Rational numbers

Every number you know belongs to one family

You know whole numbers: 3, 0, −7. You know fractions: 1/2, 3/4. You know decimals: 0.5 or −2.25. They look like three different worlds. They are not.

All of them together are called rational numbers. A rational number is any number you can write as a fraction. The number 3 is the fraction 3/1. The number 0.5 is the fraction 1/2. Even negative −2 is the fraction −2/1.

That is the whole trick. If you can rewrite a number as a fraction, it is rational. And because that works for almost every number you meet in Year 7, you will work with them all year.

Why bother? Because rational numbers can be compared and lined up on a number line. And if you can find a number on the line, you will not mix up a sign, a discount or a temperature. Today we sort that line out in your head.

Ema and Adam look at the outdoor thermometer: −2.5 °C. Adam claims: “−3 is more than −2.5, three is bigger than two.” Ema draws a number line on the fogged glass. Zero in the middle, frost to the left. She points: −3 sits further left than −2.5. “The further left, the smaller. So −3 is smaller.” Adam is quiet for a moment. Then he nods: “So when it warms from −3 to −2.5, that is warming.” Exactly. The line does not lie, even when signs confuse you.

How to compare and convert rational numbers

The number line. Draw a line, put zero in the middle. Positive numbers grow to the right, negatives fall to the left. Every rational number has its place: 1/2 sits between 0 and 1, the number −2.5 between −3 and −2.

Comparing. Further left = smaller. That is why −7 < −2, even though seven looks bigger. With fractions and decimals, convert them to the same form first.

Fraction to decimal: divide the numerator by the denominator. 3/4 = 3 : 4 = 0.75.

Decimal to fraction: read it out loud. 0.2 is “two tenths”, so 2/10, simplified 1/5. The number 0.75 is 75/100, simplified 3/4.

The opposite number is the same distance from zero, just on the other side: the opposite of 4 is −4. The distance from zero is called the absolute value — for 4 and for −4 it is 4.

This conversion triangle (fraction ↔ decimal ↔ place on the line) you will use the whole level. Percentages, ratios and proportions sit on it.

Convert 3/8 to a decimal

A fraction is hidden division. So 3/8 = 3 : 8.

The division: 3 : 8 = 0, remainder 3. Add a decimal point and a zero: 30 : 8 = 3, remainder 6. Next: 60 : 8 = 7, remainder 4. Next: 40 : 8 = 5, remainder 0. Done.

Result: 3/8 = 0.375. Check with an estimate: 3/8 is a bit less than a half (4/8 = 0.5). And 0.375 really is a bit less than 0.5. That checks out.

Order from smallest: −1.5; 1/2; −2; 0.4

Step 1: convert everything to decimals. 1/2 = 0.5. The rest already is decimal: −1.5; −2; 0.4.

Step 2: negatives are always smaller than positives. So on the left will be −2 and −1.5, on the right 0.4 and 0.5.

Step 3: of −2 and −1.5, the smaller is −2 (further left from zero). Of 0.4 and 0.5, the smaller is 0.4.

Result: −2 < −1.5 < 0.4 < 1/2.

Write 0.45 as a fraction in simplest form

Step 1: read the number out loud. 0.45 is “forty-five hundredths”. Write: 45/100.

Step 2: simplify. Divide numerator and denominator by five: 45 : 5 = 9, 100 : 5 = 20.

Result: 0.45 = 9/20. The fraction 9/20 cannot be simplified more — 9 and 20 have no common divisor except one. That is called simplest form. Check backwards: 9 : 20 = 0.45. That checks out.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Mulai with ten. When it goes well, add more.

Revise calculating with negatives. Write only the number, with a minus if it is negative.

Practise the questions

On paper: the line and conversions

Draw a big number line from −3 to 3. It will serve the whole exercise.

1. Mark these numbers on the line: −2.5; 3/4; −1/2; 1.25. Remember: 3/4 sits between 0 and 1, closer to one.

2. Convert to decimals by dividing: 1/4, 2/5, 7/8. Write the whole method for each, not just the result.

3. Convert to fractions in simplest form: 0.6; 0.25; 0.08. First read them out loud (tenths, hundredths), then simplify.

4. Order from smallest: −0.7; 2/3; −3/4; 0. Write the conversion steps too.

Now you 💪

1. I can explain what a rational number is: any number you can write as a fraction.
2. I convert a fraction to a decimal by dividing, and a decimal to a fraction by reading it out.
3. With two negatives I can tell which is smaller — the one further left from zero.

What to take from this lesson

• A rational number = a number you can write as a fraction. Whole numbers, fractions and decimals belong here.
• Number line: left is smaller, right is bigger. That holds for negatives too.
• Fraction to decimal: divide the numerator by the denominator.
• Decimal to fraction: read it out loud (tenths, hundredths) and simplify.
• The opposite number sits equally far from zero, just on the other side.

© 2026 Ing. Martin Polak / AlgoRhino · Ketentuan penggunaan konten