GED Math: The Number Line and Signed Numbers
The GED test may ask you to work with signed numbers in spreadsheets or to reason about the sign of an answer without calculating it fully. In a spreadsheet, a cell is named by its column and row (A1 is column A, row 1), the symbol * means multiplication, and / means division. Substitute the number from each cell, keep the parentheses, and then use the order of operations. To judge the sign of a long product, count the negative factors. When writing an expression from words (“the product of 2 and 8 divided by −8”), build it piece by piece and use parentheses or a fraction bar so the operations happen in the right order.
1. Signed Numbers and the Number Line
Signed numbers include zero, all positive numbers, and all negative numbers. Zero is neither positive nor negative. On a number line, positive numbers are to the right of zero and negative numbers are to the left, so numbers get greater as you move right. A positive number may be written with a plus sign (+4), but if there is no sign it is positive. A negative number must have a minus sign (−5). A signed number gives two facts: the sign tells the direction from zero, and the number tells the distance from zero. The farther left a number is, the smaller it is, so −8 is less than −3 even though 8 is greater than 3.
Worked Example: Place −5 and +4 on a number line, then put −4, 2, −1, and 0 in order from least to greatest.
−5 is five units left of zero and +4 is four units right of zero. For the order, read left to right on the number line: −4, −1, 0, 2.
1. Which number is farthest to the left on a number line?
Explanation: The farther left, the smaller the number. −7 is the least of the four.
2. Which statement is true?
Explanation: −3 is to the right of −8 on the number line, so −3 is greater.
3. Where is the number −5 on a number line?
Explanation: The negative sign means left of zero, and 5 is the distance from zero.
4. Which list is in order from least to greatest?
Explanation: Reading the number line from left to right gives −4, −1, 0, 2.
5. At night the temperature in Town A is −12°F and in Town B it is −9°F. Which town is colder, and why?
Explanation: −12 is to the left of −9 on the number line, so it is the lower temperature.
2. Adding Signed Numbers
You can model addition on a number line: start at the first number, then move right to add a positive number or left to add a negative number. Without a number line, use two rules. Like signs: add the numbers and keep the same sign. Unlike signs: subtract the smaller number from the larger and use the sign of the number that is larger (farther from zero). The online calculator has a change-sign key, (−), which turns a positive number into a negative one and back. Do not confuse it with the subtraction key.
Worked Example: Find (a) −5 + 4, (b) −2 + (−3), and (c) 15 + (−25).
(a) Unlike signs: 5 − 4 = 1, and 5 is the larger number, so the answer is −1. (b) Like signs: 2 + 3 = 5 and both are negative, so −5. (c) Unlike signs: 25 − 15 = 10, and 25 is larger and negative, so −10.
1. Find 8 + (−3).
Explanation: Unlike signs: 8 − 3 = 5, and 8 is larger and positive, so 5.
2. Find −14 + (−6).
Explanation: Like signs: 14 + 6 = 20, keep the negative sign.
3. Find −9 + 13.
Explanation: Unlike signs: 13 − 9 = 4, and 13 is larger and positive.
4. Find 7 + (−12).
Explanation: 12 − 7 = 5, and 12 is larger and negative, so −5.
5. Find −55 + 20.
Explanation: 55 − 20 = 35, and 55 is larger and negative.
6. A number line shows a jump that starts at 2 and moves 3 units to the left. Which expression does it show?
Explanation: Starting at 2 and moving left 3 is adding −3: 2 + (−3) = −1.
3. Subtracting Signed Numbers
Subtraction is the opposite of addition. To subtract a signed number, change the operation to addition and change the sign of the number you are subtracting; then use the addition rules. Subtracting a negative is the same as adding a positive, because it takes away a debt. The second number is the only sign that changes, and the first number stays as it is.
Worked Example: Find 3 − 8, then 11 − (−2), then −4 − (−5).
3 − 8 = 3 + (−8) = −5. 11 − (−2) = 11 + 2 = 13. −4 − (−5) = −4 + 5 = 1.
1. Find 6 − 9.
Explanation: 6 − 9 = 6 + (−9) = −3.
2. Find 11 − (−2).
Explanation: Subtracting −2 is adding 2: 11 + 2 = 13.
3. Find −7 − (−3).
Explanation: −7 + 3 = −4.
4. Find −5 − 6.
Explanation: −5 − 6 = −5 + (−6) = −11.
5. Find 50 − (−15).
Explanation: 50 + 15 = 65.
6. The temperature was −4°F and then rose to 11°F. How many degrees did it rise?
Explanation: The change is 11 − (−4) = 11 + 4 = 15 degrees.
