Comparing Decimal Numbers

I. Comparison of decimal places

1. First, we look at the "pre-decimal numbers". These are the numbers that come before the decimal point. If they differ, the number whose pre-decimal point number is already larger is the larger one. In that case, you are done. If both pre-decimal numbers are identical, then you need to go to step two.

2. Next, compare the tenths digits. The tenth digit is the first number that comes after the decimal point. If a number has a tenth digit which is larger then the tenth digit of the other number, then the total number is larger then the other number. If both tenth digits are the same, go to step three.

3. Next, compare the hundredths digits of the two numbers, i.e. the respective second digits after the decimal point. Proceed as in step two for the comparison. If this is also the same, then compare the thousandths digits, then the ten-thousandths digits, and so on.

4. If all digits are identical, then both numbers are the same.

Examples

Compare each pair of numbers with each other. Use the above procedure to decide which of the two numbers is greater.

• $23.6$ and $24.6$

Here the two pre-decimal numbers are $23$ and $24$. Since $24$ is greater than $23$, you already know that $24.6$ is greater than $23.6$.

• $23.6$ and $23.7$

Here, the pre-decimal number is $23$ for both numbers. So you have to go to step 2, i.e. comparing the tenths digits. These are $6$ and $7$, respectively. Since $7$ is greater than $6$, you finally know that $23.7$ is greater than $23.6$.

• $23.026$ and $23.0265$

Now it gets a little more difficult: The pre-decimal number is $23$ in both numbers. Also the tenths digits agree (both are $0$), and so do the hundredths digits (both are $2$) and the thousandths digits (both are $6$). Now, in $23.026$ there is no further digit coming. But remember that adding zeros in the end of a number behind the decimal point does not change the value of the number (e.g., $1 = 1.0 = 1.00$).

$\rightarrow$ So you may as well compare $23.0260$ and $23.0265$

If you now look at the ten-thousandths digit, you finally see that $5$ is greater than $0$. Thus, $23.0265$ is greater than $23.0260 = 23.026$.

Common mistakes

• Omitting the decimal point

• Pretending, there was a seperate number after the decimal point

II. Comparison by shift of the decimal point

If we compare two numbers with each other, the relation (greater/smaller) does not change if we multiply both numbers by $10$.

For example, instead of comparing $0.3$ with $0.4$, we could also compare $3$ with $4$.

Now, multiplying by $10$ just shifts the decimal point one place to the right. We may do this arbitrarily often. That is, we shift the decimal point in both numbers simultaneously to the right, until both numbers are natural numbers:

Then, compare the two resulting natural numbers.

Examples

• $0.035$ and $0.056$

Multiply by $10$:

$\rightarrow$ You obtain $0.35$ and $0.56$

Multiply by $10$:

$\rightarrow$ You obtain $3.5$ and $5.6$

Multiply by $10$:

$\rightarrow$ You obtain $35$ and $56$

$35$ is smaller than $56$

Accordingly, $0.035$ is smaller than $0.056$.

Note: With a bit of practice, you may also directly multiply by $1000$ instead of multiplying by $10$ three times.

Common mistakes