Digits show how precisely we know

The idea

The number of digits in the coefficient reflects how precisely a number is known; after the exponent the first digit of the coefficient matters most, and the following digits are small corrections.

How many people live in Kohima?

The population of Kohima is 1,42,395. Suppose we are sure only up to about 1 lakh 42 thousand.

We then write 1.42 × 10^5. If we are sure only up to about 1 lakh 40 thousand, we write 1.4 × 10^5.

Which one says we know the population more precisely?

• 1.4 × 10^5
• 1.42 × 10^5
• Both the same

Add digits one by one

Press + digit. The pink band shows where the real number could be. Watch the band shrink.

More digits, more precision

The number of digits in the coefficient reflects how precisely a number is known. After the exponent, the first digit of the coefficient matters most, and the following digits are small corrections.

Sun–Saturn: 1.4335 × 10^12 m. Saturn–Uranus: 1.439 × 10^12 m. Same exponent, same first digits: the later digits are small corrections.

• Written | Means about
• 1 × 10^5 | 1 lakh
• 1.4 × 10^5 | 1 lakh 40 thousand
• 1.42 × 10^5 | 1 lakh 42 thousand

Notes

The number of digits in the coefficient shows how precisely a number is known. After the exponent, the first digit matters most; the rest are small corrections.

Check yourself

Which one is written most precisely?

Sun–Saturn is 1.4335 × 10^12 m. Saturn–Uranus is 1.439 × 10^12 m. Which distance is greater?

Write the population 1,42,395 as a coefficient with 2 digits (× 10^5).

Answer: 1.4

1.42395 rounded to 2 digits is 1.4, so 1,42,395 ≈ 1.4 × 10^5.

1.4 × 10^5 is about how many thousand?

Answer: 140

1.4 × 10^5 = 1,40,000 = 140 thousand.

• 1.4 × 10^5. It has only 2 digits, so it is the roughest of the three.
• 1.42 × 10^5. It has 3 digits, but another one has more.
• 1.4239 × 10^5 — correct. Yes! It has the most digits, so it pins the number down the most.
• Sun–Saturn. 1.4335 is a little less than 1.439. Compare the digits from the left.
• Saturn–Uranus — correct. Yes! The exponents are equal. Then 1.439 > 1.4335, so Saturn–Uranus is a little greater.
• They are equal. Both are about 1.4 × 10^12, but the later digits (small corrections) show that they are not equal.