4. Combining Several Signed Numbers
You can use the same rules to combine many terms. Step 1: rewrite each subtraction as addition of the opposite. Step 2: add all the positive terms and, separately, add all the negative terms. Step 3: combine the two results with the unlike-signs rule. This keeps the signs organized, and you can also work left to right one pair at a time.
Worked Example: Find (−5) + 6 − 4 − (−2).
Rewrite: (−5) + 6 + (−4) + 2. Positives: 6 + 2 = 8. Negatives: −5 + (−4) = −9. Combine: 8 + (−9) = −1.
1. Find 7 + (−3) + (−5) − 10.
Explanation: Positives: 7. Negatives: −3, −5, −10 total −18. 7 + (−18) = −11.
2. Find 66 + (−22) − 33.
Explanation: 66 − 22 = 44, and 44 − 33 = 11.
3. Find −14 − (−6) + 18.
Explanation: −14 + 6 + 18 = −14 + 24 = 10.
4. Find 80 − (−15) − 20.
Explanation: 80 + 15 − 20 = 95 − 20 = 75.
5. Find −23 + (−11) − (−15) + 21.
Explanation: −23 − 11 + 15 + 21 = −34 + 36 = 2.
6. At noon the temperature was 92°F. Over the next two hours it changed by +12, −5, +6, −3, and +13 degrees. What was the temperature at the end?
Explanation: The changes total 12 − 5 + 6 − 3 + 13 = 23, and 92 + 23 = 115°F.
5. Multiplying and Dividing Signed Numbers
In algebra, the times sign (×) can look like the variable x, so multiplication is written with a dot or by placing numbers next to each other, often in parentheses. The expressions −5·6, −5(6), and (−5)(6) all mean “−5 times 6.” Division is usually written as a fraction bar, so 20/−5 and 20 over −5 both mean 20 divided by −5. The sign rules are the same for both operations: same signs give a positive answer, and different signs give a negative answer. To multiply several factors, count the negatives: an even number of negative factors gives a positive product and an odd number gives a negative product. If any factor is 0, the product is 0.
Worked Example: Find (a) 4(−25), (b) −160 ÷ −8, and (c) (−5)(6)(−1)(−2)(2).
| Signs | Result | Example |
|---|---|---|
| same | positive | (−3)(−4) = 12 |
| different | negative | (−3)(4) = −12 |
(a) Multiply 4 × 25 = 100; the signs differ, so −100. (b) 160 ÷ 8 = 20; the signs are the same, so 20. (c) 5 × 6 × 1 × 2 × 2 = 120, and three of the factors are negative (odd), so −120.
Notice: 4 − 5 means four minus five (−1), but 4(−5) means four times negative five (−20).
1. Find (7)(−3).
Explanation: Different signs give a negative product: −21.
2. Find (−2)(−9).
Explanation: Same signs give a positive product: 18.
3. Find −25 ÷ 5.
Explanation: 25 ÷ 5 = 5, and the signs differ, so −5.
4. Find (−1)(2)(−3)(2)(−1).
Explanation: 1 × 2 × 3 × 2 × 1 = 12. There are three negative factors (odd), so the answer is −12.
5. Find (−4)(−6)(−5).
Explanation: 4 × 6 × 5 = 120, and there are 3 negatives, so −120.
6. Find 25(4) ÷ (−5).
Explanation: 25(4) = 100, and 100 ÷ (−5) = −20.
7. Find (−5)(−2)(0)(−1).
Explanation: Any product with a factor of 0 is 0.
6. Order of Operations with Signed Numbers
When a problem has more than one operation, follow the order of operations. First do any operations inside parentheses, and for a fraction bar do the work above and below the bar first and then divide. Then do all multiplication and division from left to right. Finally do all addition and subtraction from left to right. Handle each sign carefully at every step, and rewrite subtraction as addition of the opposite when it helps.
Worked Example: Find (−4)(6) − (3 + (−9)) ÷ (−2).
1. Multiply: −24. 2. Above the bar: 3 + (−9) = −6, so the fraction is −6 ÷ (−2) = 3. 3. Subtract: −24 − 3 = −27.
1. Find 2 + 3(−4).
Explanation: Multiply first: 3(−4) = −12. Then 2 + (−12) = −10.
2. Find (−3)(2) + 10 ÷ (−5).
Explanation: Multiply and divide first: −6 and −2. Then −6 + (−2) = −8.
