Comprehensive th Grade Mathematics 8
Study Guide
Master Every Topic with Deep Explanations and Examples
Table of Contents
1. The Number System
2. Exponents and Radicals
3. Linear Equations and Inequalities
4. Systems of Linear Equations
5. Functions and Relations
6. Geometry: Transformations and Congruence
7. The Pythagorean Theorem
8. Volume of 3D Shapes
9. Statistics and Data Analysis
10. Probability
1. The Number System
1.1 Understanding Rational Numbers
What are Rational Numbers?
A rational number is any number that can be expressed as a fraction in the form pq , where:
p is an integer (the numerator)
q is an integer (the denominator)
q=
0 (we cannot divide by zero)
Examples of Rational Numbers:
, Type Examples Explanation
−5 0 3
Integers −5, 0, 3, 100 Can be written as 1 , 1, 1
1 3 7
Positive Fractions 2, 4, 8
Already in fraction form
Negative Fractions − 23 , − 56
Negative numerator or denominator
Terminating Decimals 0.5, 0.25, 1.75 0.5 = 12 , 0.25 = 14 , 1.75 =
7
4
Repeating Decimals 0.3, 0.6, 0.16 0.3 = 0.333... = 13 , 0.6 =
2
3
Why These Are Rational:
The key characteristic is that rational numbers either:
1. Terminate (stop) after a certain number of decimal places, OR
2. Have a repeating pattern in their decimal expansion
Converting Repeating Decimals to Fractions:
Let’s convert 0.3 = 0.333... to a fraction.
Let x = 0.333...
Multiply both sides by 10: $10x = 3.333...$
Subtract the original equation from this new equation: $10x − x = 3.333... − 0.333...9x = 3
x = 39 = 13 $
Another Example: Convert 0.16 = 0.1666... to a fraction.
Let x = 0.1666...
Multiply by 10: 10x = 1.666...
Multiply by 100: 100x = 16.666...
Subtract: 100x − 10x = 16.666... − 1.666...
90x = 15
15 1
x= =
90 6
1.2 Understanding Irrational Numbers
What are Irrational Numbers?
,An irrational number is a number that CANNOT be expressed as a simple fraction pq . Their
decimal representations:
Never terminate (don’t stop)
Never repeat (no pattern)
Go on forever without repeating
Common Irrational Numbers:
Number Decimal Why It’s Irrational
Approximation
π 3.14159265358979... Represents the ratio of a circle’s circumference to
diameter; no repeating pattern
e 2.71828182845904... Euler’s number; fundamental in mathematics and
science
2 1.41421356237309... The square root of 2; no perfect square equals 2
3 1.73205080756887... The square root of 3; no perfect square equals 3
ϕ (Golden 1.61803398874989... 1+ 5
2
;
appears in nature and art
Ratio)
Why 2 is Irrational (Proof by Contradiction):
Assume 2 is rational, so
2=
p
q
where p and q are integers in lowest terms (no common
factors).
p2 2
Square both sides: $2 = q 2 2q
= p2 $
This means p2 is even, so p must be even. Let p = 2k for some integer k .
Substitute: $2q 2 = (2k)2 = 4k 2 q 2 = 2k 2 $
This means q 2 is even, so q must be even.
But if both p and q are even, they share a common factor of 2, which contradicts our assumption
that they’re in lowest terms. Therefore, 2 cannot be rational—it must be irrational.
1.3 Approximating Irrational Numbers
Since we can’t write the complete decimal expansion of irrational numbers, we estimate them
using rational numbers.
, Method 1: Using Perfect Squares
To approximate 20:
First, identify perfect squares near 20:
42 = 16
52 = 25
Since 16 < 20 < 25, we know: $4 < 20 < 5$
To narrow it down further, test values between 4 and 5:
4.42 = 19.36 (too small)
4.52 = 20.25 (too large)
4.472 = 19.9809 (very close!)
4.4722 = 19.998784 (even closer!)
So 20 ≈ 4.472
Method 2: Using a Calculator or Estimation
For 50:
72 = 49
82 = 64
Since 49 < 50 < 64: $7 < 50 < 8$
Since 50 is very close to 49, 50 is close to 7. Testing: 7.072 = 49.9849
So 50 ≈ 7.071
Practice Problems:
1. Approximate 30 to two decimal places.
Between which two integers? 52 = 25, 62 = 36, so between 5 and 6
Test: 5.52 = 30.25, 5.482 = 30.0304
Answer: 30 ≈ 5.48
2. Approximate 75 to two decimal places.
Between which two integers? 82 = 64, 92 = 81, so between 8 and 9
Test: 8.72 = 75.69, 8.662 = 74.9956
Answer: 75 ≈ 8.66
Study Guide
Master Every Topic with Deep Explanations and Examples
Table of Contents
1. The Number System
2. Exponents and Radicals
3. Linear Equations and Inequalities
4. Systems of Linear Equations
5. Functions and Relations
6. Geometry: Transformations and Congruence
7. The Pythagorean Theorem
8. Volume of 3D Shapes
9. Statistics and Data Analysis
10. Probability
1. The Number System
1.1 Understanding Rational Numbers
What are Rational Numbers?
