This material is based upon work supported by the National Science Foundation under Grant No

This material is based upon work supported by the National Science Foundation under Grant No. DUE-0336493

STEP- Science, Technology, Engineering, and Mathematics Talent Expansion Program

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I. Know the language--SYMBOLS AND SETS

The Number Line, Graphing Numbers on the Number Line, Order Relation, Sets and Set Notation, The Negative Symbol & the Opposite, Absolute Value, Grouping Symbols, Order of Operations Agreement, Operations With Real Numbers

THE NUMBER LINE

-2 -1 0 1 2

The number line is a visual representation of the set of Real Numbers. Positive numbers lie to the right of zero and negative numbers lie to the left. Zero is neither positive nor negative. The fractions (which include the decimal numbers) lie between the whole numbers. Not every number can be labeled on the number line. It is helpful to make the distance between the numbers (the scale) uniform. For example, since 1 is the same distance from 0 as it is from 2, it should be located mid way between 0 and 2 on the number line.

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GRAPHING NUMBERS ON THE NUMBER LINE

-2 -1 0 1 2

The number paired with a point on the number line is called the coordinate of that point on the line.

The point paired with a number is called the graph of that number.

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ORDER RELATION ON THE NUMBER LINE

The order of points on the number line is the order of the numbers. Larger numbers are located to the right of smaller numbers on the number line. Therefore, –1 is larger than –2 because –1 lies to the right of –2 on the number line.

The mathematical symbols for order relation are as follows:

Greater than: >; Greater than or equals to: >; Less than: <; Less than or equals to: <

The following statements are true: –2 < –1; –1 > –2; –2 < –1; –1 > –2

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SETS AND SET NOTATION

A set is a collection of objects. The objects in a set are called elements or members and are written between set braces. For example, the set containing the whole numbers between 0 and 6 is written as: {1,2,3,4,5}. Zero and 6 are not listed because the description of the set does not include these numbers.

Sets can have a definite number of elements or they can have an infinite number of elements. For example, the set of all whole numbers has an infinite number of elements. That set can be written as: {0,1,2,3,4,5...}. The three dots indicate that the roster continues without coming to an end and is called an infinite set.

A finite set is a set in which the elements can be counted.

Using a list is not the only way to describe a set. It is not possible to list all the elements of the infinite set: {all fractions between 0 and 1}. The finite set: {all even numbers between 2 and 12} could also be described using the following list: {4,6,8,10}. Two and twelve are not listed because those numbers are not between 2 and 12.

A set can contain no elements and is called the empty set or the null set. The empty set is designated using braces without any elements in between them { } or with a special symbol Ø, written without braces.

The set {0} is not the empty set. This set is the set containing 0 as its only element.

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THE NEGATIVE SYMBOL & THE OPPOSITE

There are several names for the negative symbol. Other names include minus, subtract and opposite. The negative symbol is a separate symbol from the number. A negative sign in front of a number means that number is found to the left of zero on the number line and is called negative. Two negative signs in front of a number mean the opposite of a negative. Since the opposite of negative is positive, the opposite of a negative number is a positive number.

For example: –(–) = + 2.

A positive number may or may not be written with a + sign before the number. A negative number must have the – symbol in front of the number..

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ABSOLUTE VALUE

The symbol for absolute value is two parallel vertical bars which contain a numerical or algebraic expression.

| 2 | is read the absolute value of 2.

| –2 | is read the absolute value of negative 2.

Absolute value is the distance from zero to the expression between the vertical bars on the number line. Since distance is always positive, absolute value will always be positive.

For example: | 2 | = 2 and | –2 | = 2

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GROUPING SYMBOLS

Grouping symbols are used to indicate the order in which the operations should be performed in a math problem. Grouping symbols include: parenthesis ( ), brackets [ ], braces { }, and the fraction bar.

For example: 6 + (3 × 5) indicates that the multiplication is completed before the addition.

When grouping symbols are absent, operations are performed according to the Order of Operations Agreement.

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ORDER OF OPERATIONS AGREEMENT

The Order of Operations Agreement contains four steps:

1--Perform all operations within parenthesis or other grouping symbols,

2--Simplify exponents,

3--Do all multiplications and divisions in the order in which they occur, working from left to right,

4--Do all additions and subtractions in the order in which they occur, working from left to right.

For example: 8 • 4 – 3(5 + 2) = 8 • 4 – 3(7) = 32 – 21 = 11

Work the problem inside the parenthesis first, then complete the multiplication as it occurs from left to right, then complete the addition from left to right.

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OPERATIONS WITH REAL NUMBERS

ADDITION AND SUBTRACTION

To add two numbers on the number line, start at the position of the first number, then move a distance equal to the second number in the direction associated with the sign of the second number. Move left if the second number is negative. Move right if the second number is positive. The number at the resulting position is the required sum.

3 + 4 = 7 and 3 + –4 = –1 are illustrated below.

The steps for addition of two number with like signs are as follows:

Add the numbers.
Keep their sign.

For example: 6 + 9 = 15 and –6 + –9 = –15

If both numbers are positive, the answer is positive. If both numbers are negative, the answer is negative.

The steps for addition of two number with unlike signs are as follows:

Subtract the numbers.
Attach the sign of the number with the larger absolute value to the difference.

For example: –6 + 9 = 3 and 6 + –9 = –3

Problems that look like subtraction are addition problems with negative numbers.

Use the rules for addition of unlike signs.

For example: 6 – 9 = – 3 because 6 – 9 is the same as 6 + – 9

For example: 6 – –9 = 6 + 9 = 15, because the opposite of –9 is +9.

MULTIPLICATION AND DIVISION

Because division can be defined as multiplication by the reciprocal of the divisor, multiplication and division problems have the same rules.

When multiplying or dividing two numbers with the same signs, the answer is always positive.

For example: 6 • 9 = 54, –6 • –9 = 54, and

When multiplying or dividing two numbers with unlike signs, the answer is always negative.

For example: –6 • 9 = –54, 6 • –9 = –54, and

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