1. 𝐏𝐓𝐒 𝟑
Bridge to Calculus Workshop
Summer 2020
Lesson 1
The Real Number
System
“Mathematics is the language in
which the gods speak to people.“
- Plato -

2. Lehman College, Department of Mathematics
Definition of Real Numbers (1 of 4)
Let us review the types of numbers that make up the
real number system. We start with the natural numbers:
The integers consist of the natural numbers together
with their negatives and the number zero:
The rational numbers are constructed by taking ratios of
integers:
Where 𝑚 and 𝑛 are integers and 𝑛 ≠ 0. Division by zero
is undefined. Examples of rational numbers:
1
2
, −
5
3
, 0.17.
𝟏, 𝟐, 𝟑, 𝟒, 𝟓, …
… , −𝟑, −𝟐, −𝟏, 𝟎, 𝟏, 𝟐, 𝟑, …
𝒓 =
𝒎
𝒏

3. Lehman College, Department of Mathematics
Definition of Real Numbers (2 of 4)
Every integer is a rational number, for example:
There exist real numbers that cannot be written as the
ratio of two integers. These are the irrational numbers:
The set of real numbers is designated by the symbol: ℝ
Every real number has a decimal representation. If the
number is rational, then the corresponding decimal
representation is repeating. For example:
𝟐 =
𝟐
𝟏
𝟐, 𝝅, 𝒆
𝝅 𝟐
𝟐
,
𝟏
𝟐
= 𝟎. 𝟓𝟎𝟎 … =
𝟐
𝟑
= 𝟎. 𝟔𝟔𝟔 … =𝟎. 𝟓, 𝟎. 𝟔

4. Lehman College, Department of Mathematics
Definition of Real Numbers (3 of 4)
The bar in 𝟎. 𝟔 indicates that the sequence of digits
repeats forever. For example:
If the number is irrational, the decimal representation is
nonrepeating:
If we stop the decimal expansion of any number at a
certain place, we get an approximation to the number.
For instance, we can write:
Where the symbol ≈ is read “is approximately equal to”
𝟐𝟏
𝟐𝟐
= 𝟎. 𝟗𝟓𝟒𝟓𝟒𝟓 … = 𝟎. 𝟗𝟓𝟒𝟓
𝟐 = 𝟏. 𝟒𝟏𝟒𝟐𝟏𝟑𝟓𝟔𝟐𝟑𝟕 … , 𝝅 = 𝟑. 𝟏𝟒𝟏𝟓𝟗𝟐𝟔𝟓𝟑𝟓𝟖 …
𝝅 ≈ 𝟑. 𝟏𝟒𝟐

5. Lehman College, Department of Mathematics
Definition of Real Numbers (4 of 4)
ℕ ⊆ ℤ ⊆ ℚ ⊆ ℝ ⊆ ℂ

6. Lehman College, Department of Mathematics
Properties of Real Numbers (1 of 9)
We know that 2 + 3 = 3 + 2, and 7 + 5 = 5 + 7, etc. In
general, we write this as:
Where 𝑎 and 𝑏 are any real numbers. This is called the
commutative property of addition. Similarly, 3 ⋅ 2 = 2 ⋅ 3,
and we have the commutative property of multiplication:
We also have the associative properties which states
the order in which we add or multiply three numbers:
𝒂 + 𝒃 = 𝒃 + 𝒂
𝒂𝒃 = 𝒃𝒂
𝒂 + 𝒃 + 𝒄 = 𝒂 + 𝒃 + 𝒄
𝒂 𝒃𝒄 = 𝒂𝒃 𝒄

7. Lehman College, Department of Mathematics
Properties of Real Numbers (2 of 9)
An important property of the number zero is given by
the following example:
In general, for any real number 𝑎, we have:
Another important property of zero:
In general, we have:
Last, the cancellation properties: 𝑥 + 3 = 5 + 3, so 𝑥 = 5
𝒂 + 𝟎 = 𝟎 + 𝒂 = 𝒂
𝟑 + 𝟎 = 𝟎 + 𝟑 = 𝟑
𝟑 ⋅ 𝟎 = 𝟎 ⋅ 𝟑 = 𝟎
𝒂 ⋅ 𝟎 = 𝟎 ⋅ 𝒂 = 𝟎
𝒂 + 𝒄 = 𝒃 + 𝒄 𝒂 = 𝒃If then

8. Lehman College, Department of Mathematics
Properties of Real Numbers (3 of 9)
Now, 2 + −2 = 0, and −3 + 3 = 0. So, in general:
is true for any real number 𝑎. Let us now define the
operation of subtraction. Now 8 = 3 + 5. If we add −5:
So 8 + −5 = 3. We define the operation 𝑎 − 𝑏 as:
for any real numbers 𝑎 and 𝑏. Let us look how to
combine real numbers involving negatives.
𝒂 + −𝒂 = −𝒂 + 𝒂 = 𝟎
𝟖 + (−𝟓) = 𝟑 + 𝟓 + (−𝟓)
𝟖 + (−𝟓) = 𝟑 + 𝟎 = 𝟑
𝒂 − 𝒃 = 𝒂 + −𝒃

9. Lehman College, Department of Mathematics
Example Rule
Properties of Real Numbers (4 of 9)
−𝟏 ⋅ 𝟓 = −𝟓
−(−𝟓) = 𝟓
−𝟏 𝒂 = −𝒂
−(−𝒂) = 𝒂
−𝟓 ⋅ 𝟕 = 𝟓 ⋅ −𝟕 =−𝟑𝟓 −𝒂 𝒃 = 𝒂 −𝒃 = −𝒂𝒃
−𝒂 (−𝒃) = 𝒂𝒃−𝟑 (−𝟒) = 𝟏𝟐
−(𝟐 + 𝟑) = −𝟓 − 𝒂 + 𝒃 = −𝒂 − 𝒃
−(𝟓 − 𝟕) = 𝟕 − 𝟓 − 𝒂 − 𝒃 = 𝒃 − 𝒂
−𝟐 − 𝟑 =

