A difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. “The difference of a and b” means a − b, in that order.
The difference answers “how many more?”: Lena has 5 more than Omar. On a number line it is the gap between the two numbers, 12 − 7 = 5.
Formal definition and notation
For numbersa and b, the difference of a and b is a − b, the unique number d with b + d = a. Here a is the minuend and b the subtrahend. The difference is negative when b > a. The distance between a and b on the number line is the absolute value|a − b|, which does not depend on the order.
Symbol
Read it as
Meaning
a − b
a minus b
the difference of a and b
|a − b|
the absolute value of a minus b
the positive difference, or distance between two numbers
Δ y
delta y
a change, the difference between a new value and an old value
Key facts
The difference of 20 and 8 is 20 − 8 = 12; the difference of 8 and 20 is 8 − 20 = −12. Order matters.
Check any difference by adding: 12 + 8 = 20.
Word clues for a difference: “how many more”, “how many fewer”, “how much less”, “how far apart” and “change”.
Adding the same amount to both numbers keeps the difference: 52 − 29 = 53 − 30 = 23. This is a quick mental-math trick.
A difference of 0 means the two numbers are equal.
In statistics, the range is a difference: maximum minus minimum.
Try it: difference as distance
Interactive
Drag the two points. The readout shows the difference and the distance between them. Put one point below zero to see that the distance still counts every step.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Difference examples, step by step
Example 1: How many more?
Warm-up
A red ribbon is 45 cm long and a blue ribbon is 28 cm long. How much longer is the red ribbon?
“How much longer” asks for a difference: 45 − 28.
Count up from 28: 2 to reach 30, then 15 to reach 45, so 2 + 15 = 17.
Check: 28 + 17 = 45.
Answer: The red ribbon is 17 cm longer.
Example 2: Order matters
Core
Find the difference of 15 and 42, then the difference of 42 and 15.
“The difference of 15 and 42” means 15 − 42 = −27.
“The difference of 42 and 15” means 42 − 15 = 27.
The two answers are opposites; both are 27 units apart.
Answer:15 − 42 = −27 and 42 − 15 = 27.
Example 3: Temperature change
Test-style
At 6 a.m. the temperature was −11degrees F. By 2 p.m. it was 19 degrees F. By how many degrees did it rise?
Change = final − start = 19 − (−11).
Subtracting a negative means adding: 19 + 11.
19 + 11 = 30.
Answer: The temperature rose 30 degrees.
Common mistakes
✗ Mistake“The difference of 9 and 25” is 25 − 9 = 16.
✓ InsteadIt is 9 − 25 = −16. Keep the order given: first number minus second.
Students often put the bigger number first out of habit. The phrase sets the order.
✗ Mistake63 − 27 = 44, by taking 3 from 7 in the ones column.
✓ InsteadRegroup: 63 is 5 tens and 13 ones. 13 − 7 = 6 and 5 − 2 = 3, so the difference is 36.
Each column subtracts the bottom digit from the top digit. Taking the smaller from the larger ignores regrouping.
✗ MistakeThe difference between −4 and 6 is 2.
✓ Instead6 − (−4) = 10. They are 10 units apart on the number line.
Dropping the negative sign makes the gap look smaller. Count across zero.
The range of a data set is one special difference: largest minus smallest. For 3, 9, 14 it is 14 − 3 = 11.
Quick check: 3 questions
Question 1 of 3
What is the difference of 81 and 46?
Show answer and explanation
Answer: C.81 − 46 = 35. 127 is the sum; 45 skips regrouping in the ones column; −35 reverses the order.
Question 2 of 3
Sam read 132 pages. Ana read 87 pages. How many more pages did Sam read?
Show answer and explanation
Answer: A. “How many more” is a difference: 132 − 87 = 45. 55 is a regrouping slip, 219 adds the numbers, and 155 subtracts the wrong digits.
Question 3 of 3
What is the difference −7 − 5?
Show answer and explanation
Answer: D. Start at −7 and move 5 more to the left: −12. −2 and 2 treat it like −7 + 5 or 7 − 5; 12 drops the sign.
Difference: frequently asked questions
What does difference mean in math?
Difference means the result of subtraction. To find the difference of two numbers, subtract the second from the first: the difference of 50 and 18 is 50 − 18 = 32. In word problems the difference answers “how many more?”, “how many fewer?” or “how far apart?”. It is one of the four result words: sum, difference, product, quotient.
Can a difference be negative?
