Course Material 1.6 Absolute Value and Triangle inequality
Definition. (Absolute Value of a number)
For every real number 𝑥 the absolute value of 𝑥 is defined as
𝑥 𝑖𝑓 𝑥 ≥ 0
𝑥 =
−𝑥 𝑖𝑓 𝑥 < 0
Note.
1. 𝑥 = max 𝑥, −𝑥
2. The function 𝑓 𝑥 = 𝑥 is a non negative real valued function defined on ℝ
Theorem (properties)
For any two real numbers 𝑎 and 𝑏,
1. 𝑎𝑏 = 𝑎 𝑏
2
2. 𝑎 = 𝑎2 and 𝑎 = 𝑎2
3. 𝑎 ≤ 𝑎 and −𝑎 ≤ 𝑎
4. - 𝑎 ≤ 𝑎 ≤ 𝑎
5. 𝑥 < 𝜖 iff 𝑥 ∈ −𝜖, 𝜖
6. 𝑥 − 𝑎 < 𝜖 iff 𝑥 ∈ 𝑎 − 𝜖, 𝑎 + 𝜖
7. 𝑎+𝑏 ≤ 𝑎 + 𝑏
8. 𝑎 − 𝑏 ≥ 𝑎 − 𝑏 and 𝑎 − 𝑏 ≥ 𝑏 − 𝑎
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9. max 𝑎, 𝑏 = 2 𝑎 + 𝑏 + 𝑎 − 𝑏 and min 𝑎, 𝑏 = 2 𝑎 + 𝑏 − 𝑎 − 𝑏
Proof.
7. The inequality 𝑎 + 𝑏 ≤ 𝑎 + 𝑏 is usually called the triangle inequality. To prove
the triangle inequality,
We have 𝑎 ≤ 𝑎 and 𝑏 ≤ 𝑏 so that 𝑎+𝑏 ≤ 𝑎 + 𝑏
Also we have −𝑎 ≤ 𝑎 and −𝑏 ≤ 𝑏 so that – 𝑎 − 𝑏 ≤ 𝑎 + 𝑏
But 𝑎 + 𝑏 = max 𝑎 + 𝑏, −𝑎 − 𝑏 ≤ 𝑎 + 𝑏 This proves the triangle inequality.
Definition. (Absolute Value of a number)
For every real number 𝑥 the absolute value of 𝑥 is defined as
𝑥 𝑖𝑓 𝑥 ≥ 0
𝑥 =
−𝑥 𝑖𝑓 𝑥 < 0
Note.
1. 𝑥 = max 𝑥, −𝑥
2. The function 𝑓 𝑥 = 𝑥 is a non negative real valued function defined on ℝ
Theorem (properties)
For any two real numbers 𝑎 and 𝑏,
1. 𝑎𝑏 = 𝑎 𝑏
2
2. 𝑎 = 𝑎2 and 𝑎 = 𝑎2
3. 𝑎 ≤ 𝑎 and −𝑎 ≤ 𝑎
4. - 𝑎 ≤ 𝑎 ≤ 𝑎
5. 𝑥 < 𝜖 iff 𝑥 ∈ −𝜖, 𝜖
6. 𝑥 − 𝑎 < 𝜖 iff 𝑥 ∈ 𝑎 − 𝜖, 𝑎 + 𝜖
7. 𝑎+𝑏 ≤ 𝑎 + 𝑏
8. 𝑎 − 𝑏 ≥ 𝑎 − 𝑏 and 𝑎 − 𝑏 ≥ 𝑏 − 𝑎
1 1
9. max 𝑎, 𝑏 = 2 𝑎 + 𝑏 + 𝑎 − 𝑏 and min 𝑎, 𝑏 = 2 𝑎 + 𝑏 − 𝑎 − 𝑏
Proof.
7. The inequality 𝑎 + 𝑏 ≤ 𝑎 + 𝑏 is usually called the triangle inequality. To prove
the triangle inequality,
We have 𝑎 ≤ 𝑎 and 𝑏 ≤ 𝑏 so that 𝑎+𝑏 ≤ 𝑎 + 𝑏
Also we have −𝑎 ≤ 𝑎 and −𝑏 ≤ 𝑏 so that – 𝑎 − 𝑏 ≤ 𝑎 + 𝑏
But 𝑎 + 𝑏 = max 𝑎 + 𝑏, −𝑎 − 𝑏 ≤ 𝑎 + 𝑏 This proves the triangle inequality.
