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Compare negative numbers

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Compare negative numbers

A negative number is a number that is less than zero. It is an opposite of positive numbers (two integers that are the same distance from 0 in opposite directions are called opposites).

Positive numbers are always greater than 0 and 0 is always greater than all the negative numbers.

With negative numbers, we have to remember that as the digit gets bigger, the number gets smaller (moved further from 0).

Example:
$$2>(−2)>(−3)$$
02-2-3

It's the same for decimals and fractions:

$$0.5>(−0.5)$$
$$(−1/4)>(−1/2)$$


Compare negative decimals, fractions and mixed numbers

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Compare negative numbers

A negative number is a number that is less than zero. It is an opposite of positive numbers (two integers that are the same distance from 0 in opposite directions are called opposites).

Positive numbers are always greater than 0 and 0 is always greater than all the negative numbers.

With negative numbers, we have to remember that as the digit gets bigger, the number gets smaller (moved further from 0).

Example:
$$2>(−2)>(−3)$$
02-2-3

It's the same for decimals and fractions:

$$0.5>(−0.5)$$
$$(−1/4)>(−1/2)$$


Adding and subtracting negative numbers

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Adding and subtracting negative numbers

If a number has no sign it means that it is a positive number (positive number are higher than 0). We can write it with a plus sign, e.g. 5=+5. We add and subtract, as we normally do. The sum of two positives numbers is always positive (higher than zero).

Subtraction of two positive numbers can take us beyond zero in some cases: 3−10=(−7) To add and subtract with negative numbers, at first add positive signs to positive numbers, e.g. 20−5=20−+5

For each combination of two signs follow these rules:

RuleExample

Two like signs (++ or −−) become a positive sign.

5+(+3)=5+3=8
−5+(+3)=−5+3=−2
2−(−3)=2+3=5
−4−(−3)=−4+3=−1

Two unlike signs (+− or −+) become a negative sign.

4+(−3)=4−3=1
−8+(−4)=−8−4=−12
1−(+3)=1−3=−2
−5−(+4)=−9


Negative numbers, fractions and decimals

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Order of operations

When calculations with negative numbers have more than one operation, we have to follow rules for the order of operations:

Step 1:
Do all operations that lie inside the parentheses first:

$$−3−8×(2+3) = −3−8×5$$

Step 2:
From left to right, do all multiplication and division.
$$−3−8×5 = −3−40$$

Step 3:
From left to right, do all addition and subtraction.
$$−3−40 = −43$$

Adding and subtracting negative numbers

If a number has no sign it means that it is a positive number (positive number are higher than 0). We can write it with a plus sign, e.g. 5=+5. We add and subtract, as we normally do. The sum of two positives numbers is always positive (higher than zero).

Subtraction of two positive numbers can take us beyond zero in some cases: 3−10=(−7) To add and subtract with negative numbers, at first add positive signs to positive numbers, e.g. 20−5=20−+5

For each combination of two signs follow these rules:

RuleExample

Two like signs (++ or −−) become a positive sign.

5+(+3)=5+3=8
−5+(+3)=−5+3=−2
2−(−3)=2+3=5
−4−(−3)=−4+3=−1

Two unlike signs (+− or −+) become a negative sign.

4+(−3)=4−3=1
−8+(−4)=−8−4=−12
1−(+3)=1−3=−2
−5−(+4)=−9

Multiplying and dividing negative numbers

To multiply and divide two numbers follow these rules:
When the signs are different the result is negative:

$$(−1)×3=(−3)$$
$$(−6)÷3=(−2)$$
$$8/{(−4)}=(−2)$$

When the signs are the same the result is positive:
$$(−1)×(−3)=3$$
$$4×3=12$$
$${(−2)}/{(−3)}=2/3$$

If there is a long multiplication with some negatives, just cancel off "minus" signs in pairs:
E.g. simplify
$$ (–1)×(–2)×(–1)×(–3)×(–4)×(–2)×(–1)$$

At first count up the minus signs: seven minus signs.
So there are three pairs that can be eliminated, with one left over. As a result, the final answer will be negative: (−48)

In other words, if there is an even number of negative factors (2 factors, 4 factors, 6 factors, etc.) , the result is positive. If there is an odd number of factors with negative sign, the result is negative.

Example:

$$(–1)×(–2)×4×(–1)×(–3)=?$$

There are four negative numbers → result will be positive:
$$(–1)×(–2)×4×(–1)×(–3)=24$$



Absolute value

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Absolute value

The absolute value of a number is its distance from zero without considering which direction from zero the number lies. We denote it with ||. E.g. the absolute value of 5 is 5 (|5|=5), and the absolute value of −5 is also 5 (|−5|=5). As you can see, the absolute value of a number is never negative.

When doing calculations with absolute value, you should respect order of operations and evaluate it at first as parentheses, e.g:

$$−5+7×|3−8|=−5+7×|−5|=$$
$$=−5+7×5=−5+35=30$$


   
   

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