The words for numbers
Introduced| English | Meaning |
|---|---|
| factor/ˈfæktə/ | A number that divides another number exactly, or an expression multiplied with others to form a product. |
| integer/ˈɪntɪdʒə/ | A whole number that may be positive, negative or zero. |
| rational number/ˈræʃənl ˈnʌmbə/ | A number expressible as an integer divided by a nonzero integer. |
| irrational number/ɪˈræʃənl ˈnʌmbə/ | A real number that cannot be expressed as a ratio of two integers. |
| prime number/praɪm ˈnʌmbə/ | A positive integer greater than 1 with exactly two positive factors. |
| significant figures/sɪɡˈnɪfɪkənt ˈfɪɡəz/ | The meaningful digits in a measured or rounded number, beginning with its first non-zero digit. |
| highest common factor/ˈhaɪɪst ˈkɒmən ˈfæktə/ | The greatest positive integer that divides each of the given integers exactly. |
| lowest common multiple/ˈləʊɪst ˈkɒmən ˈmʌltɪpl/ | The smallest positive integer divisible by each of the given positive integers. |
Explain a number as well as calculate it
- A factor 因数 divides a number exactly; a multiple is obtained by multiplying it by an integer. Naming the relationship helps another person check a calculation.
- A terminology logbook can combine a definition and example. Follow your centre's brief for submitted work; these original practice tasks do not establish official assessment weights.
Distinguish number families
- An integer 整数 includes zero and negative whole numbers. A rational number 有理数 is a ratio of integers with a nonzero denominator; an irrational number 无理数 cannot be written that way.
- A prime number 质数 is a positive integer greater than 1 with exactly two positive factors. A terminating or eventually repeating decimal is rational; an irrational decimal neither terminates nor eventually repeats.
Sort each number into its family
Say which family a number belongs to, and why.
Why is 1 not a prime number?
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A prime is a positive integer greater than 1 with exactly two positive factors. The number 1 has only one positive factor.
Match each term to its definition.
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Use precise definitions; a finite displayed approximation does not establish whether the underlying exact number is rational.
Work through operations in order
- Calculate brackets and powers first. Multiplication and division share a level and go left to right; addition and subtraction share the next level.
- In $18-3(4-7)$ the bracket is negative 3, so the product is negative 9. Subtracting that negative gives 27, not 9.
Evaluate 5 + 3 × 2² − 4.
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Indices first (2² = 4), then multiplication (3 × 4 = 12), then left to right: 5 + 12 − 4 = 13.
Keep the percentage base visible
- Increase by 15% uses multiplier 1.15; decrease by 15% uses 0.85. Reverse a change by dividing by its multiplier.
- Significant figures 有效数字 state retained precision, not proof of measurement accuracy. $0.004072$ becomes $0.00407$ to three significant figures; keep unrounded values for intermediate steps.
A price of 240 rises by 15%, then the new price falls by 15%. What is the final price?
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240 × 1.15 = 276, then 276 × 0.85 = 234.60. The two percentages act on different amounts, so you do not return to 240.
Write 0.004072 to three significant figures.
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Significant figures start at the first non-zero digit, so 4, 0 and 7 are the three. The leading zeros only fix the size.
A price that rises 20% and then falls 20% returns to its original value.
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It falls to 96% of the original: 1.20 × 0.80 = 0.96. The second percentage acts on the larger amount.
Find the lowest common multiple of 24 and 36.
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24 = 2³ × 3 and 36 = 2² × 3². Take the higher power of each prime: 2³ × 3² = 72.
Factors connect with repeating schedules. For positive integers 24 and 36, $24=2^3\times3$ and $36=2^2\times3^2$. The highest common factor 最大公因数 uses the shared lower powers, giving 12. The lowest common multiple 最小公倍数 uses the highest powers needed, giving 72. Two cycles of 24 and 36 minutes next coincide after 72 minutes.
Compare cost per item before comparing offers. Divide total cost by item count, with the same units for each offer. A lower unit price may require a higher total purchase. Sheet 1.1 combines arithmetic, schedules and reverse percentages.
A 20% fall followed by a 20% rise multiplies by $0.8\times1.2=0.96$. The second percentage uses a different base. Name that base before deciding whether two changes cancel.