Mathematics

See also Geometry, and Measurement: Numbers.

Numeral Systems

As every school student should know, the system that most of humankind uses for counting is called the decimal system. This means that it uses ten digits: 1, 2, 3, 4, 5, 6, 7, 8, 9 and 0. It's no coincidence that we call them digits; the reason we have ten of them in our numeral system is because we have ten fingers (or eight fingers and two thumbs, if you prefer).

If we didn't have fingers, but only two hands, it's possible that we might use a system with only two digits. This is in fact the system that computers have been designed to use in their most basic code – because a bit (the smallest unit of storage available) is either full or empty: it either contains data or it doesn't. Or, if you prefer: it's either on or off. The boffins use zero to represent no data (off), and 1 to represent data (on).

A system that has only two digits is called called a binary system.

The number of digits that a numeral system uses is known as the base of that system. For example, the decimal system has a base of 10 and the binary system has a base of 2.

You can in theory use any integer as the base of your numeral system. For example: if we had three hands, with three fingers on each, we'd probably use a nonary system – in which the number ten would be represented as 11.

Question setters (in my experience) never ask about the nonary system. Apart from the decimal system, which is obviously a special case, there are only two numeral systems that quizzers need to worry about. The following table introduces them, along with the decimal system for completeness, and also one other (in row 2) that's occasionally used in computing.

		Name		Digits used
Numeral system with a base of 2		Binary		0, 1
Numeral system with a base of 8		Octal		0, 1, 2, 3, 4, 5, 6, 7
Numeral system with a base of 10		Decimal		0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Numeral system with a base of 16		Hexadecimal		0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

The hexadecimal system is also used (as well as the binary) in computing; each hexadecimal digit represents four bits, and this makes the data easier for people to read than if presented in binary.

You could ask the question the other way round from the way I've presented it in my table, by giving the name of the system and asking how many digits there were in it. Some might question – as I did on one occasion – whether the use of the word 'digit' is correct in this context. In normal usage, 1, 2, 3 etc. are digits, but A, B, C etc. are letters. Are there 16 digits in the hexadecimal system, or are there ten digits and six letters?

Wikipedia seems to agree with me, because it refers to them as 'symbols' rather than 'digits'. According to dictionary.com, however, 'digit' is correct, because a digit can be "any of the symbols of other number systems, as 0 or 1 in the binary."

Binary numbers (selected)

Decimal		1		2		3		4		5		6		7		8		9		10
Binary		1		10		11		100		101		110		111		1000		1001		1010

It's not unusual to get asked to convert binary numbers to decimal (denary), or vice versa. For example: "What's the decimal equivalent of the binary number 1101?"

To convert from binary to decimal: take each digit in turn, starting at the right. If the rightmost digit is 1, start with 1; if it's zero, start with zero.  If the second digit from the right is 1, add 2; otherwise carry on. If the third number from the right is 1, add 4; and so on (increasing powers of two each time). Then add all the results together. For example: 1101 is 1 + 0 + 4 + 8, which is 13.

It's less common to get asked to convert from decimal to binary, but it does happen. First of all you need to work out what is the highest number that's a power of two and is less than the number in question. For example: if asked what the decimal number 13 is in binary: the highest power of 2 that you need to worry about is 8 (which is 2 to the power of 3), because the next highest is 16, which is more than 13. Write down a 1 for the 8 (in the 8 column, if you like), then subtract it from the 13; this leaves 5. If the remainder is greater than or equal to the next highest power of 2, write down another 1. Five is indeed greater than 4, so we have a second 1. When you take the 4 away from the 5 you get 1, which is less than the next highest power of 2 (two itself), so we have a zero. We're now down to the '1' column, and we still have 1 left (from when we subtracted 4 from 5), so we write down a 1. So 13 in binary is 1101 (1 for the 8, 1 for the 4, 0 for the lack of the 2, and 1 for the last 1).

One useful tip is that an odd number will always end in a 1 in binary, and an even number will always end in a zero.

It sounds complicated; and if you need to have it explained then it may well be that you wouldn't be able to do it in a quiz. But at least I can say I've got it covered!

Converting Hexadecimal numbers

It's just possible that you may be asked to convert a hexadecimal number into decimal – and I'd say that you're equally likely (i.e. not very likely) to be asked to do the reverse. If you are, the principle is exactly the same as in binary – but the good news is that you're not going to have as many digits to deal with.

The thing to remember is that in any numeral system, the number that is the base of that system is written as 10. So in binary the number two is written as 10, in decimal the number ten is written as 10, and so on. In hexadecimal, the number sixteen is written as 10.

Next, we have to deal with the numbers from ten to fifteen. Because we don't have numeric digits to represent them, we use letters: A to represent ten, B to represent eleven, and so on up to fifteen which is represented by F.

So the first sixteen numbers in the hexadecimal system are 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F and 10.

Let's look now at how we convert hexadecimal numbers of seventeen and over into decimal. Just as above, to convert from decimal to binary we divided by powers of two, in hexadecimal we divide by powers of sixteen.

The square of sixteen is 256 (decimal), so 256 is represented in the hexadecimal system as 100. (Just as 100 in decimal is the square of ten, and 100 in binary is the square of two, which is four.) It's most unlikely that you'll be asked to convert any number higher than this (unless it's something simple like decimal 257 – 101 in hexadecimal).

For numbers between 17 and 255 (inclusive), all we have to do is divide by sixteen. For example: suppose we're asked to convert the decimal number 99 into hexadecimal. 99 divided by 16 is 6, remainder 3; so decimal 99 in hexadecimal is 63.

What could be simpler?!

What could be slightly more complicated is if we were asked to convert a number that's greater than 160 into hexadecimal: for example, 199. 199 divided by 19 is 12, remainder 1; so the first digit is the hexadecimal symbol for 12, which is C. Decimal 199 in hexadecimal is C1.

