# The A to Z of mathematics a basic guide - Sidebotham T.H.

ISBN 0-471-15045-2

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References: Finite, Line, Line Segment.

INFINITE DECIMALS

These are decimals that have no end to the number of decimal places. There are two kinds of infinite decimals:

1. Recurring decimals, like | = 0.33333333..., which goes on forever. We usually exclude decimals with recurring zeros, because they are regarded as terminating decimals. For example, 0.3 could be written as 0.300000000..., but it is not normally regarded as a recurring decimal.

2. Irrational numbers, like it = 3.14159265..., which go on forever, but there is no recurring pattern to the numbers.

References: Decimal, Finite Decimals, Irrational Numbers, Pi, Recurring Decimal. INFINITY

This is the name given to a value that is infinite, too large to be counted, and cannot be calculated. The symbol for infinity is oo.

References: Infinite.

INFLECTION, POINT OF

Reference: Concave.

INRADIUS

Reference: Incircle.

INSCRIBE

References: Circumscribe, Incenter.

254 INTEGERS

INSTALLMENT

Reference: Hire Purchase.

INTEGERS

The following numbers are explained under this entry:

♦ Natural numbers

♦ Whole numbers

♦ Integers

♦ Rational numbers

♦ Irrational numbers

♦ Real numbers

Natural Numbers The symbol for the natural numbers is N. Another name for them is counting numbers. N is an infinite set of numbers; some of them are listed here:

N = {1,2, 3, 4, 5,...}

For example, natural numbers are used to count the chapters in a book. There is not a zero chapter.

Whole Numbers The symbol for the whole numbers is W. This set is exactly the same as N, except that it includes zero. W is an infinite set of numbers; some of them are listed here:

W = {0, 1, 2, 3, 4, 5,...}

For example, suppose you were recording the number of people in a room. There could be one, two, three, and so on, but there could also be no people in the room. Whole numbers would be required for this situation.

Integers The symbol for the integers is Z or i. It is an infinite set of numbers; some of them are listed here:

! = {•••, —3, —2, —1, 0, 1,2, 3,...}

These numbers can be described as positive and negative whole numbers, and are sometimes called directed numbers. For each positive number there is an equal and opposite negative number. In real life, directed numbers are used to show that some quantities can fall below zero. A temperature of 10 degrees below zero is written as —10 degrees, and a temperature of 5 degrees above zero is written as +5 degrees, or more simply as 5 degrees.

INTEGERS 255

Rational Numbers The symbol for the rational numbers is Q, from the word quotient, which means the result of dividing one number by another. A rational number can be expressed as a quotient of two integers, with the denominator not zero. Q is an infinite set of numbers. In addition to containing all the integers, it also includes all the positive and negative fractions. Some examples of rational numbers are listed here:

Q = {...,-4, If,-2,-2, 0, 1, 2, 3, i, 100,...)

It is William’s birthday and he has three cakes. He is cutting up the cakes into 20 equal pieces, so each piece is 3/20 of the total number of cakes. This is an example where rational numbers are needed.

Irrational Numbers The symbol for the irrational numbers is Q;, which means the complement of Q, and is all those numbers that are not rational numbers, but are real numbers. Some examples of this infinite set of irrational numbers are listed here:

{.%, and surds like-v/2, V7, VlO, V34}

For example, if the two equal sides of a right-angled isosceles triangle are of length 1 unit then the hypotenuse is of length -s/2 units. This is an irrational number.

Real Numbers The symbol for the real numbers is R; this is an infinite set of numbers. This is the family name given to all the numbers, both rational and irrational. The family members of real numbers are related to each other in the following way: The natural numbers are a subset of the whole numbers. The whole numbers are a subset of the integers. The integers are a subset of the rational numbers. The rational and irrational numbers are both subsets of the real numbers. This information is put neatly in the number tree drawn here in figure a.

Real numbers (R)

Irrational (Q') Rational (Q)

Integers (I)

Whole (W)

Natural (N)

(a)

The rules for adding, subtracting, multiplying, and dividing integers are now explained.

256 INTEGERS

Add ing Two I nteg ers Counting “steps” on a number line may used to add integers. If the integer is positive, we count steps to the right, and if it is negative, we count steps to the left.

Example. Work out the answer to ~3 + +2.

M 1 1 1 1 1 1 1 »

-3-2-10123

(b)

Solution. Draw a number line (see figure b), or imagine one in your head. Always start at zero and take 3 steps to the left, to represent “3. Then turn and take 2 steps to the right, to represent +2. Where you end up is the answer, which is _ 1:

“3 + +2 = “I

Subtracting Two Integers We can transform the problem from a subtraction into an addition problem by “adding the opposite integer,” then doing the addition in the usual way.

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