🪙 小数运算：小数点要对齐🪙 Decimal Arithmetic: Line Up the Point
整数世界里，1 就是 1，清清楚楚。可一到文具店，价签上全是 3.50、1.25 这样的“带零头的数”。小算盘机器人掏出 5 元，买得起这两样吗？这一片就来算清楚。In the integer world, 1 is just 1 — clean and clear. But in a stationery shop, price tags are full of leftovers: 3.50, 1.25. The little robot has $5 — can it afford both? Let's find out.
第 1 步：0.1 和 0.01，差在哪Step 1: 0.1 vs 0.01 — what's the difference?
先闹清楚一件事：小数点右边每一位，各值多少钱。First, let's be clear about what each place to the right of the point is worth.
0.1 元就是 1 角，0.01 元就是 1 分。小数点右边第一位叫十分位，第二位叫百分位——3.50 元 = 3 元 5 角 0 分，1.25 元 = 1 元 2 角 5 分。同一个数字 5，站在十分位是 5 角，往右挪一位就只剩 5 分：位置一变，价值差十倍。0.1 is one dime, 0.01 is one penny. The first place right of the point is the tenths place, the second is the hundredths place — $3.50 is 3 dollars and 5 dimes, $1.25 is 1 dollar, 2 dimes and 5 pennies. The same digit 5 means 5 dimes in the tenths place, but only 5 pennies one place to the right: move it, and its value drops tenfold.
知道了每位值多少，就明白为什么竖式不能从尾巴对整齐。Once you know what each place is worth, you see why we never line numbers up by their last digit.
第 2 步：加减法，小数点对齐Step 2: To add or subtract, line up the points
竖式里，个位对个位、十分位对十分位——两个小数点必须待在同一条竖线上。3.50 + 1.25：0 分加 5 分是 5 分，5 角加 2 角是 7 角，得 4.75。要是把 1.25 摆错位变成 12.5，答案立刻差出好几倍。位数不一样就先补 0：3.5 当成 3.50 再算。In a vertical sum, units face units and tenths face tenths — the two points must sit on the same vertical line. 3.50 + 1.25: 0 pennies plus 5 pennies is 5, 5 dimes plus 2 dimes is 7, giving 4.75. Lay 1.25 in the wrong column and it becomes 12.5 — the answer jumps tenfold. If the lengths differ, pad with zeros first: treat 3.5 as 3.50.
加减靠对齐，乘除有另一套顺手的办法。Adding and subtracting rely on lining up — multiplying and dividing have their own handy moves.
第 3 步：乘除数位数，量一量生活Step 3: Count places to multiply, measure life to see
乘法：先当整数乘，再数一数两个数一共几位小数，给积点上几位。0.4 × 0.3：4 × 3 = 12，共两位小数，得 0.12。除法：把除数的小数点挪成整数，被除数跟着挪同样多格：7.5 ÷ 0.25 = 750 ÷ 25 = 30。生活里也一样——量身高 1.42 − 1.28，小数点对齐着减，正好长高 0.14 米。Multiply: compute as if they were whole numbers, then count the total decimal places and give them back to the product. 0.4 × 0.3: 4 × 3 = 12, two decimal places in total, so 0.12. Divide: slide the divisor's point to make it a whole number, and slide the dividend's point the same number of steps: 7.5 ÷ 0.25 = 750 ÷ 25 = 30. Life works the same way — measuring height, 1.42 − 1.28 with points lined up gives a growth of 0.14 m.
🎮 你来当收银员（1 分钟）🎮 Your turn: cashier duty (1 minute)
道理讲完了。拖两个价格滑块凑出 4 个目标金额；再点「换个摆法看看」——亲眼看小数点错位会多花多少钱。Theory done. Drag the two price sliders to hit 4 target totals; then hit “Try a shifted layout” and watch how much a slipped decimal point costs.
一句话记住它：小数运算就是让每一位站在自己的位置上——加减对齐小数点，乘法数小数位，除法把小数点一起挪。Remember it in one line: decimal arithmetic keeps every digit in its own place — line up the points to add, count decimal places to multiply, shift the points together to divide.
十分位是 0.1，百分位是 0.01；位置差一位，价值差十倍Tenths are 0.1, hundredths are 0.01 — one place over means ten times the value加减：小数点对齐，位数不够用 0 补（3.5 → 3.50）Add/subtract: line up the points, pad with zeros when needed (3.5 → 3.50)乘除：乘法先整数乘再数小数位；除法先把除数挪成整数Multiply/divide: whole numbers first, then count places; or shift both points right
内容参考 OpenStax 等公开教材，多来源核对 · AI 生成、人工审核 · 发现错误欢迎指正，帮这片叶子长得更好。Based on OpenStax and other open textbooks, cross-checked across sources · AI-generated, human-reviewed · Spotted a mistake? Tell us — help this leaf grow.
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