Signed numbers and rational calculation

A sign gives a position relative to zero

On a number line, values increase to the right. Therefore -7 < -2 < 0 < 3. A negative number with greater magnitude lies further left: -7 is less than -2, even though 7 is greater than 2.

A temperature rises from -4°C by 7°C: -4 + 7 = 3°C. A fall of 7°C from 3°C reverses that change: 3-7 = -4°C. Always distinguish a signed value from the size of a change.

Subtracting undoes addition

5-(-3) asks for a value which, after adding -3, becomes 5. That value is 8, since 8 + (-3) = 5. Subtracting a negative therefore adds its opposite: 5-(-3) = 5 + 3 = 8.

Keep brackets around negative substituted values. -2-(-6) = 4 differs from -2-6 = -8.

Multiplication must preserve the distributive law

3 × (-4) = -12, three equal negative changes. Also 0 = (-3) × [4 + (-4)]. Distributing gives 0 = -12 + (-3)(-4), so (-3)(-4) = 12. This explains the positive product of two negatives without confusing multiplication with a direction on the line.

Division uses the inverse relationship: -12÷3 = -4, and -12÷(-3) = 4. Check by multiplying quotient and divisor. Division by zero is undefined.

Fractions and decimals follow the same signs

-3/4 + 1/2 = -1/4, using quarters. For (-0.6)(-0.5) = 0.30, the positive sign follows from two negative factors and the magnitude from six tenths times five tenths. An estimate catches a misplaced decimal point: the magnitude must be below 0.6.

Check your understanding 1 Keep the operation and sign separate

Calculate -2/3-(-1/6) and (-1.2)÷0.3. Check the division.

Your answer or drawing on paper

Show answer

-4/6 + 1/6 = -3/6 = -1/2. The quotient is -4, checked by (-4)(0.3) = -1.2. The negative divisor was subtracted in the first task; the positive divisor is used in the second.

Interpret the result

A lift starts on level 2, descends 5 levels and then rises 1. Find its final signed level under a model where adjacent level labels differ by one, including level 0.