Similarly, if the remainder is greater than 9, we need to convert it to the equivalent hexadecimal symbol. For example: decimal 47 divided by sixteen is two, remainder fifteen. Decimal 47 is represented as 2F in hexadecimal.

To convert hexadecimal into decimal is even easier. For example: convert hexadecimal B9 into decimal. B in hexadecimal represents eleven, so we have eleven times sixteen, plus nine: 176 + 9 = 185.

Fractions

		One half		One quarter		One eighth
Binary		0.1		0.01		0.001
Octal		0.4		0.2		0.1
Hexadecimal		0.8		0.4		0.2

The square root of a quarter

One half

General

The lowest prime number, and the only even one		2
The three lowest perfect numbers		6
		28
		496
Sequence of numbers where each number is got by adding a constant to the previous number (cf. Geometric progression)		Arithmetic progression
A basic principle, assumed to be true without proof		Axiom
Form of algebra where variables represent logical statements and relationships, rather than numbers; named after its inventor, a 19th century British mathematician		Boolean
Values of the variables are the truth values true and false, usually denoted as 1 and 0 respectively
The branch of mathematics that deals with the rate of change of variable quantities; named after the Latin for a pebble or small stone		Calculus
An operation where the order of the operands does not affect the result (e.g. addition but not subraction, and multiplication but not division) is said to be ...		Commutative
A number that can be written in the form x + iy, where x and y are real numbers and i is the square root of minus one (the imaginary unit)		Complex number
Pi r squared h, where h is the height: volume of a		Cylinder
Greek letter, used (in mathematics and other sciences) as the 'difference operator' – to denote a change in any quantity		Delta (Δ)
The number below the line in a fraction – cf. Numerator, Divisor		Denominator
The process of determining the derivative of a function (i.e. its sensitivity to change)		Differentiation
The number above the line in a division		Dividend
The number below the line in a division		Divisor
The product of all the integers up to and including a particular integer – represented by that integer followed by an exclamation mark (e.g. 3! = 3 x 2 x 1 = 6)		Factorial
A series of numbers where each is the sum of the previous two (1, 1, 2, 3, 5, 8, …)		Fibonacci sequence
A rational number written as one integer over another (non–zero) integer		(Common or vulgar) fraction
Sequence of numbers where each number is got by multiplying the previous number by a constant (cf. Arithmetic progression)		Geometric progression
Most common name for the ratio between two numbers, if the ratio between the larger one (a) and the smaller one (b) is the same as that between their sum and the larger one: a/b = (a+b)/a. Represented by the Greek letter phi (Φ); value is approx. 1.618		Golden ratio
The square root of a negative number, or of zero (term coined in 1637 by Descartes and meant to be derogatory); the non–real part of a complex number		Imaginary number
The name given to the square root of –1 – represented as i		The imaginary unit
A fraction where the numerator is greater than the denominator (e.g. 3/2)		Improper fraction
The reverse of differentiation (a.k.a. antidifferentiation)		Integration
A number that can't be expressed as a fraction (ratio) – e.g. π, e, the square root of 2		Irrational number
The average value of a set of numbers		Mean
The middle number in the sequence, if all the numbers in a set are sorted into order		Median
(The sum of) a whole number and a proper fraction		Mixed number
The number that occurs most frequently in a set		Mode
The number above the line in a fraction – cf. Denominator, Dividend		Numerator
Raising any number to the power zero, gives a value of		One
A quantity on which an operation is performed (for example, each of the two or more numbers that are added together in an addition)		Operand
1, 2 and 3 are cardinal numbers; 1st, 2nd and 3rd are		Ordinal numbers
Common name for a triangle representing all the binomial coefficients		Pascal's triangle
A number that's the sum of all its factors other than itself (e.g. 6 = 1 + 2 + 3, 28 = 1 + 2 + 4 + 7 + 14)		Perfect number
An expression consisting of variables and coefficients, involving only the operations of addition, subtraction and multiplication; for example, ax2 + bx + c – where x is the variable and a and b are the coefficients; c is a constant		Polynomial
The Sieve of Eratosthenes is an ancient method of finding		Prime numbers
The result of a multiplication		Product
A fraction where the numerator is less than the denominator (e.g. 2/3)		Proper fraction
"ax2 + bx + c = y" (where a, b, c and y are known numbers, and x is an unknown number) is the generalised form of a		Quadratic equation
The result of a division		Quotient
A number that can be expressed as a fraction (e.g. 2 = 4/2, 0.345 = 345/1,000)		Rational number
A number that is in one–to–one correspondence with a point on an infinite "number line" (a retronym, coined in response to "imaginary number")		Real number
The result of dividing one by a number – in other words, the value by which a number must be multiplied to make one (also known as the multiplicative inverse)		Reciprocal
Standard deviation of a Poisson distribution whose mean is n		Root n
Two equations applying to the same set of variables		Simultaneous equations
A set with only one member (a term also used in Bridge, and other card games, for a card that's the only one in a suit)		Singleton
The square root of the variance of a distribution – the root mean square (RMS) of the variation from the average. Commonly used in preference to variance because the variance is a square value (e.g. when measuring lengths in metres, the variance is in square metres)		Standard deviation
Three dots in an upright triangular formation (i.e. with two dots at the base and one above) is a symbol meaning		Therefore
The study of those properties of geometric forms that do not change under certain transformations, such as bending or stretching		Topology
Standard measure of the statistical distribution of a random variable – a measure of how spread out the values are; calculated as the mean value of the squares of the deviations – cf. Standard deviation		Variance
A diagrammatic representation of sets by means of overlapping circles		Venn diagram

© Haydn Thompson 2017–23