Mathematics · Numbers and calculation · Grade 7

The number line: from −7 to √2

−7 °C at night, +4 °C by day — what is the difference? Why can’t √2 be written exactly?

Open the experiment

The situation

On a winter night in Tashkent the thermometer showed −7 °C, and by day +4 °C — by how many degrees did the temperature rise? A lift goes from a car park on floor −2 up to floor 5, and a debt on a bank card is also written with a negative number. On the number line there are fractions between the whole numbers, and between the fractions there is… something else: the diagonal of a square yard cannot be measured exactly with any ruler. Why? Let us take a journey along the number line.

Topic

Negative numbers and the number line; integers, rational, irrational and real numbers. Thermometer, lift floors and a bank debt: negative numbers; the diagonal of a square, √2 — a number that cannot be written as a finite decimal

Key ideas

• Negative numbers
• Rational numbers
• Irrational numbers

Formulas

The modulus is the distance from a number to zero. Subtraction is adding the opposite number.

The rule of signs: the product (quotient) of numbers with different signs is negative, with the same signs positive.

The decimal expansion of a rational number $\frac{p}{q}$ either terminates or recurs.

An irrational number is an infinite non-recurring decimal; it cannot be written as a fraction.

Theory

Negative numbers and the number line

A temperature below zero, an underground floor, a debt — these are negative numbers. On the number line, positive numbers lie to the right of 0 and negative numbers to the left. $7$ and $-7$, which differ only in sign, are opposite numbers: they are the same distance from 0. That distance is the modulus: $|-7|=|7|=7$. On the line the number on the left is smaller: $-7<-2<0<4$. Among negative numbers, the one with the larger modulus is smaller!

Adding and subtracting integers

Adding a positive number is a jump to the right on the line, adding a negative number is a jump to the left: $(-7)+11=4$. Subtracting is adding the opposite number: $3-5=3+(-5)=-2$, $(-3)-(-8)=-3+8=5$. That is why the temperature difference is $4-(-7)=4+7=11$ degrees. The rule: for numbers with the same sign the moduli are added and the sign is kept; for different signs the moduli are subtracted and the sign is that of the number with the larger modulus.

Multiplying and dividing

$(-3)\cdot 4$ means adding $-3$ four times: 4 jumps to the left, the result is $-12$. Multiplying by a negative number turns the jump the other way: $(-3)\cdot (-4)=12$. The rule: the product of numbers with different signs is negative, of numbers with the same sign positive; division is the same: $12:(-4)=-3$, $(-12):(-4)=3$. That $(-1)\cdot (-1)=1$ follows from the distributive law: $(-1)\cdot (1+(-1))=0$. Division by zero is impossible.

Rational numbers

Numbers that can be written in the form $\frac{p}{q}$ ($p$ an integer, $q$ a natural number) are rational numbers. Integers are rational too: $3=\frac{3}{1}$. If we divide the numerator by the denominator by long division we get a decimal expansion: $\frac{3}{8}=0.375$ is terminating, $\frac{1}{3}=0.333\dots =0.(3)$ is recurring (a remainder repeats). Between any two rational numbers there is another rational number — on the line they are very dense, but they do not fill the line.

Irrational and real numbers

The diagonal of a square with side 1 satisfies $d^2=1+1=2$, $d=\sqrt{2}$. Suppose $\sqrt{2}=\frac{p}{q}$ is a fraction in lowest terms; then $p^2=2q^2$, so $p$ is even, $p=2k$, and then $q^2=2k^2$ — $q$ is even too. The fraction could be reduced — a contradiction! So $\sqrt{2}$ is irrational: its expansion is infinite and not recurring; so is $\pi$. Rational and irrational numbers together are the real numbers — they fill the line without gaps: $\mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}\subset ℝ$. Operations on them are carried out with approximate values: $\sqrt{2}+\sqrt{3}\approx 1.414+1.732=3.146$; when comparing, the one on the left of the line is smaller.

About the model

The line in the drawing is limited (at most $\pm 40$), but the real line is infinite. Irrational numbers are shown on the screen to 7–8 places — this is an approximate value, not the number itself. The thermometer uses the Celsius scale; in nature there is no temperature below $-273.15$ °C, but on the line the negative numbers go on for ever. Constructing $\sqrt{n}$ as the diagonal of a rectangle with whole-number sides is shown only for $n=a^2+b^2$; the others can be constructed with a compass too.

Did you know?

• The rules for calculating with negative numbers were set out by the Indian mathematician Brahmagupta in his work “Brahmasphutasiddhanta” of 628: he called positive numbers “fortunes” and negative numbers “debts”. In Europe the signs “+” and “−” were first printed in 1489 in Johannes Widmann’s book on commercial arithmetic.
• The irrationality of $\sqrt{2}$ was discovered by the Pythagoreans in the 5th century BC; according to legend, Hippasus, who revealed the secret, was thrown into the sea. By 2024 the expansion of $\pi$ had been calculated to more than 100 trillion digits — there is no recurrence and there never will be: the irrationality of $\pi$ was proved by Johann Lambert in 1761.

Inside the experiment

The topic is explained step by step with animation, from simple to advanced: from a real-life picture to the formula and the proof. Then you drag the points yourself to check. Inside are missions, a quiz and a guided tour.

More experiments in this section

• Fractions: from pizza to formula — Grade 5
• Prime numbers, HCF and LCM — Grade 7
• Powers: from folding to a formula — Grade 7
• Roots: from squares to powers — Grade 8