Compare numbers written in scientific notation

Key Notes:

Scientific Notation :

• Numbers in scientific notation are expressed as a×10na \times 10^na×10n, where aaa is a number between 1 and 10, and nnn is an integer.
• Example: 3.25×1043.25 \times 10^43.25×104 means 32,500.

Comparison Steps:

Compare the coefficients (the numbers aaa) directly.

If the coefficients are equal, compare the exponents nnn:

• Larger exponent means a larger number.
• Smaller exponent means a smaller number.

Examples:

Compare 2.1×1052.1 \times 10^52.1×105 and 3.5×1043.5 \times 10^43.5×104:

• Coefficient comparison: 2.1>3.52.1 > 3.52.1>3.5 (so 2.1×1052.1 \times 10^52.1×105 is larger).

Compare 4.2×1034.2 \times 10^34.2×103 and 4.5×1034.5 \times 10^34.5×103:

• Coefficients are close, compare exponents: 4.2×103<4.5×1034.2 \times 10^3 < 4.5 \times 10^34.2×103<4.5×103 (so 4.5×1034.5 \times 10^34.5×103 is larger).

You can compare two numbers written in scientific notation by following these steps:

• Compare the powers of 10. If one of the powers of 10 is greater than the other, that number is greater.
• If the powers of 10 are the same, compare the first factors. If one of the first factors is greater than the other, that number is greater.

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In any other situation

Let’s try it! Compare 4.7×10⁸ and 3.6×10⁹.

Start by comparing the powers of 10. Since 10⁸ < 10⁹, then 4.7×10⁸ < 3.6×10⁹.

You can see why this works by rewriting each number in standard form.

4.7×10⁸ = 470,000,000

3.6×10⁹ = 3,600,000,000

Since 470,000,000 < 3,600,000,000, then 4.7×10⁸ < 3.6×10⁹.

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Another example

Compare 5.1×10⁻⁷ and 2.8×10⁻⁷.

Start by comparing the powers of 10. Since they are the same, compare the first factors.

Compare the first factor in each number. Since 5.1 > 2.8, then 5.1×10⁻⁷ > 2.8×10⁻⁷.

You can see why this works by rewriting each number in standard form.

5.1×10⁻⁷ = 0.00000051.

2.8×10⁻⁷ = 0.00000028.

Since 0.00000051 > 0.00000028, then 5.1×10⁻⁷ > 2.8×10⁻⁷.

Learn with an example

Let’s practice!