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Knowing Our Numbers - Comparing Numbers

Grade 10CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The Law of Trichotomy: For any two real numbers aa and bb, exactly one of the following relations holds: a<ba < b, a=ba = b, or a>ba > b.

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Comparison by Place Value: When comparing two large numbers, the number with more digits is greater. If the number of digits is the same, compare the digits starting from the leftmost place (highest place value).

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Comparing Rational Numbers: To compare two fractions ab\frac{a}{b} and cd\frac{c}{d}, we can use cross-multiplication. ab>cd\frac{a}{b} > \frac{c}{d} if and only if ad>bcad > bc, provided b,d>0b, d > 0.

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Comparing Irrational Numbers: To compare square roots like x\sqrt{x} and y\sqrt{y}, we compare the values under the radical. If x>y≥0x > y \geq 0, then x>y\sqrt{x} > \sqrt{y}.

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Scientific Notation Comparison: For numbers expressed as a×10na \times 10^n and b×10mb \times 10^m, if n>mn > m, then a×10n>b×10ma \times 10^n > b \times 10^m. If n=mn = m, compare the coefficients aa and bb.

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The Density Property: Between any two distinct real numbers, there exists another real number, which allows for infinite refinement in comparison.

📐Formulae

ab>cd  ⟺  ad>bc (for b,d>0)\frac{a}{b} > \frac{c}{d} \iff ad > bc \text{ (for } b, d > 0\text{)}

If a>b and c>0, then ac>bc\text{If } a > b \text{ and } c > 0, \text{ then } ac > bc

If a>b and c<0, then ac<bc\text{If } a > b \text{ and } c < 0, \text{ then } ac < bc

xn>xm  ⟺  n>m (for x>1,n,m∈R)x^n > x^m \iff n > m \text{ (for } x > 1, n, m \in \mathbb{R}\text{)}

an>bn  ⟺  a>b (for a,b≥0,n∈N)\sqrt[n]{a} > \sqrt[n]{b} \iff a > b \text{ (for } a, b \geq 0, n \in \mathbb{N}\text{)}

💡Examples

Problem 1:

Which is greater: 353\sqrt{5} or 535\sqrt{3}?

Solution:

35=32×5=9×5=453\sqrt{5} = \sqrt{3^2 \times 5} = \sqrt{9 \times 5} = \sqrt{45}. 53=52×3=25×3=755\sqrt{3} = \sqrt{5^2 \times 3} = \sqrt{25 \times 3} = \sqrt{75}. Since 75>4575 > 45, 75>45\sqrt{75} > \sqrt{45}. Therefore, 53>355\sqrt{3} > 3\sqrt{5}.

Explanation:

To compare surds of the same order, move the coefficients inside the square root by squaring them and then compare the resulting radicands.

Problem 2:

Compare the numbers A=4.5×1012A = 4.5 \times 10^{12} and B=9.2×1011B = 9.2 \times 10^{11}.

Solution:

The exponent of 1010 in AA is 1212 and in BB is 1111. Since 12>1112 > 11, A>BA > B.

Explanation:

In scientific notation, the power of 1010 dictates the magnitude of the number. A higher positive exponent always results in a larger number regardless of the decimal coefficient (mantissa).

Problem 3:

Arrange the following in ascending order: π\pi, 227\frac{22}{7}, and 3.143.14.

Solution:

We know π≈3.14159265...\pi \approx 3.14159265..., 227≈3.142857...\frac{22}{7} \approx 3.142857..., and the given decimal is 3.140000...3.140000.... Comparing these values: 3.14<3.14159...<3.14285...3.14 < 3.14159... < 3.14285.... Thus, 3.14<π<2273.14 < \pi < \frac{22}{7}.

Explanation:

By expanding the numbers to a sufficient number of decimal places, we can compare them digit by digit from left to right.