Integers and Signed Numbers

• Three negative factors (odd), so the product is negative: \(1\times 2\times 3 = 6\), hence \(\textcolor{colorprop}{(-1)}\times \textcolor{colorprop}{(-2)}\times \textcolor{colorprop}{(-3)} = \textcolor{colorprop}{-6}\).
• Three negative factors (odd) and one positive factor, so the product is negative: \(4\times 1\times 5\times 2 = 40\), hence \(\textcolor{colordef}{(+4)}\times \textcolor{colorprop}{(-1)}\times \textcolor{colorprop}{(-5)}\times \textcolor{colorprop}{(-2)} = \textcolor{colorprop}{-40}\).

Both numbers are negative: the one with the larger distance to zero is the smaller number. Here \(-2.4\) is \(2.4\) from zero and \(-2.15\) is \(2.15\) from zero, and \(2.4 > 2.15\), so: $$-2.4 < -2.15.$$ Watch out: \(2.4 > 2.15\) for positive numbers, but the order reverses for the negatives.

Subtracting is adding the opposite: $$(-3.5) - (-1.2) = (-3.5) + (+1.2) = -2.3.$$

$$\textcolor{colorprop}{(-2.5)}\times \textcolor{colorprop}{(-1.2)}= \textcolor{colordef}{+3}\quad (2.5\times 1.2 = 3 \text{ and } \textcolor{colorprop}{(-)}\times \textcolor{colorprop}{(-)} = \textcolor{colordef}{(+)}) $$

$$\textcolor{colorprop}{(-8.4)}\div \textcolor{colorprop}{(-0.4)}= \textcolor{colordef}{+21}\quad (8.4\div 0.4 = 84\div 4 = 21 \text{ and } \textcolor{colorprop}{(-)}\div \textcolor{colorprop}{(-)} = \textcolor{colordef}{(+)}) $$

• \(3\div 2 = 1.5\) and \(\textcolor{colorprop}{(-)}\div \textcolor{colordef}{(+)} = \textcolor{colorprop}{(-)}\), so \(\textcolor{colorprop}{(-3)}\div \textcolor{colordef}{(+2)} = \textcolor{colorprop}{-1.5}\): the quotient is not a whole number, but it is a decimal one.
• \(1\div 3\) is not a decimal number: its exact value is written as a fraction, \(\textcolor{colordef}{(+1)}\div \textcolor{colorprop}{(-3)} = \textcolor{colorprop}{-\frac{1}{3}}\). A calculator shows \(-0.333\dots\), and \(-0.33\) is only an approximation rounded to the hundredth.

A quotient of two signed numbers is therefore not always a whole number, nor even a decimal one: the exact writing is kept, and it is said each time a value is an approximation.

$$\begin{aligned}[t] A&=\textcolor{colordef}{(+8)}-\big(\textcolor{colorprop}{(-3)}+\textcolor{colordef}{(+5)}\big)&&\text{(innermost brackets first)}\\ &=\textcolor{colordef}{(+8)}-\textcolor{colordef}{(+2)}&&\text{(}\textcolor{colorprop}{(-3)}+\textcolor{colordef}{(+5)}=\textcolor{colordef}{+2}\text{)}\\ &=\textcolor{colordef}{(+8)}+\textcolor{colorprop}{(-2)}&&\text{(add the opposite)}\\ &=\textcolor{colordef}{+6}&&\text{(opposite signs)} \end{aligned}$$

$$\begin{aligned}[t] B&=\textcolor{colorprop}{(-4)}+\textcolor{colordef}{(+9)}+\textcolor{colordef}{(+3)}+\textcolor{colorprop}{(-6)}&&\text{(add the opposite)}\\ &=\textcolor{colordef}{(+5)}+\textcolor{colordef}{(+3)}+\textcolor{colorprop}{(-6)}&&\text{(left to right)}\\ &=\textcolor{colordef}{(+8)}+\textcolor{colorprop}{(-6)}&&\text{(left to right)}\\ &=\textcolor{colordef}{+2}&&\text{(opposite signs)} \end{aligned}$$

$$\begin{aligned}[t] A&=\textcolor{colordef}{(+3)}+\textcolor{colorprop}{(-2)}\times\textcolor{colordef}{(+4)}&&\text{(given)}\\ &=\textcolor{colordef}{(+3)}+\textcolor{colorprop}{(-8)}&&\text{(multiply first)}\\ &=\textcolor{colorprop}{-5}&&\text{(opposite signs)} \end{aligned}$$

$$\begin{aligned}[t] B&=\big(\textcolor{colordef}{(+3)}+\textcolor{colorprop}{(-2)}\big)\times\textcolor{colordef}{(+4)}&&\text{(given)}\\ &=\textcolor{colordef}{(+1)}\times\textcolor{colordef}{(+4)}&&\text{(innermost brackets first)}\\ &=\textcolor{colordef}{+4}&&\text{(same signs)} \end{aligned}$$ The brackets force the addition before the multiplication, so \(B\neq A\).

Division and multiplication have the same priority: compute from left to right. $$\begin{aligned}[t] \textcolor{colordef}{(+12)}\div \textcolor{colorprop}{(-3)}\times \textcolor{colordef}{(+2)} &= \textcolor{colorprop}{(-4)}\times \textcolor{colordef}{(+2)} && \text{(division first, left to right)}\\ &= \textcolor{colorprop}{-8}. \end{aligned}$$

$$\begin{aligned}[t] C&=\textcolor{colordef}{(+18)}\div\textcolor{colorprop}{(-3)}+\textcolor{colordef}{(+2)}\times\textcolor{colorprop}{(-4)}&&\text{(given)}\\ &=\textcolor{colorprop}{(-6)}+\textcolor{colorprop}{(-8)}&&\text{(divide and multiply first)}\\ &=\textcolor{colorprop}{-14}&&\text{(same sign)} \end{aligned}$$

1. Each step wraps the previous result: $$A = (-5)\times(+3)+(+12).$$ The multiplication is done first, the addition last: \(A\) is a sum — the sum of the product \((-5)\times(+3)\) and the number \((+12)\). Its value: \(A = (-15)+(+12) = \textcolor{colorprop}{-3}\).
2. Program: “choose \((+6)\); subtract \((-4)\); divide by \((+5)\)”. The division comes last: \(B\) is a quotient — the difference \((+6)-(-4)\) divided by \((+5)\). Its value: \(B = (+10)\div(+5) = \textcolor{colordef}{+2}\).

The keys are $$\boxed{(-)}\;4.8\;\boxed{\times}\;2.5\;\boxed{-}\;\boxed{(-)}\;6.3\;\boxed{\div}\;\boxed{(-)}\;0.9\;\boxed{=}$$ and the calculator shows \(-19\). Only the second minus sign, between \(2.5\) and \(6.3\), is a subtraction; the three others are signs, typed with the sign key, and the brackets around the negative numbers are not typed.
Size: \((-5) \times 2.5\) is about \(-12\) and \((-6) \div (-1) = 6\), so the result is about \(-12 - 6 = -18\): \(-19\) has a sensible size.
Check: the multiplication and the division come first, and the sign rules give $$\begin{aligned}[t] (-4.8) \times 2.5 - (-6.3) \div (-0.9) &= (-12) - (+7) && \text{(} \times \text{ and } \div \text{ first; sign rules)}\\ &= -19. && \text{(subtracting is adding the opposite)} \end{aligned}$$ The calculator result is correct.