Therefore Symbol Copy and Paste (∴ ∵ ∎ ⇒)
Therefore (hence)
Unicode U+2234 · HTML ∴
Tap any symbol to copy it. Each tap also adds it to the box above, so you can copy several together.
Codes for every symbol on this page (Unicode, HTML, Alt, Mac, Word)
| Symbol | Name | Unicode | HTML | Windows Alt code | Mac | MS Word |
|---|---|---|---|---|---|---|
| ∴ | Therefore (hence) | U+2234 | ∴ | - | - | 2234 then Alt+X |
| ∵ | Because (since) | U+2235 | ∵ | - | - | 2235 then Alt+X |
| ∎ | End of proof | U+220E | ∎ | - | - | 220E then Alt+X |
| ⇒ | Implies | U+21D2 | ⇒ | - | - | 21D2 then Alt+X |
The therefore symbol ∴ is three dots arranged like a triangle, pointing up, and it means "so" or "hence". Its partner, the because symbol ∵, is the same three dots turned upside down and means "since".
Indian school maths answers use them all the time, especially in geometry: you write the reason with ∵ and the result with ∴. Tap either one in the grid above to copy it.
Therefore vs Because: Easy Way to Remember
| Symbol | Name | Dots | Read as |
|---|---|---|---|
| ∴ | Therefore | one dot on top, two at the bottom | therefore, so, hence |
| ∵ | Because | two dots on top, one at the bottom | because, since |
A memory trick: ∴ stands firmly on two legs, like a conclusion that is settled. ∵ is balanced on one point, like a reason that holds the next step up. Another trick: "because" comes first in a sentence, so the reason with ∵ usually comes on the line before ∴.
Therefore Symbol in Maths: Worked Examples
Algebra
2x + 3 = 11
2x = 8
∴ x = 4
Geometry
In △ABC, AB = AC.
∵ angles opposite equal sides are equal,
∴ ∠B = ∠C.
Proof style
∵ the sum of angles in a triangle is 180°, and ∠A = 50°, ∠B = 60°,
∴ ∠C = 180° − 50° − 60° = 70°.
Teachers like this layout because each step shows its reason. In board exams, writing "∴" before the final answer also makes it easy for the examiner to find. For the signs used inside these steps, see the angle symbol and degree symbol pages.
Hence Proved Symbol and Other Proof Signs
| Symbol | Name | Meaning |
|---|---|---|
| ∴ | Therefore | a result follows from what came before |
| ∵ | Because | gives the reason |
| ⇒ | Implies | if the left is true, the right is true |
| ∎ | End of proof (tombstone) | the proof is complete |
| Q.E.D. | Latin quod erat demonstrandum | "which was to be shown" |
| H.P. | Hence proved | common in Indian classrooms |
There is no single official hence symbol. In Indian schools, "∴" is often used to mean hence, and many students finish a proof with "Hence proved" or "H.P.". In college books you will see the hence proved symbol written as a solid square ∎ at the right end of the last line, or as the letters Q.E.D.
Do not mix up ⇒ and ∴. ⇒ joins two statements ("x = 2 ⇒ x² = 4"), while ∴ starts a new line with a conclusion.
How to Type ∴ and ∵
Neither symbol has a Windows Alt code or a Mac Option key, so use one of these:
| Where | Therefore ∴ | Because ∵ |
|---|---|---|
| Microsoft Word | 2234 then Alt+X | 2235 then Alt+X |
| Word equation (Insert > Equation) | type \therefore then Space | type \because then Space |
| Google Docs | Insert > Special characters, search "therefore" | search "because" |
| Mac | Control+Command+Space, search "therefore" | search "because" |
| LaTeX (amssymb package) | \therefore | \because |
| HTML | ∴ | ∵ |
For ∎ in Word type 220E then Alt+X, and for ⇒ type 21D2 then Alt+X.
On a phone: these symbols are not on Android or iPhone keyboards. Copy them from this page, or save a shortcut. On iPhone use Settings > General > Keyboard > Text Replacement, and on many Android keyboards use the personal dictionary, so typing "thr" gives ∴.
Common Mistakes with the Symbol of Therefore
- Typing three full stops (...) instead. That is an ellipsis and means something is left out.
- Using ∴ for every line. Use it only when the line really follows from the one before.
- Swapping ∴ and ∵. Check the dots: two at the bottom means therefore.
- Using ∴ in formal essays. In English writing, spell out "therefore".
More signs used in proofs and working, such as ≠, ≈ and ∝, are on our math symbols page. You may also like the proportional symbol and not equal to symbol guides.
Frequently asked questions
What is the symbol of therefore?
It is ∴, three dots in a triangle with one on top. It is read as "therefore" or "hence" and introduces a conclusion.
What is the because symbol?
The because symbol is ∵, three dots with two on top and one below. It gives the reason for the next step, for example "∵ AB = AC".
Can I use ∴ in my board exam answers?
Yes. ∴ and ∵ are standard in maths answers and teachers accept them. Use them neatly and only where a step follows from the one before.
What is the Word shortcut for the therefore symbol?
Type 2234 and press Alt+X in Word for Windows. Inside an equation box you can also type \therefore followed by a space.