2.X

Chapter 2 — Let's Sum Up · Review · Coding · Journal

🎯 LET'S SUM UP!

Operations on Real Numbers

We can extend the four operations on integers to involve all real numbers.

Operations on Integers

Addition:

5 + (−8) = 5 − 8 = −3
(−6) + 10 = −6 + 10 = 4
(−3) + (−4) = −3 − 4 = −7

Subtraction:

8 − (−3) = 8 + 3 = 11
(−2) − 5 = −2 − 5 = −7
(−5) − (−6) = −5 + 6 = 1

Multiplication:

9 × (−2) = −18
(−3) × 4 = 4 × (−3) = −12
(−5) × (−6) = 5 × 6 = 30

Division:

15 ÷ (−3) = 15/(−3) = −5
(−20) ÷ 5 = −20/5 = −4
(−36) ÷ (−9) = −36/(−9) = 4

Order of Operations

First, evaluate the expressions in brackets, starting with the innermost pair if there are more than one pair.
Next, evaluate the powers and roots.
Then, multiply and divide from left to right.
Finally, add and subtract from left to right.

Brackets → Powers & Roots → × ÷ → + −

Example. −7 + 3² × (5 − 7) = −7 + 9 × (−2) = −7 + (−18) = −25.

Classification of Numbers

            Real Numbers
   e.g., −9, −5, 0, 0.4, 1.3̇, 3, 11, 3/8, 9/4, 3 4/9, −2 1/7, π, √2, 0.232 232 223…
           /                                 \
   Rational Numbers                     Irrational Numbers
   (terminating + recurring             (non-terminating, non-recurring)
    decimals)                            e.g., π, √2, 0.232 232 223…
   e.g., −9, −5, 0, 0.4, 1.3̇, 3, 11, 3/8, …
        /            \
   Integers        Fractions
   e.g., …          e.g., …
     /    \
 Whole   Negative
 Numbers Integers
 (0+pos) (−1,−2,…)

Rational Numbers vs Irrational Numbers

Rational Numbers

Rational numbers can be expressed in the form a/b, where a and b are integers and b ≠ 0.
Rational numbers consist of terminating decimals and recurring decimals.

Examples: 1/2 = 0.5; 1/3 = 0.3̇.

Irrational Numbers

Irrational numbers cannot be expressed in the form a/b.
Irrational numbers consist of non-terminating and non-recurring decimals.

Examples: π ≈ 3.141 592 6…; √2 ≈ 1.414 213 56…

Number Line

Every real number can be represented by a point on the number line.
The number line shows the order of real numbers. Every number is less than any number on its right and greater than any number on its left.

For example: −3 < −2.6 < −2 and 1.6 > −2.6.

🔁 REVIEW EXERCISE 2

本章 14 道综合复习题——结合 2.1–2.5 的所有内容（整数、有理数、无理数、实数运算）。

(a) Find the values of (−2)² and (7/6)² without using a calculator.

(−2)² = (7/6)² =

(b) On a number line, represent −2, 7/6, (−2)² and (7/6)² (open answer — sketch).

(c) Inequality connecting −2 and 7/6:

(d) Inequality connecting (−2)² and (7/6)²:
Evaluate without calculator.

(a) (−16) × (−3) − (−8) × 5 =

(b) [−2 + (−7)]³ =

(c) 3½ × (−5 1/7) ÷ 1 4/5 =

(d) 2/3 × (1/4 − 1/8) =

(e) (−2½) − [(−½)³ − 1/4] × (−2 2/3) =

(f) [1½ − (−8 1/3) × (−1 1/5)] ÷ (−1 2/15) =

(g) 3.2 ÷ (−1.6) + (−0.8) × (−5) =

(h) (−1.2)² × 100 ÷ [−1/3 + (−1/6)] =
Use a calculator to evaluate:

(a) [3 2/3 − (−4/11)] ÷ (√8 ÷ ∛4) ≈ (b) −(1 1/9)² + (0.6 − 2/3) ÷ 2 1/6 ≈ (c) (−3 1/8 + 1 1/6 − 3 1/4) × (−2/5)² =
Is each result always negative? If not, provide a counterexample.

(a) positive + negative →

(b) positive × negative →

(c) negative ÷ negative →

(d) ∛negative →

💡 反例 / 说明
(a) 5 + (−2) = 3，正；不总为负。
(b) 正×负 = 负，总是。
(c) 负÷负 = 正，从不为负。
(d) ∛(负数) = 负实数（如 ∛(−8) = −2）；总是。
6 packets of fruit juice, supposed 375 ml each. Inspection results show amount below/above (ml):

Packet 1 2 3 4 5 6

Amount (ml) −5 +12 −5 −9 +7 −2

(a) Which packet has the largest volume? Packet

(b) Total volume of 6 packets = ml
4 golfers in 3 rounds (strokes below/above standard):

Ali Ben Chetan Doris

R1 −5 +3 +10 −3

R2 −1 −2 +2 −4

R3 −2 −1 −3 +5

(a) Lowest score in Round 1:

(b) Totals — Ali: Ben: Chetan: Doris:

(c) Winner (lowest total):
Liquid air = N₂ + O₂. Boiling points: N₂ = −196 °C; O₂ = −183 °C. From −210 °C heating up, which evaporates first?

=

Explain briefly:

ENN₂ evaporates first. N₂'s bp = −196 °C < O₂'s bp = −183 °C. Heating up from −210 °C, the temperature reaches −196 °C before −183 °C, so N₂ boils (evaporates) first.