3. Find 50 ÷ 2(−5).
Explanation: Work left to right: 50 ÷ 2 = 25, and 25(−5) = −125.
4. Find 6(5) over (−3)(2), that is, [6(5)] ÷ [(−3)(2)].
Explanation: Top: 30. Bottom: −6. 30 ÷ (−6) = −5.
5. Find 4(−4) over −8(−2).
Explanation: Top: −16. Bottom: 16. −16 ÷ 16 = −1.
6. Find −20 + (−3)(−4).
Explanation: Multiply first: (−3)(−4) = 12. Then −20 + 12 = −8.
7. Applying Signed Numbers: Spreadsheets and Reasoning
The GED test may ask you to work with signed numbers in spreadsheets or to reason about the sign of an answer without calculating it fully. In a spreadsheet, a cell is named by its column and row (A1 is column A, row 1), the symbol * means multiplication, and / means division. Substitute the number from each cell, keep the parentheses, and then use the order of operations. To judge the sign of a long product, count the negative factors. When writing an expression from words (“the product of 2 and 8 divided by −8”), build it piece by piece and use parentheses or a fraction bar so the operations happen in the right order.
Worked Example: In the spreadsheet, A1 = −3, C1 = 7, A3 = −1, and B3 = 3. Find A1*C1*A3 / (B3*A3).
| A | B | C | |
|---|---|---|---|
| 1 | −3 | 4 | 7 |
| 2 | 2 | −5 | −8 |
| 3 | −1 | 3 | −2 |
Top: (−3)(7)(−1) = 21. Bottom: (3)(−1) = −3. Then 21 ÷ (−3) = −7.
1. Using the spreadsheet above, what is A1*C1*A3 / (B3*A3)?
Explanation: Top is 21 and bottom is −3, so the value is −7.
2. Using the spreadsheet, what is A2*B2 + C2?
Explanation: A2*B2 = 2(−5) = −10, and −10 + (−8) = −18.
3. Using the spreadsheet, what is A1 + B1*C3?
Explanation: Multiply first: 4(−2) = −8. Then −3 + (−8) = −11.
4. The product of 2 and 8 is divided by −8. Which expression shows this?
Explanation: Multiply 2 and 8 first, then divide that product by −8: [2(8)] ÷ (−8) = −2.
5. Which is true about the value of (52)(−103)(−45)(−8)(3)?
Explanation: There are three negative factors (an odd number), so the product is negative. You do not need to multiply it out.
6. Which statement is true about 4 − 5 and 4(−5)?
Explanation: 4 − 5 is subtraction (−1). 4(−5) is multiplication (−20).
Common Mistakes to Avoid
- Treating the minus sign as always meaning subtract. A minus sign can mean a negative number, a subtraction, or the opposite of a number.
- Forgetting to change the sign of the second number when rewriting subtraction as addition. 11 − (−2) is 11 + 2, not 11 − 2.
- Adding like signs and then losing the sign: −14 + (−6) is −20, not 20.
- With unlike signs, using the sign of the first number instead of the sign of the larger number.
- Thinking −8 is greater than −3 because 8 is greater than 3. On the number line, −3 is farther right, so it is greater.
- Applying the “two negatives make a positive” rule to addition. It works for multiplying and dividing, but −3 + (−4) is −7.
- Miscounting negative factors in a long product. Count them first: even gives positive, odd gives negative.
- Doing addition before multiplication. In 2 + 3(−4), multiply first.
- Confusing 4 − 5 (subtraction) with 4(−5) (multiplication).
Frequently Asked Questions
What is the rule for adding signed numbers?
With like signs, add the numbers and keep the sign. With unlike signs, subtract the smaller from the larger and use the sign of the larger number.
How do I subtract a negative number?
Change the subtraction to addition and change the sign of the second number. Subtracting −2 is the same as adding 2, so 11 − (−2) = 13.
Why is a negative times a negative positive?
Each pair of negative factors cancels out. Two negatives give a positive, three give a negative, four give a positive, and so on.
How do I find the sign of a long product?
Ignore the numbers and count the negative factors. An even count gives a positive result and an odd count gives a negative result, unless a factor is zero.
How does the GED calculator handle negative numbers?
Use the change-sign key, written (−), to turn a number negative. Do not use the subtraction key for a negative number.
What is the order of operations with signed numbers?
Do parentheses and fraction bars first, then multiplication and division from left to right, then addition and subtraction from left to right. Keep track of every sign at each step.
Where can I practice more problems like these?
The GED Math practice in the GED Math QBank gives you more practice on this topic, and you can browse every other topic in the full GED Math QBank.