Foundations of Geometry 2.1 and 2.2

Real Numbers Whole numbers = counting numbers & 0

Integers = whole numbers and their negatives

Rational Numbers = can be written as a fraction of integers Terminating – has a finite number of digits, ends

1/8 = 0.125

Nonterminating – repeat, but never end

1/7 = 0.142857142857…. = 0.142857

Irrational Numbers – never repeat and do not endπ = 3.141592654…. √2 = 1.414213562….

Properties of Equality for Real Numbers

Reflexive a = a

Symmetric If a = b, then b = a

Transitive If a = b and b=c, then a=c

Addition and Subtraction If a = b, then a + c = b + c and a - c = b - c

Multiplication and Division If a = b, thenac = bc and if c≠0 then a/c = b/c

Substitution If a = b, then a may be substituted for b in any equation.

Number Lines Real numbers can be superimposed on a line to

provide coordinates of points.

A point is selected as the starting point, or “origin.”

Numbers to the left or the origin are negative, to the right positive.

-2 -1 0 1 2 3

A B C D

Distance on a Number Line The distance between points is found by taking the

positive difference in their coordinates

Length (distance) A(Xa) to B (Xb)

d = AB = | Xb – Xa |

Absolute value | a |

Means to take what ever is inside and make it positive

| 8 | = 8

| -8 | = 8

Segments and Bteweenness B is between A and C if it is on the same line and to

the right of one, and to the left of the other.

For any B between A and C, segment addition:

AB + BC = AC

For endpoints A and B, the Midpoint (M):

M = 1/2(A + B)

A B C

References Glencoe: Geometric Concepts and Applications

Section 2.1 and 2.2