Number and Finance in IB Mathematics: Applications and Interpretation | IB数学应用与解释：数与金融

📚 Number and Finance in IB Mathematics: Applications and Interpretation | IB数学应用与解释：数与金融

In the IB Mathematics: Applications and Interpretation course, the module 'Number and finance' equips students with essential quantitative tools for modelling real-world financial situations. From understanding numerical precision to evaluating loans, annuities, and currency conversions, these topics form a foundation for informed decision-making in personal and professional contexts. This article provides a structured revision guide, covering all key concepts, formulas, and problem-solving strategies required for success in exams and internal assessments.

在IB数学：应用与解释课程中，“数与金融”模块为学生提供了处理现实世界金融情境的核心量化工具。从理解数值精度到评估贷款、年金和货币兑换，这些主题为个人与专业背景下做出明智决策奠定了基础。本文提供了一份结构化的复习指南，涵盖了考试和内部评估所需的所有关键概念、公式与解题策略。

1. The Real Number System | 实数系统

The real number system is the foundation of all quantitative finance. Natural numbers (ℕ) are used for counting periods; integers (ℤ) can represent profit and loss; rational numbers (ℚ) appear in interest rates and proportions; irrational numbers arise in continuous growth models, and the set of real numbers (ℝ) contains them all. In finance, we assume amounts can be any real number, although currencies are rounded to two decimal places in practice.

实数系统是所有量化金融的基石。自然数 (ℕ) 用于计算期数；整数 (ℤ) 可以表示盈亏；有理数 (ℚ) 出现在利率和比例中；无理数则出现于连续增长模型，而实数集 (ℝ) 包含了上述所有数。在金融中，我们假设金额可以是任意实数，尽管实际货币金额会四舍五入到两位小数。

Understanding number sets helps in interpreting results, for instance, when an equation yields a negative square root that must be rejected because a time period cannot be a non-real number.

理解数集有助于解释结果，例如，当方程产生负平方根时必须舍去，因为期数不能是非实数。

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2. Scientific Notation and Significant Figures | 科学记数法与有效数字

Scientific notation expresses numbers as a × 10^k handling very large or very small numbers in finance, such as national debt (e.g. 2.5 × 10¹²) or transaction fees (e.g. 1.5 × 10⁻³ as a decimal fraction).

科学记数法将数字表示为 a × 10^k，其中 1 ≤ a < 10，k 为整数。处理金融中的极大或极小数至关重要，例如国债 (2.5 × 10¹²) 或交易费用 (如小数形式的 1.5 × 10⁻³)。

Significant figures reflect the precision of a measurement. The rules: all non-zero digits are significant, zeros between non-zeros are significant, and trailing zeros after a decimal point are significant. In finance, rounding to a given number of significant figures affects the accuracy of further calculations.

有效数字反映测量的精度。规则为：所有非零数字均为有效，非零数字之间的零有效，小数点后末尾的零有效。在金融中，按给定有效数字舍入会影响后续计算的准确性。

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3. Approximation, Rounding and Error | 近似、舍入与误差

is the absolute difference between the exact value and the approximation: absolute error = |v − v_approx|. For example, if π is approximated by 3.14, the absolute error is about 0.00159.

当我们对数值进行近似时，就引入了误差。绝对误差是精确值与近似值之差的绝对值：绝对误差 = |v − v_approx|。例如，用 3.14 近似 π，绝对误差约为 0.00159。

The relative error is absolute error divided by the exact absolute value, and percentage error turns this into a percentage. In financial contexts, percentage error helps gauge the impact of rounding on profit margins or interest calculations.

相对误差是绝对误差除以精确值的绝对值，而百分比误差则将其表示为百分比。在金融环境中，百分比误差有助于衡量舍入对利润率或利息计算的影响。

Upper and lower bounds arise from rounding. If a length is given as 3.2 m to 1 decimal place, the true value lies in [3.15, 3.25). This concept is used in tolerances of financial forecasts.

上界和下界源自舍入。如果长度显示为 3.2 m（精确到 1 位小数），实际值位于 [3.15, 3.25) 区间内。此概念用于财务预测的公差范围。

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4. Percentage Error and Bounds | 百分比误差与界限

Percentage error = (|v − v_approx| / |v|) × 100% For a reported value R rounded to a certain unit, the bound width is the unit of rounding.

百分比误差 = (|v − v_approx| / |v|) × 100%。处理界限时，我们可以利用半区间宽度求出最大可能百分比误差。对于舍入到某一单位的报告值 R，界限宽度即为舍入单位。

Example: An investment is reported as €12 300 to the nearest € 350. The maximum absolute error is 50, and the maximum percentage error is (50/12 300)×100% ≈ 0.407%.

