B1. Number Sense

Example

Write 2500000 and 0.0042 in scientific notation, and compare $3 \times 10^{5}$ with $9 \times 10^{4}$.

1. 2500000 → move the point until one digit remains in front: $2.5 \times 10^{6}$.
2. 0.0042 → $4.2 \times 10^{-3}$ (three places right).
3. Compare exponents first: $10^{5}$ beats $10^{4}$, so $3 \times 10^{5}$ is larger despite the smaller coefficient.

Example

Order **$\sqrt{5}$, 2.4, $\frac{7}{3}$, −1.5** from least to greatest.

1. −1.5 is the only negative: least.
2. $\sqrt{5}$: 2² = 4 and 3² = 9, so between 2 and 3. Compare squares: 2.4² = 5.76 > 5, so $\sqrt{5}$ < 2.4.
3. $\frac{7}{3}$ = 2.333…, less than 2.4; and 2.333² ≈ 5.44 > 5, so $\frac{7}{3}$ > $\sqrt{5}$. Order: −1.5, $\sqrt{5}$, $\frac{7}{3}$, 2.4.

Example

Find $\sqrt{81}$. Then estimate $\sqrt{41}$ with whole-number brackets.

1. $\sqrt{81} = 9$, because 9 × 9 = 81.
2. 41 sits between 36 = 6² and 49 = 7², so $\sqrt{41}$ is between 6 and 7 — nearer 6, since 41 is nearer 36.
3. Brackets are the estimate; a calculator gives more digits when needed.

Example

Find 150% of 60 and 0.5% of 600, and explain each as a fraction and a decimal.

1. 150% = 1.5 = $\frac{3}{2}$: 1.5 × 60 = 90. More than the whole.
2. 0.5% = 0.005 = $\frac{1}{200}$: 0.005 × 600 = 3. Half of 1%.
3. Switch forms freely: whichever makes the arithmetic easiest.

Summary

• Coefficient in [1, 10) × a power of ten; negative exponents for small.
• Irrationals: bracket, never a rounded decimal.
• √ of a perfect square is exact; otherwise bracket.
• Percents above 100 and below 1 are just decimals and fractions.