⏱ 15 秒后查看中文版

中文N₂ 先蒸发。N₂ 沸点 −196°C < O₂ 沸点 −183°C。从 −210°C 升温，温度先到 −196 → N₂ 先沸腾。

自评：
(a) Jet fighter at 324 m above sea level; submarine at 58 m below. Vertical distance the missile travelled:

= m

(b) A bird is midway between jet and submarine. Height of bird above sea level:

= m
Science books shelf: 3/8 Biology, 1/3 of remaining = Chemistry, rest = Physics.

(a) Fraction of Physics:

(b) As decimal:

(c) If 10 Physics books, total books =
Henry's method to convert recurring decimals into fractions:

Let ★ = 0.7̇
Then 10★ = 7.7̇
Hence 9★ = 7 (use subtraction)
★ = 7/9
Since at first ★ = 0.7̇, ∴ 0.7̇ = 7/9.

Use the same method to convert:

(a) 0.8̇ = (b) 0.9̇5̇ =

💡 步骤

(a) 0.8̇: 令 ★=0.8̇, 10★=8.8̇, 9★=8, ★=8/9。
(b) 0.9̇5̇: 令 ★=0.9̇5̇=0.95 95 95…, 100★=95.9̇5̇, 99★=95, ★=95/99。
Find the largest possible negative integer n such that n × (−1/3) × (−2/8) × (−6/12) gives a positive integer.

n =

💡 推导
(−1/3)(−2/8)(−6/12) = (−1/3)(−1/4)(−1/2) = −1/24（三负相乘 = 负）。
所以 n × (−1/24) = 正整数 → −n/24 = 正整数 → n = −24k（k 正整数）。
最大（最接近 0 的负数）n = −24（k=1）。
验证：−24 × (−1/3)(−1/4)(−1/2) = −24 × (−1/24) = 1 ✓。
Triathlon: run 20/103 of total; cycle = 4 × run; swim = 1.5 km.

(a) Swim fraction of race =

(b) Total race distance = km

💡 推导
run + cycle + swim = total。run = 20/103 T；cycle = 4 × 20/103 T = 80/103 T。
swim = T − 20/103 T − 80/103 T = T × 3/103 = 1.5 → T = 1.5 × 103/3 = 51.5 km。
Four numbers a, b, c, d are on the number line at approximately a ≈ −1.7, b ≈ −0.3, c ≈ 0.7, d ≈ 1.3.

Choose any two so that:

(a) Their sum is the greatest:

(b) Their difference is the smallest (most negative):

(c) Their product closest to 0:

(d) Their quotient largest positive:

💡 思路
(a) 两个最大的 = c (0.7) + d (1.3) = 2。
(b) 差最小（最负）= 最小 − 最大 = a − d = −1.7 − 1.3 = −3。
(c) 积最接近 0 = b × c（小数 × 小数）≈ −0.21。
(d) 正商最大 = 同号的 a/b（两负）= −1.7 / −0.3 ≈ 5.67。
Three numbers a, b, c are on the number line with a < 0, 0 < b < 1, c > 1.

(a) Determine sign of each expression:

(i) a × c →

(ii) b × c →

(iii) a/b →

(iv) b ÷ √c × a →

(b) Given c > 1, describe a possible position of a × c on the number line.

ENSince c > 1 and a < 0, we have |a × c| = |a| × c > |a|. So a × c is more negative than a — that is, a × c lies to the LEFT of a on the number line.

⏱ 15 秒后查看中文版

中文由 c > 1，a < 0，所以 |a×c| = |a|×c > |a|，即 a×c 比 a "更负"——a×c 在数轴上 a 的左边。

自评：

Packet	1	2	3	4	5	6
Amount (ml)	−5	+12	−5	−9	+7	−2

	Ali	Ben	Chetan	Doris
R1	−5	+3	+10	−3
R2	−1	−2	+2	−4
R3	−2	−1	−3	+5

💻 CODING — Collatz Conjecture flowchart

In Computer Science, a flowchart shows a sequence of steps needed to perform a process. Each step is described in a symbol:

Symbol	Interpretation
⬭ rounded	Start / End Terminal
▱ parallelogram	Input / Output
▭ rectangle	Process
◇ diamond	Decision

The flowchart below illustrates the computation of the Collatz conjecture:

Input a positive integer n.
If n is even, set n = n / 2; if odd, set n = 3n + 1.
Repeat until n = 1; output the result.

Q1. Verify that the flowchart eventually ends if the initial input is

(a) 2 →

(b) 5 →

💡 三条 Collatz 序列

(a) 2 → 1 ✓
(b) 5 (odd) → 3×5+1 = 16 → 8 → 4 → 2 → 1 ✓
(c) 12 → 6 → 3 (odd) → 10 → 5 → 16 → 8 → 4 → 2 → 1 ✓
Collatz 猜想：对所有正整数，按这个规则迭代最终都会到 1。但至今没有证明——是个著名的未解数学问题（自 1937 年 Lothar Collatz 提出）。

Q2. Can you draw a flowchart to find all the prime numbers from 1 to 100?

自评：

📖 MATHS JOURNAL

把第 2 章学到的整数/有理数/无理数/实数概念用自己的话整理。

1 Describe in your own words how the negative and positive sign rules work for the four operations on integers. Use examples.

自评：

2 Explain the difference between rational and irrational numbers, and give 2 examples of each from real life or measurements.

自评：