Comparing Numbers

There are certain rules for comparing numbers. Let us look at the following example.

Yesterday the temperature was 9°C, and today it is 14°C. Today is warmer than yesterday. The number 9 is less than the number 14, so we can write: 9 < 14. If we represent these numbers on a number line, the point with the value 9 will be located to the left of the point with the value 14.

Now let us consider negative temperatures. Yesterday the temperature was −15°C, and today it is −10°C. Today is warmer than yesterday. Therefore, the number −15 is considered less than the number −10. On a horizontal number line, the point with the value −15 is located to the left of the point with the value −10. We can write: −15 < −10.

So, when comparing numbers using a horizontal number line, the smaller of two numbers is the one whose image on the number line is located further to the left, and the greater number is the one located further to the right.

On the number line, positive numbers are located to the right of zero, and negative numbers are located to the left of zero. Every positive number is greater than zero, and every negative number is less than zero. Therefore, every negative number is less than every positive number.

Comparing Negative Numbers

If we compare two negative numbers, we need to compare their absolute values: the greater number is the one with the smaller absolute value, and the smaller number is the one with the greater absolute value.

For example, compare −9 and −4. Both numbers are negative. Compare their absolute values: 9 and 4. Since 9 is greater than 4, then −9 < −4.

Comparing Common Fractions

Of two fractions with the same denominator, the fraction with the smaller numerator is smaller, and the fraction with the greater numerator is greater.

Algorithm for Comparing Common Fractions:

If a fraction has a whole part, begin the comparison with it. The fraction with the greater whole part is greater.
If the fractions have different denominators, convert them to a common denominator.
Compare the numerators of the fractions. The fraction with the greater numerator is greater.

Comparing Decimal Fractions

Decimal fractions can only be compared when they have the same number of digits to the right of the decimal point.

Algorithm for Comparing Decimal Fractions:

Pay attention to the number of digits to the right of the decimal point. If the number of digits is the same, comparison can begin. If not, add the required number of zeros.
Compare the decimal fractions from left to right: whole numbers with whole numbers, tenths with tenths, hundredths with hundredths, and so on.

For example, compare the decimal fractions 42.5 and 42.518:

Add the necessary number of zeros: 42.500 and 42.518.

Begin comparing from left to right: whole numbers: 42 = 42; tenths: 5 = 5; hundredths: 0 < 1. Since the hundredths in the first fraction are smaller: 42.500 < 42.518.

Equivalent Fractions

If the numerator and denominator of a fraction are multiplied or divided by the same natural number, the result is an equivalent fraction: 2/7 = 6/21, because (2 × 3) / (7 × 3).

Reducing a fraction means dividing the numerator and denominator by their common divisor. For example, 18/24 = (18÷6)/(24÷6) = 3/4.

To compare fractions with different denominators, convert them to a common denominator and compare the numerators. For example: 2/3 ? 5/8. Common denominator 24: 16/24 ? 15/24. Since 16 > 15: 2/3 > 5/8.

Comparing Numbers