示例：一笔投资报告为 12 300 欧元，精确到百位。下界为 12 250，上界为 12 350。最大绝对误差为 50，最大百分比误差为 (50/12 300)×100% ≈ 0.407%。

Using bounds, we can also determine the error in compound calculations like interest earned, by propagating the extreme values through the formula.

利用界限，我们还可以通过将极端值代入公式来计算复利计算中的误差，比如利息收入。

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5. Simple Interest and Linear Growth | 单利与线性增长

s, the interest I is: I = P × r × t. The amount A after t years is A = P + I = P(1 + r t).

单利仅根据原始本金 P 计算。如果年利率为 r（以小数表示），时间 t 以年为单位，则利息 I 为：I = P × r × t。t 年后的总额 A 为 A = P + I = P(1 + r t)。

A = P(1 + r t)

This represents linear growth. It is commonly used for short-term savings or loans where interest is not compounded, but it underestimates growth over long periods compared to compound interest.

这代表线性增长。通常用于短期储蓄或不复利的贷款，但相比复利，它在长期内低估了增长。

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6. Compound Interest and Exponential Growth | 复利与指数增长

erest earns interest on the interest. With annual compounding, the formula is A = P(1 + r)^t. For compounding n times per year, the formula becomes A = P(1 + r/n)^{n t}. This models exponential growth, where the amount accelerates over time.

复利可以利滚利。年复利下，公式为 A = P(1 + r)^t。对于每年复利 n 次，公式变为 A = P(1 + r/n)^{n t}。这模拟了指数增长，随着时间推移，总额加速增加。

A = P(1 + r/n)^{n t}

Example: €5000 invested at 4% per annum compounded monthly for 5 years yields A = 5000 × (1 + 0.04/12)^12×5 ≈ €6104.98.

示例：5000 欧元以年利率 4% 每月复利投资 5 年，得到 A = 5000 × (1 + 0.04/12)^12×5 ≈ 6104.98 欧元。

al equivalent rate (AER) or effective annual rate (EAR) allows comparison: AER = (1 + r/n)ⁿ − 1. For the above, AER ≈ (1 + 0.04/12)¹² − 1 = 0.04074 or 4.074%.

年等效利率 (AER) 或有效年利率 (EAR) 便于比较：AER = (1 + r/n)ⁿ − 1。上例中，AER ≈ (1 + 0.04/12)¹² − 1 = 0.04074，即 4.074%。

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7. Compound Interest with Different Compounding Periods | 不同计息期的复利

al amount grows, but it approaches a limit. Continuous compounding uses the formula A = Pe^{r t} required in IB AI, understanding the effect of more frequent compounding reinforces the concept of limits in growth.

复利可以是年、半年、季度、月、日甚至连续复利。随着复利频率的增加，最终金额会增长，但趋于一个极限。连续复利用公式 A = Pe^{r t}，其中 e ≈ 2.71828。尽管 IB AI 并不总是要求，但理解更频繁复利的影响可以强化增长中的极限概念。

Using a GDC TVM (Time Value of Money) solver simplifies these calculations. Key variables: N (total number of payments), I% (annual interest rate), PV (present value), PMT (payment per period), FV (future value), P/Y (payments per year), C/Y (compounding periods per year). One must be careful to match P/Y and C/Y.

使用图形计算器 (GDC) 的 TVM（资金时间价值）求解器可简化这些计算。关键变量：N（总付款次数）、I%（年利率）、PV（现值）、PMT（每期付款额）、FV（终值）、P/Y（每年付款次数）、C/Y（每年复利次数）。必须注意匹配 P/Y 和 C/Y。

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8. Depreciation: Straight Line and Reducing Balance | 折旧：直线法与余额递减法

Depreciation is the decrease in value of an asset over time. The straight-line method reduces the book value by a fixed amount each year: V_t = P − d t, where d is the annual depreciation charge. This is a linear model.

折旧是资产价值随时间减少的过程。直线法每年按固定金额减少账面价值：V_t = P − d t，其中 d 为年折旧费用。这是线性模型。

The reducing balance method applies a constant percentage r to the remaining value: V_t = P(1 − r)^t. This reflects greater loss in early years and is often used for vehicles and technology.

余额递减法则对剩余价值应用固定百分比 r：V_t = P(1 − r)^t。这反映了早期较大损失，常用于车辆和技术资产。

Straight line: V_t = P − d t    Reducing balance: V_t = P(1 − r)^t

Choosing the method depends on the asset type and accounting rules. Students must be able to switch between models and interpret the book value at any time.