A rational number is any number that can be expressed as a fraction in the form pq , where:
p is an integer (the numerator)
q is an integer (the denominator)
q=
0 (we cannot divide by zero)
Examples of Rational Numbers:
, Type Examples Explanation
−5 0 3
Integers −5, 0, 3, 100 Can be written as 1 , 1, 1
1 3 7
Positive Fractions 2, 4, 8
Already in fraction form
Negative Fractions − 23 , − 56
Negative numerator or denominator
Terminating Decimals 0.5, 0.25, 1.75 0.5 = 12 , 0.25 = 14 , 1.75 =
7
4
Repeating Decimals 0.3, 0.6, 0.16 0.3 = 0.333... = 13 , 0.6 =
2
3
Why These Are Rational:
The key characteristic is that rational numbers either:
1. Terminate (stop) after a certain number of decimal places, OR
2. Have a repeating pattern in their decimal expansion
Converting Repeating Decimals to Fractions:
Let’s convert 0.3 = 0.333... to a fraction.
Let x = 0.333...
Multiply both sides by 10: $10x = 3.333...$
Subtract the original equation from this new equation: $10x − x = 3.333... − 0.333...9x = 3
x = 39 = 13 $
Another Example: Convert 0.16 = 0.1666... to a fraction.
Let x = 0.1666...
Multiply by 10: 10x = 1.666...
Multiply by 100: 100x = 16.666...
Subtract: 100x − 10x = 16.666... − 1.666...
90x = 15
15 1
x= =
90 6
1.2 Understanding Irrational Numbers
What are Irrational Numbers?
,An irrational number is a number that CANNOT be expressed as a simple fraction pq . Their
decimal representations:
Never terminate (don’t stop)
Never repeat (no pattern)
Go on forever without repeating
Common Irrational Numbers:
Number Decimal Why It’s Irrational
Approximation
π 3.14159265358979... Represents the ratio of a circle’s circumference to
diameter; no repeating pattern
e 2.71828182845904... Euler’s number; fundamental in mathematics and
science
2 1.41421356237309... The square root of 2; no perfect square equals 2
3 1.73205080756887... The square root of 3; no perfect square equals 3
ϕ (Golden 1.61803398874989... 1+ 5
2
;
appears in nature and art
Ratio)
Why 2 is Irrational (Proof by Contradiction):
Assume 2 is rational, so
2=
p
q
where p and q are integers in lowest terms (no common
factors).
p2 2
Square both sides: $2 = q 2 2q
= p2 $
This means p2 is even, so p must be even. Let p = 2k for some integer k .
Substitute: $2q 2 = (2k)2 = 4k 2 q 2 = 2k 2 $
This means q 2 is even, so q must be even.
But if both p and q are even, they share a common factor of 2, which contradicts our assumption
that they’re in lowest terms. Therefore, 2 cannot be rational—it must be irrational.
1.3 Approximating Irrational Numbers
Since we can’t write the complete decimal expansion of irrational numbers, we estimate them
using rational numbers.
, Method 1: Using Perfect Squares
To approximate 20:
First, identify perfect squares near 20:
42 = 16
52 = 25
Since 16 < 20 < 25, we know: $4 < 20 < 5$
To narrow it down further, test values between 4 and 5:
4.42 = 19.36 (too small)
4.52 = 20.25 (too large)
4.472 = 19.9809 (very close!)
4.4722 = 19.998784 (even closer!)
So 20 ≈ 4.472
Method 2: Using a Calculator or Estimation
For 50:
72 = 49
82 = 64
Since 49 < 50 < 64: $7 < 50 < 8$
Since 50 is very close to 49, 50 is close to 7. Testing: 7.072 = 49.9849
So 50 ≈ 7.071
Practice Problems:
1. Approximate 30 to two decimal places.
Between which two integers? 52 = 25, 62 = 36, so between 5 and 6
Test: 5.52 = 30.25, 5.482 = 30.0304
Answer: 30 ≈ 5.48
2. Approximate 75 to two decimal places.
Between which two integers? 82 = 64, 92 = 81, so between 8 and 9
Test: 8.72 = 75.69, 8.662 = 74.9956
Answer: 75 ≈ 8.66
