives a positive result, while multiplying a negative and a positive number together gives a negative result.
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Zero, or 0, represents nothing, no quantity or a base point on a scale. A negative number is a real number which is less than zero.
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Negative numbers and the number zero underpin fundamental arithmetic, and hence mathematics, but are also commonplace in everyday life. On a lift control panel, the ground floor is often labelled 0, while lower floors have negative numbers. In finance, negative numbers are widely used to denote debit amounts.
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BHASKARA II
1114–85
Indian mathematician who wrote about calculus hundreds of years before it was developed in Europe and analysed negative numbers in his Crown of Treatises (1150)
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John McDermott
Zero may represent freezing point or sea level, with negative numbers representing lower levels.
POWERS & ROOTS
the 30-second calculation
A power is the result of repeated multiplication of a number by itself. Three rows of three objects is nine. This is expressed as ‘three squared is nine’ or ‘three to the power two is nine’ and written mathematically as 32 = 9, since there are two threes multiplied together. The superscript number is called the index. The term ‘squared’ is used because three rows of three forms a square. The first power of any number is just itself: 31 = 3. The square root of a number is a second number which, multiplied by itself, gives the first. The square root of 9, often written as √9, is 3, and √25 is 5. The square root symbol √ can be adapted for higher roots by decoration with a small number, but is often replaced by fractional indices: 91/2 = 3 and 811/4 = 3. Powers of a number may be combined when multiplying: 37 = 3 × 3 × 3 × 3 × 3 × 3 × 3 = 34 × 33, for example, and 35 × 32, and so on, with the sum of the two indices always 7. Generally, two powers of a number multiplied together will give that number to the power of the sum of the two indices. The same power of two, possibly different, numbers may be combined: 22 × 32 = 4 × 9 = 36 = 62 = (2 × 3)2. Negative indices represent fractions of 1 divided by that power, so that 3-4 = 1/34 = 1/81. This is consistent for combining: 37 × 3-4 = 37+(-4) = 33 = 27.
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The second, third or higher power of a number is the result of multiplying two, three or more of that number together. Roots are the opposite of powers.
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Powers of numbers, or exponents as they are also known, grow really fast (‘exponentially’). For example, while 37 is just over 2,000, 312 is already over half a million. A ‘googol’ or 10100, is massively more than the estimated number of atoms in the universe – a mere 1080.
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John McDermott
We get a square number as the result when we multiply a number by itself.
