Chapter 3: Real Numbers and the Coordinate Plane

Square Roots

The square root of a number is another number that when multiplied by itself is equal to the original number

I know , so the 39 933

Square Root

If a number has a square root, it has two

933933

SO…

39

39

Square roots

Remember: positive x positive = positivenegative x negative = negative

Therefore…

9 This is not a real number!(It is imaginary, but we are not getting into that)

Perfect Squares

When the square root comes out as a whole number it is a perfect square

6,636

5,525

4,416

3,39

2,24

1,11

12,12144

11,11121

10,10100

9,981

8,864

7,749

Use Perfect Squares to Estimate Other Square Roots

14 169

3 4...7.3NOTE: This number is irrational, it can not be written as a fraction and will not stop or repeat, just like Pi. This is true for all non-perfect squares.

Rational Vs. Irrational

• Remember rational numbers can be written as a fraction– Repeating decimals – Integers (positive and negative whole numbers)– Fractions– Terminating decimals (the decimal stops)

• Irrational– Square roots that are not perfect squares– Decimals that don’t stop or repeat

Square roots in Algebra

• The square root is the opposite of squaring a number

• Just like division undoes multiplication, the square root will undo squaring a number

• Remember to do order of operations backwards (easy operations first)

Square Roots in Algebra

t

t

t

t

t

51

51

5116

1616816

16816

2

2

2

2

Opposite of multiplying by 16 is dividing by 16

Calculate 816/16=51

Square root and squaring cancel out

Homework

• Page 109 9-49 odd

The Pythagorean Theorem

It is a formula! 𝑎2+𝑏2=𝑐2

hypotenuseclegblega

a

b

c

NOTE: The hypotenuse is always the longest side, and the side opposite the right angleNOTE: this only works for right

triangles

Using the Formula

• Just like every other formula, copy it down and list your variables

5cm

10cm

c

?105

ccmbcma

𝑎2+𝑏2=𝑐2

Substitute into the Formula

𝑎2+𝑏2=𝑐2

?105

ccmbcma

52+102=𝑐2

Solve the Formula

c

c

c

c

c

125

125

125

10025

105

2

2

2

222

cm125 Don’t forget your units!

Homework

• Page 114 3-24

Pythagorean theorem with unknown Leg

Variables (a) and (b) are interchangeable, But if you are given the hypotenuse and a leg, you need to find the other leg with algebra

9cm

3cm

x

9?3

222

cba

cba 222 93 x

Substitute into the formula

Solving with algebra

cmx

x

x

x

x

72

72

72

9 9819

93

2

2

2

222

Opposite of adding 9 is subtracting 9

Don’t forget units!

Tips

• Give yourself plenty of space to work• Draw pictures to illustrate situations from

word problems• If the answer is not a perfect square, leave it

under the radical sign ( )

Homework

• Page 120 3-21

Graphing in the Coordinate Plane

• Ordered Pairs: give the location of a point on my plane

3,2 The first number is the x coordinate

The second number is the y coordinate

Pythagorean theorem with the Coordinate Plane

𝑎2+𝑏2=𝑐2

?units 3units 7

cba

units 58

37 222

c

c

See notes on Pythagorean Theorem for details on solving

Homework

• Page 126 11-31

Equations, Tables, and Graphs

• All three can be used to represent the same data

Suppose you save three dollars a week

Table:Number of weeks (x)

Total savings (y) expression

0 0

1

2

3

4

x

Suppose you save three dollars a week

Table:Number of weeks (x)

Total savings (y) Expression

0 0

1 3

2

3

4

x

Suppose you save three dollars a week

Table:Number of weeks (x)

Total savings (y) Expression

0 0

1 3

2 6

3

4

x

Suppose you save three dollars a week

Table:Number of weeks (x)

Total savings (y) Expression

0 0

1 3

2 6

3 9

4

x

Suppose you save three dollars a week

Table:Number of weeks (x)

Total savings (y) Expression

0 0

1 3

2 6

3 9

4 12

x

Suppose you save three dollars a week

Table:Number of weeks (x)

Total savings (y) Expression

0 0

1 3

2 6

3 9

4 12

x y

Suppose you save three dollars a week

Look for patterns. What do I do to one column to get the next?

Number of weeks (x)

Total savings (y) Expression

0 0

1 3

2 6

3 9

4 12

x y

Rule:Multiply by 3

Suppose you save three dollars a week

Look for patterns. What do I do to one column to get the next?

Number of weeks (x)

Total savings (y) Expression

0 0 3(0)

1 3 3(3)

2 6 3(2)

3 9 3(3)

4 12 3(4)

x y 3(x)

Rule:Multiply by 3

Graph ItNumber of weeks (x)

Total savings (y)

Expression Ordered Pair

0 0 3(0) (0,0)

1 3 3(3) (1,3)

2 6 3(2) (2,6)

3 9 3(3) (3,9)

4 12 3(4) (4,12)

x y 3(x) (x,y)

Plot these points on the coordinate plan and connect the dots with a straight line.

For lines you only need 2 points

Homework

• Page 133 5-18