选择方法取决于资产类型和会计准则。学生需要能够在模型之间切换，并解释任意时刻的账面价值。

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9. Loans and Amortization: Present Value and Payments | 贷款与摊销：现值与付款

A loan involves borrowing a present value (PV) and repaying it with regular payments (PMT) over a certain number of periods N at an interest rate r per period. The fundamental equation linking these is the present value of an ordinary annuity:

贷款涉及借入现值 (PV)，并按每期利率 r 在 N 期内用定期付款 (PMT) 偿还。连接这些要素的基本方程是普通年金的现值：

PV = PMT × [1 − (1 + r)^−N] / r

rtization splits each payment into interest and principal repayment; the interest portion decreases over time.

适用于期末付款。使用 GDC 的 TVM 求解器时，将 PV 设为负（现金流出），PMT 设为正（银行角度的流入），FV = 0，然后求解未知数。摊销将每笔付款分为利息和本金偿还，利息部分随时间减少。

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10. Annuities and Sinking Funds | 年金与偿债基金

An annuity is a series of equal payments made at regular intervals. To find the future value (FV) of an annuity where you invest PMT each period, use the formula:

年金是定期支付的一系列等额款项。若要计算每期投入 PMT 的年金终值 (FV)，可使用公式：

FV = PMT × [(1 + r)^N − 1] / r

of annuity set up to reach a target future amount. For example, saving for a car: you need €20 000 in 4 years; if the annual interest rate is 5% thly, the monthly deposit PMT can be found by solving the FV formula with N=48, r=0.05/12.

偿债基金是为达到未来目标金额而设立的一种特殊年金。例如，为买车储蓄：4年内需要 20 000 欧元；若年利率 5% 每月复利，可通过求解 FV 公式（N=48，r=0.05/12）算出每月存款 PMT。

GDC TVM: set PV=0, FV=20000, P/Y=12, C/Y=12, N=48, I%=5, and solve for PMT. It will be negative as an outflow.

GDC TVM 设置：PV=0, FV=20000, P/Y=12, C/Y=12, N=48, I%=5，求解 PMT。结果为负值代表流出。

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11. Inflation and Real Value of Money | 通货膨胀与货币实际价值

, given annual inflation rate i, is obtained by discounting: Real Value = Amount / (1 + i)^T.

通货膨胀降低了货币的购买力。给定年通胀率 i，未来 T 年一笔金额的实际价值通过折现获得：实际价值 = 金额 / (1 + i)^T。

The relationship between nominal interest rate (r_nom), real interest rate (r_real), and inflation (i) is given by the Fisher equation: 1 + r_nom = (1 + r_real)(1 + i). For small rates, r_real ≈ r_nom − i.

名义利率 (r_nom)、实际利率 (r_real) 和通货膨胀 (i) 之间的关系由费雪方程给出：1 + r_nom = (1 + r_real)(1 + i)。对于较小的比率，r_real ≈ r_nom − i。

Example: A salary increase of 3% when inflation is 2% yields a real increase of only about 1% (using exact formula (1.03/1.02) − 1 = 0.0098 or 0.98%).

示例：通胀率为 2% 时加薪 3%，实际增幅仅约为 1%（使用精确公式 (1.03/1.02) − 1 = 0.0098，即 0.98%）。

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12. Currency Conversions | 货币兑换

hen to convert GBP to USD you multiply by 1.25. Conversely, from USD to GBP you divide by 1.25 or multiply by the reciprocal 0.8. Banks apply commission and offer different buy/sell rates, so the actual rate for you will be less favourable than the mid-market rate.

汇率使得货币间可以兑换。若 1 英镑 = 1.25 美元，那么英镑兑美元需乘以 1.25。相反，从美元兑英镑则除以 1.25 或乘以倒数 0.8。银行会收取手续费并提供不同的买入/卖出价，因此您的实际汇率会差于中间市场汇率。

centage. A typical problem: convert 500 EUR to USD with a rate of 1 EUR = 1.10 USD and a 2% commission. First convert 500 × 1.10 = 550 USD, then commission reduces it by 2%: 550 × 0.98 = 539 USD. The effective rate becomes 539/500 = 1.078 USD per EUR.

旅行时，兑换手续费可能是固定费用或百分比。典型问题：按 1 欧元 = 1.10 美元且 2% 手续费兑换 500 欧元。先兑换 500 × 1.10 = 550 美元，然后手续费减少 2%：550 × 0.98 = 539 美元。有效汇率变为 539/500 = 1.078 美元/欧元。

Remember to treat exchange rates with appropriate significant figures and consider the direction of conversion when solving problems.

记住要以适当的有效数字处理汇率，并在解题时考虑兑换方向。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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