Yes. If the second number is larger, the difference is negative: 4 − 9 = −5. When a problem asks how far apart two numbers are, use the positive difference, the absolute value |4 − 9| = 5. Questions such as “how many more” usually expect a positive answer, so subtract the smaller number from the larger.
Number LineA number line is a straight line with numbers placed at equal distances along it, so every point stands for a number. Zero sits in the middle, positive numbers go to the right and negative numbers to the left. Numbers increase from left to right, so a number line shows order, distance and adding as moving.Number System · Grade 2, Grade 3, Grade 6NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2MinuendThe minuend is the number you subtract from: the first number in a subtraction. In 17 − 6 = 11, the minuend is 17, the subtrahend is 6 and the difference is 11. The minuend is the whole you start with, so it equals the difference plus the subtrahend.Operations · Grade 1, Grade 2, Grade 3SubtrahendThe subtrahend is the number being subtracted: the number written after the minus sign. In 12 − 5 = 7, the subtrahend is 5. It is taken away from the minuend (12) to leave the difference (7). To find a missing subtrahend, subtract the difference from the minuend: 12 − 7 = 5.Operations · Grade 1, Grade 2, Grade 3Absolute ValueThe absolute value of a number is its distance from zero on the number line, so it is never negative. It is written with two vertical bars: |−6| = 6 and |6| = 6. A positive number and 0 keep their value; a negative number loses its minus sign.Number System · Grade 6, Grade 7, Grade 8RangeThe range of a data set is the difference between the greatest and least values: maximum minus minimum. For 2, 5, 9, the range is 9 − 2 = 7. Range is a simple measure of spread: it tells how far apart the extreme values are, though one outlier can make it misleading about the rest of the data.Statistics · Grade 5, Grade 6, Grade 7DegreeA degree is the standard unit for measuring an angle, written with the symbol °. One degree is 1360 of a full turn, so a full turn is 360°, a straight angle is 180° and a right angle is 90°. An angle that turns through 25 one-degree angles measures 25°.Geometry · Grade 4, Grade 5, Grade 7RegroupingRegrouping is trading 10 of one place value for 1 of the next larger place, or 1 for 10 of the next smaller place, so that column addition or subtraction works. In 47 + 38, the 15 ones become 1 ten and 5 ones; in 52 − 27, 1 ten becomes 10 ones. It is also called carrying and borrowing.Operations · Grade 1, Grade 2, Grade 3SumA sum is the result you get when you add two or more numbers. In 8 + 5 = 13, the numbers 8 and 5 are the addends and 13 is the sum. The order does not change it: 5 + 8 is also 13. Word problems ask for a sum with words like total, in all and altogether.Operations · Grade 1, Grade 2, Grade 3QuotientA quotient is the answer you get when you divide one number by another. In 20 ÷ 4 = 5, the quotient is 5. The number being divided is the dividend and the number you divide by is the divisor. When the division is not exact, the quotient comes with a remainder, or as a decimal or fraction.Operations · Grade 3, Grade 4, Grade 5DataData are facts or values collected to answer a question, such as survey answers, measurements or counts. Numerical data are numbers you measure or count, like heights; categorical data are labels or groups, like favorite colors. Data are organized in tables and shown in graphs so patterns are easy to see.Statistics · Grade 2, Grade 3, Grade 6SubtractionSubtraction is the operation that finds how much is left when you take one amount away from another, or how far apart two numbers are. It is written with a minus sign: 9 − 4 = 5. The answer is called the difference. Subtraction undoes addition, so you can check 9 − 4 = 5 by adding 5 + 4 = 9.Operations · Grade 1, Grade 2, Grade 3Negative NumberA negative number is a number less than zero, written with a minus sign, such as −3, −0.5 or −23. On a number line, negative numbers sit to the left of 0. They describe amounts below a starting point: temperatures below zero, depths below sea level, or money owed. Zero itself is neither negative nor positive.Number System · Grade 6, Grade 7Percent ChangePercent change measures how much a value grew or shrank relative to where it started: the difference divided by the original amount, written as a percent. From 50 to 60 is a 20% increase; from 80 to 60 is a 25% decrease. The original value is always the base, which is the step people get wrong.Ratios and Proportions · Grade 7, Grade 8, Algebra 1ProductThe product is the answer you get when you multiply two or more numbers. In 4 × 6 = 24, the product is 24, and 4 and 6 are the factors. In word problems, the words “product of” always mean multiply, so “the product of 3 and 7” is 21.Operations · Grade 3, Grade 4, Grade 5