ICSE Class 8 Maths Number Systems Chapter 05 Directed Numbers

Directed Numbers

Operations Involving Directed Numbers

Introduction

Seven boys A, B, C, D, E, F, and G live in houses, 50 m from each other, on the same side of a straight road.

Each distance, along with its direction, in the above example, is known as a directed number. The reference point in each case is known as the zero point.

Remember: Integers are directed numbers, where the reference point is always the digit 0 on the number line.

G's house is 300 m from A's house and F's house is 200 m from B's house. The distance of A's as well as G's house is 150 m from D's house. We can say that A stays 150 m away to D's left and G stays 150 m away to D's right.

Thus, to describe a location, along with the distance, the direction is important too. As we see from the diagram above, if D's house is considered as the reference point, E, F, and G's houses lie to the right and A, B, and C's houses lie to the left. If the right direction is denoted by a positive sign and the left direction by a negative sign, then from D's house,

A = - 150 m, B = - 100 m, C = - 50 m and E = + 50 m, F = + 100 m, and G = + 150 m

If B's house is the reference point, then from B's house,

A = - 50 m, C = + 50 m, D = + 100 m, E = + 150 m, F = + 200 m, and G = + 250 m

Example 1

The maximum temperature recorded in the third week of May 2004 in a place is given in the following table:

Taking the average maximum temperature during the week as the zero point, represent the temperature of each day of the week in °C above or below average.

The average maximum temperature of the week

\[\frac{35 + 39 + 37 + 40 + 36 + 34 + 38}{7} = \frac{259}{7} = 37 \text{°C} = \text{zero point}\]

Teacher's Note

Temperature readings in weather forecasts use directed numbers to show how many degrees above or below a reference point conditions are expected to change.

Operations Involving Directed Numbers

1. Positive directed numbers are numerically greater than negative directed numbers.

2. The zero point is greater than all negative numbers and less than all positive numbers.

3. Every directed number (other than 0) has another number of opposite sign, known as its additive inverse.

Addition

1. The sum of two directed numbers of the same sign is the sum of the absolute numbers with the same sign as the addends.

2. The sum of two directed numbers of different signs is the difference of the absolute numbers with sign of the addend with the greater numerical value.

Example 2: (-71) + (-44) = -115

Example 3: (+105) + (+27) = +132

Example 4: (+85) + (-99) = -14

Example 5: (+261) + (-205) = +56

Try this! 1. (+183) + (-85) = 2. (-364) + (-36) =

Subtraction

Subtraction of a directed number involves addition of its opposite or additive inverse.

Example 6: (+109) - (+54) = (+109) + (-54) = +55

Example 7: (+64) - (-46) = (+64) + (+46) = +110

Example 8: (-219) - (+29) = (-219) + (-29) = -248

Example 9: (-102) - (-88) = (-102) + (+88) = -14

Try this! 1. (-183) - (-85) = 2. (-364) - (-36) =

Multiplication

1. The product of two directed numbers of the same sign is positive.

2. The product of two directed numbers of different signs is negative.

Example 10: (-36) \(\times\) (-5) = +180

Example 11: (+42) \(\times\) (+6) = +252

Example 12: (-212) \(\times\) (+3) = -636

Example 13: (+3) \(\times\) (-81) = -243

Division

1. The quotient of two directed numbers of the same sign is positive.

2. The quotient of two directed numbers of different signs is negative.

Example 14: (+544) \(\div\) (+8) = +68

Example 15: (-623) \(\div\) (-7) = +89

Example 16: (+264) \(\div\) (-11) = -24

Example 17: (-306) \(\div\) (+9) = -34

Try this! 1. (-23) \(\times\) (+11) = 2. (-23) \(\times\) (-9) = 3. (-99) \(\div\) (+11) = 4. (-99) \(\div\) (-9) =

Simplification

1. When there is a (+) sign before a grouping symbol or bracket, the bracket may be removed without changing the sign of any number within it.

2. When there is a (-) sign before a bracket, the bracket may be removed, changing the sign of all the numbers within it.

3. When there is a directed number before a bracket, the bracket may be removed, multiplying the directed number with all the numbers within.

4. The order of operations involving different grouping symbols is as follows:

vinculum - operate first () parentheses - operate second { } braces - operate third [ ] square brackets - operate fourth

5. The order of operations in simplification is as follows:

B - first operate within grouping symbols, or brackets.

O - operate on 'of' between two directed numbers by multiplying

D - divide

M - multiply

A - add

S - subtract

Example 18: Simplify (+1) \(\times\) (-1) + (+1) \(\div\) (-1) - (-1) + (-1) \(\times\) (-1) \(\div\) (-1)

= (+1) \(\times\) (-1) + (-1) - (-1) + (+1) \(\times\) (-1) \(\div\) (-1) (first divide)

= (-1) + (-1) - (-1) + (-1) + (-1) (then multiply)

= (-2) + (-1) + (-2) (then add)

= (-4) + (-1) (finally subtract)

= (-4) + (+1) = -3

Example 19: Simplify: {7 of 6 + 2 - 4(8 \(\times\) 12 \(\div\) 3 + -3 of + 6 - 2)} \(\div\) (-3)

= {7 of 6 + 2 - 4(8 \(\times\) 12 \(\div\) 3 + -18 - 2)} \(\div\) (-3)

= {7 of 6 + 2 - 4(8 \(\times\) 12 \(\div\) 3 - 20)} \(\div\) (-3)

= {7 of 6 + 2 - 4(8 \(\times\) 4 - 20)} \(\div\) (-3)

= {7 of 6 + 2 - 4(32 - 20)} \(\div\) (-3)

= {7 of 6 + 2 - 4 \(\times\) 12} \(\div\) (-3)

= {42 + 2 - 48} \(\div\) (-3)

= {21 - 48} \(\div\) (-3)

= -27 \(\div\) (-3) = + 9

Try this! 1. 24 + (-8) + 3 \(\times\) (-2)

Teacher's Note

Order of operations with directed numbers is essential in computer programming, where the sequence of calculations determines whether a result is correct or produces an error.

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