10. Lehman College, Department of Mathematics
Penrose Triangle (1 of 1)

11. Lehman College, Department of Mathematics
Properties of Real Numbers (5 of 9)
We can prove all the previous statements using the
distributive property of multiplication over addition.
Example 1. Perform the following algebraic operation:
Solution. Using PEMDAS, we obtain:
However, by the distributive property:
Distributive Property. Let 𝑎, 𝑏, 𝑐 be real numbers, then
Example 2. Show that the following is true (FOIL):
6 ⋅ 2 + 3
6 ⋅ 5 = 30
6 ⋅ 2 + 3 = 6 ⋅ 2 + 6 ⋅ 3 = 12 + 18 = 30
𝑎 𝑏 + 𝑐 = 𝑎𝑏 + 𝑎𝑐 and 𝑎 + 𝑏 𝑐 = 𝑎𝑐 + 𝑏𝑐
𝑎 + 𝑏 𝑐 + 𝑑 = 𝑎𝑐 + 𝑎𝑑 + 𝑏𝑐 + 𝑏𝑑

12. Lehman College, Department of Mathematics
Properties of Real Numbers (6 of 9)
Solution. By the distributive property:
Example 9. Prove the difference of two squares rule:
Solution. By the distributive property:
Exercise 1. Prove the square of a binomial rule:
𝑎 + 𝑏 𝑐 + 𝑑 =
= 𝑎𝑐 + 𝑎𝑑 +
𝑎 𝑐 + 𝑑 + 𝑏 𝑐 + 𝑑
𝑏𝑐 + 𝑏𝑑
𝑎 + 𝑏 𝑎 − 𝑏 = 𝑎2
− 𝑏2
𝑎 + 𝑏 𝑎 − 𝑏 = 𝑎 𝑎 − 𝑏 + 𝑏 𝑎 − 𝑏
= 𝑎2 − 𝑎𝑏 + 𝑏𝑎 − 𝑏2 = 𝑎2 −𝑏2
𝑎 + 𝑏 2 = 𝑎2 + 2𝑎𝑏 + 𝑏2
= 𝑎2
+ 𝑎(−𝑏) + 𝑏𝑎 + 𝑏(−𝑏)

13. Lehman College, Department of Mathematics
Properties of Real Numbers (7 of 9)
Example 11. Perform the following operations:
Solution.
Example 12. Use the square of the binomial rule:
to show that:
𝑥 + 2 𝑥 − 3(a) (b) 𝑥 + 4 𝑥 − 4 (c) 𝑥 + 2 2
𝑥 + 2 𝑥 − 3(a) = 𝑥 𝑥 − 3 + 2 (𝑥 − 3)
= 𝑥2
− 3𝑥 + 2𝑥 − 6
= 𝑥2
− 𝑥 − 6
(b) 𝑥 + 4 𝑥 − 4 = 𝑥2
− 42
= 𝑥2
− 16
(c) 𝑥 + 2 2
= 𝑥2
+ 2 𝑥 2 + 22
= 𝑥2
+ 4𝑥 + 4
𝑎 + 𝑏 2
= 𝑎2
+ 2𝑎𝑏 + 𝑏2
𝑎 − 𝑏 2
= 𝑎2
− 2𝑎𝑏 + 𝑏2

14. Lehman College, Department of Mathematics
Properties of Real Numbers (8 of 9)
Solution. Given the rule below:
Replace 𝑏 by −𝑏 in both sides of the equation to yield:
Example 12. Use the square of binomial rule to prove
the cube of a binomial rule:
Solution.
𝑎 + 𝑏 2
= 𝑎2
+ 2𝑎𝑏 + 𝑏2
𝑎 − 𝑏 2
=
= 𝑎2 − 2𝑎𝑏 + 𝑏2
𝑎 + (−𝑏) 2
= 𝑎2
+ 2𝑎(−𝑏) + 𝑏2
𝑎 + 𝑏 3 = 𝑎3 + 3𝑎2 𝑏 + 3𝑎𝑏2 + 𝑏3
𝑎 + 𝑏 3 = (𝑎 + 𝑏) 𝑎 + 𝑏 2 = (𝑎 + 𝑏) 𝑎2
+ 2𝑎𝑏 + 𝑏2
= 𝑎 𝑎2
+ 2𝑎𝑏 + 𝑏2
+ 𝑏 𝑎2
+ 2𝑎𝑏 + 𝑏2
= 𝑎3
+ 2𝑎2
𝑏 + 𝑎𝑏2
+ 𝑎2
𝑏 + 2𝑎𝑏2
+ 𝑏3

15. Lehman College, Department of Mathematics
Properties of Real Numbers (9 of 9)
Example 13. Determine the product of the following
binomial and trinomial:
Solution.
(2𝑥 + 3)(5𝑥3
− 𝑥 + 4)
2𝑥 + 3 5𝑥3
− 𝑥 + 4 =
= 2𝑥 5𝑥3
− 𝑥 + 4 + 3 (5𝑥3 − 𝑥 + 4)
= 10𝑥4
− 2𝑥2
+ 8𝑥 + 15𝑥3
− 3𝑥 + 12
= 10𝑥4
+ 15𝑥3
− 2𝑥2
+ 5𝑥 + 12