Sec 1 Integers, Rational and Real Numbers: Examples & Guide

Secondary 1 Mathematics · G3 focus · Study guide

Use a number line to reason about signs, distinguish types of numbers, and calculate accurately with fractions, powers and roots.

Concept → worked examples → practice14 min read · practice linked below

Know which kind of number you are using

Whole numbers are $0,1,2,3,\ldots$ . Integers also include their negatives: $\ldots,-3,-2,-1,0,1,2,3,\ldots$ . In this guide, natural numbers mean the positive counting numbers $1,2,3,\ldots$ ; some books include 0, so check the convention in a question.

A rational number can be written as $\frac{p}{q}$ , where $p$ and $q$ are integers and $q\ne0$ . Every integer is rational: $-4=\frac{-4}{1}$ . A rational decimal either terminates, such as $0.375=\frac38$ , or repeats forever, such as $0.\overline3=\frac13$ .

An irrational number cannot be written as such a fraction. Its decimal neither terminates nor repeats; examples include $\sqrt2$ and $\pi$ . Rational and irrational numbers together form the real numbers, which can be placed on a number line.

Worked example 1: a root sign does not always mean irrational

Classify the numbers $-7,\ 0,\ \frac38,\ \sqrt{81},\ \sqrt{10},\ 0.\overline6$ as rational or irrational. Identify the integers.

Evaluate any simple roots first: $\sqrt{81}=9$ , an integer.
$-7,0,\frac38,9$ are rational. The recurring decimal is also rational because $0.\overline6=\frac23$ .
Since 10 is not a perfect square, $\sqrt{10}$ is irrational. It lies between 3 and 4 because $9<10<16$ .

Answer: the only irrational number is $\sqrt{10}$ . The integers are $-7,0,\sqrt{81}$ . All six numbers are real. A calculator display of a few digits does not turn an irrational number into a rational one: it is only an approximation.

Use position to compare signed numbers

Numbers increase as you move right on a number line. Thus $-5<-2$ , even though 5 is greater than 2. A negative number with a greater distance from zero is smaller. Ascending order means smallest to largest.

The symbols $<$ and $>$ exclude equality; $\le$ and $\ge$ include it. For example, the integers satisfying $-2<x\le2$ are $-1,0,1,2$ . If $x$ can be any real number, fractions and decimals between the endpoints are included too.

Worked example 2: compare fractions and decimals

Arrange $-1.25,\ -\frac43,\ -1.2,\ -\frac32$ in ascending order.

Write comparable decimal forms: $-\frac43=-1.333\ldots$ and $-\frac32=-1.5$ .
All four are negative. On the number line, $-1.5$ is furthest left, followed by $-1.333\ldots$ , $-1.25$ , and $-1.2$ .
Give the answer in the original forms: $\boxed{-\frac32<-\frac43<-1.25<-1.2}.$

Check without rounding: over a common denominator of 60, the numbers in this order are $-\frac{90}{60},-\frac{80}{60},-\frac{75}{60},-\frac{72}{60}$ . Their numerators increase.

The positions use the same scale. The decimal

-1.25

is between

-\frac43

and

-1.2

Add and subtract by thinking about a change

Adding a positive number moves right; adding a negative number moves left. Subtraction means adding the opposite: $a-b=a+(-b)$ . Therefore $4-(-3)=4+3=7$ . The minus sign may describe a number, as in $-3$ , or an operation, as in $4-3$ ; identify which job it is doing.

Worked example 3: a temperature change

The temperature starts at $-4^\circ\mathrm C$ , rises by $7^\circ\mathrm C$ , then falls by $5^\circ\mathrm C$ . Find the final temperature and the overall change.

A rise adds 7: $-4+7=3$ .
A fall subtracts 5: $3-5=-2$ . The final temperature is $\boxed{-2^\circ\mathrm C}$ .
The overall change is final minus initial: $-2-(-4)=2$ . This is a rise of $2^\circ\mathrm C$ , even though the final temperature is below zero.

Check: the two changes combine to $+7-5=+2$ , and $-4+2=-2$ . Distinguish a temperature from a change in temperature.

Explore a signed move

Choose a starting number and an operation. Predict the direction before changing the second number. Half-unit steps let you explore fractions as well as integers.

The first dot marks the start; the arrow gives the direction and the final dot marks the answer.

−3 + 5 = 2. Adding a positive number moves right.

Multiply, divide and keep fractions exact

Multiplication or division of two numbers with the same sign gives a positive result; different signs give a negative result. For example, $(-3)(-4)=12$ and $12\div(-4)=-3$ . This sign rule is for multiplication and division, not for addition. Division by zero is undefined.

To add or subtract fractions, first use a common denominator. To multiply, multiply numerators and denominators, cancelling common factors when helpful. To divide by a non-zero fraction, multiply by its reciprocal. Convert a mixed number into an improper fraction first: $-1\frac12=-\frac32$ .

Worked example 4: a fraction calculation

Evaluate $-1\frac12+\frac34\div\left(-\frac98\right)$ .

Division comes before addition. Convert the mixed number and change division to multiplication by the reciprocal: $-\frac32+\frac34\times\left(-\frac89\right).$
Cancel factors in the product: $\frac34\times(-\frac89)=-\frac23$ .
Use denominator 6 for the addition: $-\frac32-\frac23=-\frac96-\frac46=\boxed{-\frac{13}{6}}.$

Check: $-\frac23\times(-\frac98)=\frac34$ , confirming the division. The final value is about $-2.167$ , which is smaller than $-1.5$ , as expected when adding a negative quantity.

Brackets, powers and calculator entry

Work inside brackets first, then powers and roots, then multiplication and division from left to right, then addition and subtraction from left to right. A fraction bar groups the whole numerator and denominator.

Brackets change the base of a power: $(-4)^2=16$ , but $-4^2=-(4^2)=-16$ . Similarly, $(-2)^3=-8$ . When entering a negative base on a calculator, include brackets, and check that a root covers the whole intended expression.

Worked example 5: combine roots and signed powers

Evaluate $8-3\left[\frac{\sqrt[3]{27}}2+1\div2\right]+\sqrt{85-6^2}$ .

Find the cube root and the square: $\sqrt[3]{27}=3$ , $6^2=36$ .
The square bracket is $\frac32+\frac12=2$ .
The square root is $\sqrt{85-36}=\sqrt{49}=7$ . Do not calculate $\sqrt{85}-36$ .
Finish in the correct order: $8-3(2)+7=8-6+7=\boxed9.$

Calculator check: enter the complete bracketed expression before pressing equals. Its display should agree with 9. Retain full precision in calculations that do not simplify exactly, and round only the final answer as requested.

Common mistakes and next steps

Assuming every root is irrational: simplify first; $\sqrt{81}=9$ .
Ordering negatives by ignoring their signs: compare their positions on the number line.
Using “two negatives make a positive” everywhere: $-3+(-4)=-7$ , while $(-3)(-4)=12$ .
Adding denominators: $\frac12+\frac13=\frac36+\frac26=\frac56$ , not $\frac25$ .
Calculating left to right regardless of operation: multiplication and division come before addition and subtraction.

Try the integers and rational numbers practice bank for classification, ordering and combined calculations. Write the intermediate lines even when you can check the result with a calculator.