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Mathematics

Lesson 1: Introduction to Integers

Quick check

Which of the following statement(s) is/are correct?

  1. 25 is the same as 5 • 5
  2. 7 to the 5th power = 75
  3. ten squared = 103
  4. 52 = 5 ∙ 2

What do the directions imply?

Download: Practice Worksheet

Examples

Answer true or false. If false give a counterexample.

  1. All natural numbers are whole numbers.
  2. All whole numbers are natural numbers.
  3. Zero is an even number.

Let's review some definitions. A counter example means to state a number that will show the statement is false.

Defineition and Review

Fill in the blak to make a true statement:
8 + ___ = 8

We then went to school and as early as first grade we were given a question like this. This is really the first step to algebra solving for the unknown, using common sense. Why is it hard to do this problem? This is when math anxiety occurs. There is no natural number that solves this. We now have the need for other numbers. What number do we need?

Whole Numbers

Def: A number is even if it's divisible by 2. (when divided by 2 the remainder is 0)

Def: A number is odd if it is not even.

Review:
N = {1,2,3,...} W = {0,1,2,...}

How will this information help answer the true and false questions?

Answers

True, all natural numbers are whole numbers.
{1,2,3,...} are also in {0,1,2,...}

False, not all whole numbers are natural.
Not every number in {0,1,2,...} are in {1,2,3,...}
Counterexample:
0 (0 is a whole number, but not a natural number)

True, 0 is even
0 / 2 = 0, remainder 0

Examples

Write a number that would show each situation:

  1. A profit of $15
  2. Sea level
  3. A $20 decrease in price

What are we in need of here?

Integers

Name of SetExampleSymbol
Integers{...,-3,-2,-1,0,1,2,3,...}Z

Numbers less than 0 are negative, numbers greater than 0 are positive.

Integers on a number line

Summary of the Sets of Numbers

Name of SetExampleSymbol
Natural Numbers{1,2,3,...}N
Whole Numbers{0,1,2,...}W
Integers{...,-3,-2,-1,0,1,2,3,...}Z

FYI

Notice: Integers are made up of the whole numbers and their opposites.

To indicate a positive 5, we do not have to write the sign:
+5 or 5

To indicate a negative 5, we need the negative sign: -5

0 is neither positive nor negative so it does not have a sign.

Examples

Write a number that would show each situation:

  1. A profit of $15
    15 or +15
  2. Sea Level
    0
  3. A $20 decrease in price
    -20

Let's now answer these questions.

Example

Name the integer that is represented by each of the letters on the number line.

Number line

Examples

For each integer below:

  1. Indicate how many units the given number is from zero on the real number line.
  2. Indicate if the number is to the left or right of zero.
  1. +234
  2. -11
  3. 9

Examples

Graph each on the same real number line:

  1. +3
  2. -4
  3. 0
  4. 2

Number line

Is 0 already graphed? Which number is the smallest? Biggest? explain.

Examples

Replace each ? with < or > to make a true comparison:

  1. -7 ? 5
  2. -4 ? -5
  3. 0 ? 3
  4. -4 ? 0

Examples

Tell which of the numbers in each set is between the others, then write the numbers in order, using <

  1. -4,3,0
    0 is between -4 and 3
    -4 < 0 < 3
    you can also say -4 is less than 0 and 0 is less than 3

You try these:

  1. 4,10,8
  2. 5,-6,2

Draw the number line to make this easier.

Examples

Tell whether the following is true or false:

  1. -6 > -11
  2. 4 < -9
  3. -8 > -20
  4. -5 < 2 < 5

Notation

The opposite of 3 = -3
-(3) = -3

The opposite of -4 = 4
-(-4) = 4

A negative in front of a ( ) means opposite of.

Every real number has an opposite, even zero.

The opposite of zero is zero.

-means: Subtraction, negative, and opposite of.

Example

If you drive your car 10 miles in reverse, how many miles did you drive your car?

There is a mathematical rule that only cares about distance, not direction.

Absolute Value

Definition: The absolute value of a number is the distance the number is from zero on the number line.

Example:

In words: What is the absolute value of 56?

In Symbols: Find |56|
Ans: 56

FYI

|56| = 56 ← because 56 is 56 units from 0 on the real number line

Notice:

  • The absolute value symbol is | |
  • The absolute value of a number is not the same as the opposite of a number.

Example

Evaluate each:

  1. |-7|
  2. |0|

Example

Evaluate each:

  1. |-7|
    Ans: 7
  2. |0|
    Ans: 0

Note: The absolute value of a number is never negative, but it's not always positive.

See ex. 2, above.

Examples

Put the symbols into words:

  1. -(-4)
  2. -|4|
  3. -|-4|

Examples

Put the symbols into words:

  1. -(-4)
    The opposite of -4
    4
  2. -|4|
    The opposite of the absolute value of 4
    -4
  3. -|-4|
    The opposite of the abolute value of -4
    -4

FYI: To evaluate an absolute value expression – bring down the operation symbol in front of the absolute value, then evaluate.

Ex. Find -|-65|
- 65
Ans: -65

Download: Practice Worksheet Answer Key

Assessment: Introduction to Integers

  1. Represent the quantity by an integer. 45 degrees above zero.
  2. Represent the quantity by an integer. A climb of 118 feet down into a subterranean cave.
  3. Graph the numbers on the number line. 0, 2, 4
    empty number line
  4. Insert < or > to make the statement true. 47 ___ -97
  5. Insert < or > to make the statement true. -45 ___ -65
  6. Simplify. |11|
  7. Simplify. |-7|
  8. Find the opposite of the integer. -40
  9. Find the opposite of the integer. 0
  10. Simplify. -|-13|
  11. Simplify. -|5|
  12. Simplify. -(-16)
  13. Insert <, >, or = between the pair of number to make a true statement. |-30| ___ -(-30)
  14. Insert <, >, or = between the pair of number to make a true statement. -|8| ___ -|-59|
  15. Evaluate. -(-|9|)
  16. Evaluate. -(-|-(-6)|)
  17. Determine whether the statement is true or false. The absolute value of zero is a positive number.
  18. Determine whether the statement is true or false. A positive number is always greater than a negative number.